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+arXiv:2301.02174v1  [math.PR]  5 Jan 2023
+Large time behavior of semilinear stochastic partial differential
+equations perturbed by a mixture of Brownian and fractional
+Brownian motions
+Marco Dozzi∗
+Ekaterina T. Kolkovska†
+Jos´e A. L´opez-Mimbela†
+Rim Touibi‡
+Abstract
+We study the trajectorywise blowup behavior of a semilinear partial differential equation that is
+driven by a mixture of multiplicative Brownian and fractional Brownian motion, modeling different
+types of random perturbations. The linear operator is supposed to have an eigenfunction of constant
+sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters
+of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for
+the probability of finite time blowup and of blowup before a given fixed time. Essential tools are
+the mild and weak form of an associated random partial differential equation.
+Keywords Stochastic reaction-diffusion equation; mixed fractional noise; finite-time blowup of
+trajectories
+AMS Mathematics Subject Classification 60H15 60G22 35R60 35B40 35B44 35K58
+1
+Introduction
+In this paper we study existence, uniqueness and the blowup behavior of solutions to the fractional
+stochastic partial differential equation of the form
+du(x, t)
+=
+�1
+2k2(t)Lu(x, t) + g(u(x, t))
+�
+dt + u(x, t) dNt,
+x ∈ D,
+t > 0,
+u(x, 0)
+=
+ϕ(x) ≥ 0,
+u(x, t)
+=
+0,
+x ∈ ∂D,
+t ≥ 0,
+(1.1)
+where D ⊂ Rd is a bounded Lipschitz domain, L is the infinitesimal generator of a strongly continuous
+semigroup of contractions which satisfies conditions (3.18), (3.19) below, and ϕ ∈ L∞(D), where L∞(D)
+is the space of real-valued essentially bounded functions on D. Additionally, g is a nonnegative locally
+Lipschitz function and N is a process given by
+Nt =
+� t
+0
+a(s) dB(s) +
+� t
+0
+b(s) dBH(s),
+t ≥ 0,
+(1.2)
+∗corresponding author, marco.dozzi@univ-lorraine.fr, UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy,
+France
+†Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico.
+‡UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy, France.
+1
+
+where B is Brownian motion and BH is fractional Brownian motion with Hurst parameter H > 1/2,
+a is continuous and b is H¨older continuous of order α > 1 − H. Both, B and BH, are supposed to be
+defined on a filtered probability space (Ω, F, (Ft, t ≧ 0), P) and adapted to the filtration (Ft, t ≧ 0).
+Such models have recently been studied under the name of ‘mixed models’ in the context of stochastic
+differential equations, see [19] and [20]. When N = 0, L = ∆, k = 1, g(u) = u1+β we obtain the
+classical Fujita equation which was studied in [10]. In [7] and [1] there were considered the cases when
+N is a Brownian motion, in [5] it was investigated the case when N is a fractional Brownian motion
+with Hurst parameter H > 1/2 and D ⊂ Rd, and in [6] the case of H ≥ 1/2 and D = Rd.
+The fractional Brownian motion (fBm) appears in many stochastic phenomena, where rough exter-
+nal forces are present. The principal difference, compared to Brownian motion, is that fBm is not a
+semimartingale nor a Markov process, hence classical theory of stochastic integration cannot be applied.
+Since H > 1/2, the stochastic integral with respect to BH in (1.1) can be understood as a fractional
+integral. Also the presence of both, Brownian and fractional Brownian motion in (1.1), due to their
+different analytic and probabilistic properties, modelize different aspects of the random evolution in
+time of the solution. The factor k2/2 in front of L affects dissipativity, which in several cases is in favor
+of retarding or even preventing blowup.
+We consider both, weak and mild solutions of (1.1), which we prove are equivalent and unique.
+Beyond existence and uniqueness of weak and mild solutions we are interested in their qualitative
+behaviour. In Theorem 3 below we obtain a random time τ ∗ which is an upper bound of the explosion
+time τ. In Theorem 8 we obtain a lower bound τ∗ of τ so that a.s.
+τ∗ ≤ τ ≤ τ ∗.
+The random times τ∗ and τ ∗ are given by exponential functionals of the mixture of a Brownian and a
+fractional Brownian motion. The laws of such kind of functionals presently are not known. In order to
+study the distribution of τ ∗ we use the well-known representation of BH in the form
+BH
+t =
+� t
+0
+KH(t, s) dWs,
+where the kernel KH is given in (3.23) and W is a Brownian motion defined in the same filtered
+probability space as B. In general, W can be different from the Brownian motion B appearing in the
+first integral of (1.2). We obtain estimates of the probability P(τ < ∞), and of the tail distribution
+of τ ∗. To achieve this we make use of recent results of N.T. Dung [8, 9] from the Malliavin theory for
+continuous isonormal Gaussian processes.
+In Theorem 4 we obtain upper bounds for P(τ ∗ ≤ T) in the case when B = W, and in Theorem
+5 when B is independent of W, and when B and W are general Brownian motions. In Theorem 6
+we obtain lower bounds for P(τ < ∞) when B = W. As a result in the case when W = B we get
+specific configurations of the coefficients a, b and k under which the weak solution (hence also the mild
+solution) of equation (1.1) exhibits finite time blow-up. To be concrete suppose that g(z) ≥ Cz1+β for
+2
+
+some constants C > 0, β > 0, BH
+t =
+� t
+0 KH(t, s) dBs, and
+� t
+0
+a2(r) dr ∼ t2l,
+� t
+0
+b2(r) dr ∼ t2m,
+� t
+0
+k2(r) dr ∼ t2p
+as
+t → ∞
+for some nonnegative constants l, m and p. If β ∈ (0, 1/2) and max{p, l} > H + m − 1/2, or if β = 1/2
+and p > H +m−1/2, or if β > 1/2 and p > max{l, H +m−1/2}, then all nontrivial positive solutions
+of (1.1) suffer finite-time blowup with positive probability.
+Our approach here is to transform the equation (1.1) into a random partial differential equation
+(RPDE) (2.5), whose solution blows up at the same random time τ as the solution of (1.1), and to
+work with this equation. The blowup behavior of (2.5) is easier to determine because N appears as
+a coefficient, and not as stochastic integrator as in (1.1). Such transformations are indeed known for
+more general SPDEs than (1.1), including equations whose stochastic term does not depend linearly
+on u, see [17]. But for the RPDE’s associated to more general SPDE’s it seems difficult to find explicit
+expressions for upper and lower bounds for the blowup time, and this is an essential point in our study.
+Another reason for having chosen the relatively simple form of (1.1) and (2.5) is that we consider the
+blowup trajectorywise which is a relatively strong notion compared, e.g., to blowup of the moments of
+the solution (see, e.g. [4]). The crucial ingredient in the proofs is the existence of a positive eigenvalue
+and an eigenfunction with constant sign of the adjoint operator of L. Special attention is given to the
+case H ∈ (3
+4, 1) because then the process N is equivalent to a Brownian motion [3]. This allows us to
+apply a result by Dufresne and Yor [27] on the law of exponential functionals of the Brownian motion
+to get in Theorem 7 an explicit lower bound for the probability of blowup in finite time.
+We finish this section by introducing some notations and definitions we will need in the sequel. A
+stopping time τ : Ω → (0, ∞) with respect to the filtration (Ft, t ≧ 0) is a blowup time of a solution u
+of (1.1) if
+lim sup
+tրτ
+sup
+x∈D
+|u(x, t)| = +∞
+P-a.s.
+Let (P D
+t , t ≧ 0) and ((P D)∗
+t , t ≧ 0)
+be the strongly continuous semigroups corresponding to the
+operator L and its adjoint L∗ :
+�
+D
+f(x)P D
+t g(x)dx =
+�
+D
+g(x)(P D)∗
+tf(x)dx,
+f, g ∈ L2(D).
+(1.3)
+As usual, Lf := lim
+t→0
+1
+t (P D
+t f − f) for all f ∈ L2(D) in the domain of L, denoted by Dom(L). Due to
+the Hille-Yosida theorem, Dom(L) and Dom(L∗) are dense in L2(D). Let P D
+t (x, Γ) and (P D)∗
+t (x, Γ)
+denote the associated transition functions, where t > 0, x ∈ D, and Γ ∈ B(D), the Borel sets on D.
+In the sequel we will assume that they admit densities, i.e. there exist families of continuous functions
+3
+
+(pD(t, ·, ·), t > 0) and ((pD)∗(t, ·, ·), t > 0) on D × D such that
+P D
+t g(x)
+=
+�
+D
+g(y)P D
+t (x, dy) =
+�
+D
+g(y)pD(t, x, y)dy,
+(P D)∗
+t f(x)
+=
+�
+D
+f(y)(P D)∗
+t (x, dy) =
+�
+D
+f(y)(pD)∗(t, x, y)dy.
+Due to (1.3),
+(pD)∗(t, x, y) = pD(t, y, x)
+for all t > 0 and x, y ∈ D.
+(1.4)
+2
+The weak solution of the associated random partial differential
+equation, equivalence with the mild solution
+Let us consider the random partial differential equation
+∂v
+∂t (x, t)
+=
+1
+2k2(t)Lv(x, t) − 1
+2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)),
+(2.5)
+v(x, 0)
+=
+ϕ(x), x ∈ D,
+v(x, t)
+=
+0, t ≥ 0, x ∈ ∂D.
+In this section we transform the weak form of (1.1) into the weak form of (2.5) using the transformation
+v(x, t) = exp(−Nt)u(x, t), x ∈ D, t ≥ 0. Hence, if blowup takes place in finite time, it occurs of course
+at the same time and at the same place x ∈ D for the solutions of both equations.
+In the following we write ⟨·, ·⟩D for the scalar product in L2(D).
+Definition 1. An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ∈ [0, T], x ∈ D) with values in
+L2(D) is a weak solution of (2.5) if, for all t ∈ [0, T] and all f ∈ Dom(L∗), P-a.s.
+⟨v(·, t), f⟩D
+=
+⟨ϕ, f⟩D +
+� t
+0
+�1
+2k2(s) ⟨v(·, s), L∗f⟩D − 1
+2a2(s) ⟨v(·, s), f⟩D
+�
+ds
++
+� t
+0
+exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds.
+(2.6)
+Since g is supposed to be locally Lipschitz, a blowup in finite time of v may occur, and the blowup
+time τ depends in general on ω ∈ Ω. A weak solution of (2.5) up to τ is defined as an (Ft, t ≧ 0)-
+adapted random field v that satisfies (2.6) for all t ∈ (0, T ∧ τ) P-a.s. If ω is such that v(ω, ·, ·) does
+not blowup in finite time, we set τ(ω) = ∞.
+Definition 2. An (Ft, t ≧ 0)-adapted random field u = (u(x, t), t ∈ [0, T], x ∈ D) with values in L2(D)
+is a weak solution of (1.1) up to τ if, for all t ∈ (0, T ∧ τ) and all f ∈ Dom(L∗), P-a.s.
+(i)
+� t
+0
+a2(s)
+�
+1 + ⟨u(·, s), f⟩2
+D
+�
+ds < ∞,
+b(•) ⟨u(·, •), f⟩D ∈ Cβ[0, t] for some β > 1 − H,
+4
+
+(ii)
+� t
+0
+�
+k2(s) |⟨u(·, s), L∗f⟩D| + |⟨g(u(·, s)), f⟩D|
+�
+ds < ∞,
+and
+⟨u(·, t), f⟩D = ⟨ϕ, f⟩D +
+� t
+0
+�1
+2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s), f⟩D
+�
+ds
++
+� t
+0
+⟨u(·, s), f⟩D dNs.
+(2.7)
+Conditions (i) and (ii) in the above definition are sufficient for the Itˆo, the fractional and the Lebesgue
+integrals in (2.7) to be well defined P-a.s.
+We proceed now to the relation between (2.7) and (2.6).
+Proposition 1. If u is a weak solution of (1.1) up to a random time τ, then v(x, t) = exp(−Nt)u(x, t)
+is a weak solution of (2.5) up to τ, and viceversa.
+Remark 1. We notice that ⟨v(·, s), f⟩D is absolutely continuous in s if v is a weak solution of (2.5).
+With the choice u(x, t) := exp(Nt)v(x, t) condition (i) is satisfied. In fact, for t < T ∧ τ(ω),
+� t
+0
+⟨u(·, s), f⟩D a(s) dBs =
+� t
+0
+⟨v(·, s), f⟩D exp(Ns)a(s) dBs
+is well defined since
+� t
+0(
+�
+D v(x, s)f(x) dx)2 exp(2Ns)a2(s) ds < ∞ P-a.s.
+Recall that the fractional integral
+� T
+0 f(x)dg(x) is defined (in the sense of Z¨ahle [28]) in [18, Def.
+2.1.1] for f, g belonging to fractional Sobolev spaces. If 0 < ε < H, f and g are H¨older continuous
+of exponents α and H − ε respectively, and α + H − ε > 1, this fractional integral coincides with
+the corresponding generalized Riemann-Stieltjes integral; see [18, Thm. 2.1.7]. Hence, the fractional
+integral
+� t
+0
+⟨u(·, s), f⟩D b(s) dBH
+s =
+� t
+0
+⟨v(·, s), f⟩D exp(Ns)b(s) dBH
+s
+(2.8)
+is well defined for t < T ∧ τ(ω) because, on the one hand, N· =
+� ·
+0(a(s) dBs + b(s) dBH
+s ) is P-a.s.
+H¨older continous of order 1/2 − ǫ for all ǫ > 0 by the theorem of Kolmogorov and [22, Proposition
+4.1]. On the other hand b(·) is α-H¨older continuous (with α > 1 − H) and BH is H¨older continuous
+with exponent H − ε for any ε > 0. Hence, choosing ε < min{H/2 − 1/4, α + H − 1} we get that
+the integrand on the right side of (2.8) is H¨older continuous of order min{α, 1/2 − ε}, and therefore
+H −ε+min{α, 1/2−ε} > 1 and the integral is well defined as a generalized Riemann-Stieltjes integral.
+Proof. Let T > 0. It suffices to prove the assertion for t ∈ (0, T ∧ τ). We apply (a slight generalisation
+5
+
+of) the Itˆo formula in [18, page 184]. Let
+Y 1
+t
+=
+� t
+0
+a(s) dBs ,
+Y 2
+t =
+� t
+0
+⟨u(·, s), f⟩D a(s) dBs,
+Y 3
+t
+=
+� t
+0
+b(s) dBH
+s ,
+Y 4
+t =
+� t
+0
+⟨u(·, s), f⟩D b(s) dBH
+s ,
+Y 5
+t
+=
+⟨ϕ, f⟩D +
+� t
+0
+�1
+2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D
+�
+ds,
+and let F(y1, y2, y3, y4, y5) = exp(−y1 − y3)(y5 + y2 + y4). Then
+F(Y 1
+t , . . . , Y 5
+t ) = exp(−Nt) ⟨u(·, t), f⟩D = ⟨v(·, t), f⟩D .
+The above mentioned Itˆo formula then reads
+F(Y 1
+t , . . . , Y 5
+t )
+=
+F(Y 1
+0 , . . . , Y 5
+0 ) +
+5
+�
+i=1
+� t
+0
+∂F
+∂yi
+(Y 1
+s , . . . , Y 5
+s ) dY i
+s + 1
+2
+2
+�
+i,j=1
+� t
+0
+∂2F
+∂yi∂yj
+(Y 1
+s , . . . , Y 5
+s ) d
+�
+Y i
+s , Y j
+s
+�
+.
+Since u is a weak solution of (1.1),
+⟨v(·, t), f⟩D
+=
+⟨ϕ, f⟩D −
+� t
+0
+exp(−Ns) ⟨u(·, s), f⟩D
+�
+a(s) dBs + b(s) dBH
+s
+�
++
+� t
+0
+exp(−Ns) ⟨u(·, s), f⟩D
+�
+a(s)dBs + b(s)dBH
+s
+�
++
+� t
+0
+exp(−Ns)
+�1
+2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D
+�
+ds
+−1
+2
+� t
+0
+exp(−Ns) ⟨u(·, s), f⟩D a2(s)ds
+=
+⟨ϕ, f⟩D +
+� t
+0
+�1
+2k2(s) ⟨v(·, s), L∗f⟩D − 1
+2a2(s) ⟨v(·, s), f⟩D
+�
+ds
++
+� t
+0
+exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds.
+Therefore v is a weak solution of (2.5). Similarly we obtain the viceversa result.
+In order to define the mild solutions of equations (1.1) and (2.5) we define first the evolution families
+of contractions corresponding to the generator 1
+2k2(t)L. For 0 ≤ s < t let
+K(t, s) = 1
+2
+� t
+s
+k2(r) dr,
+A(t, s) = 1
+2
+� t
+s
+a2(r) dr,
+K(t) = K(t, 0),
+A(t) = A(t, 0),
+(2.9)
+6
+
+and set pD(s, x; t, y) = pD(K(t, s), x, y), x, y ∈ D × D, 0 ≦ s < t. For f ∈ L2(D) the corresponding
+evolution families of contractions on L2(D) are given by
+U D(t, s)f(x)
+=
+�
+D
+pD(s, x; t, y)f(y)dy = P D
+K(t,s)f(x),
+(U D)∗(t, s)f(x)
+=
+�
+D
+pD(s, y; t, x)f(y)dy = (P D)∗
+K(t,s)f(x).
+Definition 3. An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ≧ 0, x ∈ D) with values in L2(D) is
+a mild solution of (2.5) on [0, T] if, for all t ∈ [0, T], P-a.s.
+v(x, t)
+=
+U D(t, 0)ϕ(x) − 1
+2
+� t
+0
+a2(s)U D(t, s)v(x, s) ds
++
+� t
+0
+exp(−Ns)U D(t, s)
+�
+g((exp Ns)v(x, s))
+�
+ds.
+Proposition 2. The mild form of (2.5) can be written as
+v(x, t)
+=
+exp(−A(t))U D(t, 0)ϕ(x)
++
+� t
+0
+exp(−Ns − A(t, s))U D(t, s)g(exp(Ns)v(·, s))(x)ds,
+(2.10)
+where A(t, s) and A(t) are given in (2.9).
+Remark 2. Since g and ϕ are supposed to be nonnegative,
+v(x, t) ≥ exp(−A(t))U D(t, 0)ϕ(x) ≥ 0 for all x ∈ D and t ≥ 0.
+Proof. Let w(x, t) = exp(A(t))v(x, t). For f ∈ L2(D), we get from the definition of the mild solution
+d
+dt⟨w(·, t), f⟩D
+=
+1
+2a2(t) exp(A(t))⟨v(·, t), f⟩D + exp(A(t)) d
+dt⟨v(·, t), f⟩D
+=
+1
+2a2(t) exp(A(t))⟨v(·, t), f⟩D
++ exp(A(t))
+�1
+2k2(t) ⟨v(·, t), L∗f⟩D − 1
+2a2(t) ⟨v(·, t), f⟩D
+�
++ exp(A(t)) exp(−Nt) ⟨g(exp(Nt)v(·, t)), f⟩D
+=
+1
+2 exp(A(t))k2(t) ⟨v(·, t), L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt)v(·, t)), f⟩D
+=
+1
+2k2(t) ⟨w(·, t), L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt − A(t))w(·, t)), f⟩D ,
+with boundary conditions w(x, 0) = ϕ(x) for x ∈ D and w(x, t) = 0 for x ∈ ∂D. Therefore w is a weak
+solution of the RPDE formally given by
+d
+dtw(x, t) = 1
+2k2(t)Lw(x, t) + exp(A(t) − Nt)g(exp(Nt − A(t))w(x, t)).
+7
+
+By the definition of the mild solution
+w(x, t) = U D(t, 0)ϕ(x) +
+� t
+0
+exp(A(s) − Ns)U D(t, s)g(exp(Ns − A(s))w(·, s))(x) ds.
+Consequently,
+v(x, t)
+=
+exp(−A(t))w(x, t)
+=
+exp(−A(t))U D(t, 0)ϕ(x) +
+� t
+0
+ds exp(−A(t, s) − Ns)U D(t, s)g(exp(Ns)v(·, s))(x).
+Theorem 1. The equation (2.10) has a unique non-negative local mild solution, i.e. there exists t > 0
+such that (2.10) has a mild solution in L∞([0, t[×D).
+Proof. Let T > 0 and denote ET = {v : [0, T] × D → L∞(D) : []v[] < ∞} , where
+[]v[] := sup
+0≤t≤T
+∥v(t, ·)∥∞.
+Let PT = {v ∈ ET : v ≥ 0} and for R > 0 let CR = {v ∈ ET : []v[] ≤ R}. Then ET is a Banach space
+and PT and CR are closed subsets of ET . Let us now define
+ψ(v)(t, x) = e−A(t)U D(t, 0)ϕ(x) +
+� t
+0
+e−A(t,s)−NsU D(t, s)g
+�
+eNsv(·, s)
+�
+(x) ds.
+We will prove that for sufficiently big R and sufficiently small T, ψ is contraction on PT ∩ CR. Let
+v1, v2 ∈ PT ∩ CR. Then
+[]ψ(v1) − ψ(v2)[] ≤
+sup
+0≤t≤T
+� t
+0
+��e−Ns �
+g
+�
+eNsv1
+�
+− g
+�
+eNsv2
+����
+∞ ds.
+Let AT = sup0≤s≤T e|Ns| and GR = sup|x|<R |g(x)|, and assume that g is locally Lipschitz with Lipschitz
+constant KR in the ball of radius R > 0 centered at 0. Then,
+sup
+0≤s≤T
+��e−Nsvi(s, ·)
+��
+∞ ≤ AT R,
+i = 1, 2,
+and
+��e−Nsg
+�
+eNsv1(s, ·)
+�
+− e−Nsg
+�
+eNsv2(s, ·)
+���
+∞ ≤ A2
+T KAT R ∥v1(s) − v2(s)∥∞ .
+Therefore,
+[]ψ(v1) − ψ(v2)[] ≤
+sup
+0≤t≤T
+� t
+0
+A2
+T KAT R []v1 − v2[] ds = TA2
+T KAtR []v1 − v2[].
+8
+
+We need
+TA2
+T KAT R < 1.
+(2.11)
+In addition, we require that CR ∩ PT be mapped by ψ into itself. Let v ∈ CR ∩ PT . Using that for 0 ≤
+s ≤ T the operator U D(t, s) is a contraction, and that ∥eNsv(·, s)∥∞ ≤ AT R, we get ∥g(eNsv(·, s))∥∞ ≤
+GAT R. It follows that
+[]ψ(v)[] ≤ ∥ϕ∥∞ + sup
+0≤t≤T
+� t
+0
+���e−N(s)���
+∞ ds GAT R ≤ ∥ϕ∥∞ + TAT GAT R.
+Hence, we need that
+∥ϕ∥∞ + TAT GAT R < R.
+(2.12)
+Let R be such that R ≥ 2∥ϕ∥∞. Since limT→0 AT = 1, we choose ε1 > 0 so that AT < 2 if T < ε1, and
+ε <
+R
+4G2R
+∧
+1
+4K2R
+∧ ε1.
+Using that GA ≤ GB and KA ≤ KB if A ≤ B, we get for R > 2∥ϕ∥∞ and T < ε,
+∥ϕ∥∞ + TAT GAT R ≤ ∥ϕ∥∞ + 2εG2R < R
+2 + R
+2 = R
+and
+TA2
+T KAT R < 4εK2R < 1.
+We proceed to prove equivalence of weak and mild solutions of (2.5). The proof of this theorem
+follows the method in [24, Theorem 9.15], where this equivalence is shown for SPDE’s with autonomous
+differential operators and driven by L´evy noise. For a comparison of weak and mild solutions of SPDEs
+driven by fractional Brownian motion we refer to [11].
+We state first the Kolmogorov backward and forward equations for U D. By the Kolmogorov back-
+ward equation for P D, the transition density pD(u, x, y) satisfies, for any y fixed,
+∂
+∂upD(u, x, y) =
+LpD(u, x, y). Then (s, x) → pD(s, x; t, y) satisfies, for (t, y) fixed, the equation
+− ∂
+∂spD(s, x; t, y) = − ∂
+∂spD(K(t, s), x, y) = − ∂
+∂upD(u, x, y) |u=K(t,s)
+∂
+∂sK(t, s)
+= 1
+2k2(s)LpD(K(t, s), x, y) = 1
+2k2(s)LpD(s, x; t, y).
+(2.13)
+Similarly, by the Kolmogorov forward equation for P D, for any x fixed, pD(u, x, y) satisfies
+∂
+∂upD(u, x, y) = L∗pD(u, x, y).
+9
+
+Then (t, y) → pD(s, x; t, y) satisfies, for (s, x) fixed, the equation
+∂
+∂tpD(s, x; t, y) = ∂
+∂tpD(K(t, s), x, y) = ∂
+∂upD(u, x, y) |u=K(t,s)
+∂
+∂tK(t, s)
+= 1
+2k2(t)L∗pD(K(t, s), x, y) = 1
+2k2(t)L∗pD(s, x; t, y).
+(2.14)
+Theorem 2. Consider the random partial differential equation (2.5). Then v is a weak solution of
+(2.5) on [0, T] if and only if v is a mild solution of (2.5) on [0, T].
+Proof. Assume that v is a weak solution of (2.5). Let h ∈ C1([0, ∞), R), f ∈ Dom(L∗), and G(x, t) :=
+− 1
+2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)). The integration by parts formula is applicable since h ∈
+C1([0, ∞), R) (see [24] Proposition 9.16) and yields
+⟨v(·, t), h(t)f(·)⟩D
+=
+⟨v(·, 0), h(0)f(·)⟩D +
+� t
+0
+⟨v(·, s), h′(s)f(·)⟩D ds
++
+� t
+0
+⟨v(·, s), 1
+2h(s)k2(s)L∗f(·)⟩D ds +
+� t
+0
+⟨G(·, s), h(s)f(·)⟩D ds.
+Since the functions h · f are dense in C1([0, ∞), Dom(L∗)), for each z ∈ C1([0, ∞), Dom(L∗)) we have
+⟨v(·, t), z(·, t)⟩D
+=
+⟨v(·, 0), z(·, 0)⟩D +
+� t
+0
+⟨v(·, s), ∂
+∂sz(·, s)⟩D ds
+(2.15)
++
+� t
+0
+⟨v(·, s), 1
+2k2(s)L∗z(·, s)⟩D ds +
+� t
+0
+⟨G(·, s), z(·, s)⟩D ds.
+For each f ∈ Dom(L∗) we define
+ψ(x, s) := (U D)∗(t, s)f(x) =
+
+
+
+⟨pD∗(s, x; t, ·), f(·)⟩D
+if s < t,
+f(x)
+if s = t,
+hence ψ ∈ C1([0, ∞), Dom(L∗)). Taking z = ψ(x, s) in (2.15) we get, for any t ∈ [0, T] fixed,
+⟨v(·, t), ψ(·, t)⟩D
+=
+⟨v(·, 0), ψ(·, 0)⟩D +
+� t
+0
+�
+v(·, s), d
+dsψ(·, s) + 1
+2k2(s)L∗ψ(·, s)
+�
+D
+ds
++
+� t
+0
+⟨G(·, s), ψ(·, s)⟩D ds.
+(2.16)
+Now we evaluate the terms above:
+⟨v(·, 0), ψ(·, 0)⟩D
+=
+�
+D
+v(x, 0)
+�
+D
+pD∗(0, x; t, y)f(y) dy dx
+=
+�
+D
+f(y)
+�
+D
+pD∗(0, x; t, y)v(x, 0) dx dy =
+�
+U D(t, 0)v(·, 0), f(·)
+�
+D .
+10
+
+By applying the Kolmogorov backward equation to (x, s) → (U D)∗(t, s)f(x) we get
+− d
+dsψ(x, s)
+=
+− ∂
+∂s
+�
+(pD)∗(s, x; t, ·), f(·)
+�
+D
+=
+1
+2k2(s)L∗ �
+(pD)∗(s, x; t, ·), f(·)
+�
+D = 1
+2k2(s)L∗ψ(x, s).
+Moreover, from Fubini’s theorem and (1.4)
+⟨G(·, s), ψ(·, s)⟩D
+=
+�
+D
+G(x, s)
+�
+D
+pD∗(s, x; t, y)f(y) dy dx
+=
+�
+D
+f(y)
+�
+D
+pD(s, y; t, x)G(x, s) dx dy =
+�
+U D(t, s)G(·, s), f(·)
+�
+D .
+Therefore, from (2.16), ⟨v(·, t), f(·)⟩D =
+�
+U D(t, 0)v(·, 0), f(·)
+�
+D +
+� t
+0⟨U D(t, s)G(·, s), f(·)⟩D ds for all
+f ∈ Dom(L∗). Since Dom(L∗) is dense in L2(D) we obtain that v is a mild solution of (2.5) on [0, T].
+To prove the converse let v be a mild solution of (2.5) on [0, T]. For f ∈ Dom(L∗),
+� t
+0
+�
+v(·, s), 1
+2k2(s)L∗f(·)
+�
+D
+ds
+=
+� t
+0
+�
+U D(s, 0)v(·, 0), 1
+2k2(s)L∗f(·)
+�
+D
+ds
++
+� t
+0
+�� s
+0
+χ[0,s](r)U D(s, r)G(·, r) dr, 1
+2k2(s)L∗f(·)
+�
+D
+ds
+=
+� t
+0
+�
+v(·, 0), (U D)∗(s, 0)1
+2k2(s)L∗f(·)
+�
+D
+ds
++
+� t
+0
+� t
+r
+�
+U D(s, r)G(·, r), 1
+2k2(s)L∗f(·)
+�
+D
+ds dr.
+(2.17)
+By applying the Kolmogorov forward equation to (U D)∗ we get for the first integral on the right side
+of (2.17):
+(U D)∗(s, 0)(1
+2k2(s)L∗f)(x) =
+�
+D
+pD∗(0, x; s, y)1
+2k2(s)L∗f(y) dy
+=
+�
+D
+(1
+2k2(s)L)pD∗(0, x; s, y)f(y) dy =
+�
+D
+∂
+∂spD∗(0, x; s, y)f(y) dy,
+and therefore
+� t
+0
+�
+v(·, 0), (U D)∗(s, 0)(1
+2k2(s)L∗)f(·)
+�
+D
+ds
+=
+� t
+0
+�
+v(·, 0),
+�
+D
+∂
+∂spD∗(0, ·; s, y)f(y) dy
+�
+D
+ds =
+�
+v(·, 0),
+�
+D
+pD∗(0, ·; t, y)f(y)dy − f(·)
+�
+D
+=
+�
+v(·, 0), (U D)∗(t, 0)f(·)
+�
+D − ⟨v(·, 0), f(·)⟩D.
+11
+
+In the same way we get for the second integral on the right side of (2.17)
+�
+U D(s, r)G(·, r), 1
+2k2(s)L∗f(·))
+�
+D
+=
+�
+G(·, r), (U D)∗(s, r)(1
+2k2(s)L∗f)(·)
+�
+D
+=
+�
+G(·, r),
+�
+D
+∂
+∂spD∗(r, ·; s, y)f(y)dy
+�
+D
+,
+and therefore
+� t
+r
+�
+U D(s, r)G(·, r), 1
+2k2(s)L∗f(·)
+�
+D
+ds =
+� t
+r
+�
+G(·, r),
+�
+D
+∂
+∂spD∗(r, ·; s, y)f(y)dy
+�
+D
+ds
+=
+�
+G(·, r),
+�
+D
+pD∗(r, ·; t, y)f(y)dy − f(·)
+�
+D
+=
+�
+G(·, r), (U D)∗(t, r)f(·) − f(·)
+�
+D
+=
+�
+U D(t, r)G(·, r), f(·)
+�
+D − ⟨G(·, r), f(·)⟩D .
+In this way we obtain
+� t
+0
+⟨v(·, s), 1
+2k2(s)L∗f(·)⟩D ds
+=
+�
+U D(t, 0)v(·, 0) +
+� t
+0
+U D(t, r)G(·, r)dr, f(·)⟩D − ⟨v(·, 0), f(·)
+�
+D
+−
+� t
+0
+⟨G(·, r), f(·)⟩D dr
+=
+⟨v(·, t), f(·)⟩D − ⟨v(·, 0), f(·)⟩ D −
+� t
+0
+⟨G(·, r), f(·)⟩D dr,
+since v is a mild solution on [0, T]. It follows that v is a weak solution on [0, T].
+Corollary 1. The equations (1.1) and (2.5) possess unique weak solutions.
+Proof. Theorem 2 and Proposition 1 show the existence and uniqueness of a local weak and mild
+solution of (2.5), and Proposition 1 shows the uniqueness of a weak solution of (1.1).
+Remark 3. We refer to [23] for an existence and uniqueness theorem of the variational solution of
+an SPDE with a nonautonomous second order differential operator and driven by fractional Brownian
+motion, and to [26] for the existence and uniqueness of the mild solution. In [20] the existence and
+uniqueness of the mild solution is shown for equations with the same differential operator and driven
+by mixed noise.
+3
+An upper bound for the blowup time and probability estimates
+3.1
+An upper bound for the blowup time
+In the remaining part of the paper we will assume that L and L∗ admit strictly positive eigenfunctions:
+there exists a positive eigenvalue λ0 and strictly positive eigenfunctions ψ0 ∈ Dom(L) for P D
+t
+and
+12
+
+ϕ0 ∈ Dom(L∗) for (P D)∗
+t with
+�
+D ψ0(x)dx =
+�
+D ϕ0(x)dx = 1 such that
+(P D
+t − e−λ0t)ψ0 = ((P D)∗
+t − e−λ0t)ϕ0 = 0,
+(3.18)
+hence
+(L + λ0)ψ0 = (L∗ + λ0)φ0 = 0.
+(3.19)
+For generators of a general class of L´evy processes, properties (3.18) and (3.19) follow from [14, 2].
+Another example are the diffusion processes: for f ∈ C2
+0(D), the set of twice continously differentiable
+functions with compact support in D, let us define the differential operator
+Lf =
+d
+�
+j,k=1
+∂
+∂xj
+�
+ajk
+∂
+∂xk
+f
+�
++
+d
+�
+j=1
+bj
+∂
+∂xj
+f − cf,
+where aj,k, bj, j, k = 1, ..., d are bounded smooth functions on D and c is bounded and continous. We
+assume that the matrix (aj,k, j, k = 1, ..., d) is symmetric and uniformly elliptic. In this case properties
+(3.18) and (3.19) follow from [12, Theorem 11, Chapter 2].
+Theorem 3. Assume (3.19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given
+constants. Let us define
+τ ∗ = inf
+�
+t > 0 :
+� t
+0
+exp [−β(λ0K(r) + A(r)) + βNr] dr ≥
+1
+Cβ ⟨ϕ, φ0⟩−β
+D
+�
+,
+(3.20)
+where the functions K and A are defined in (2.9). Then, on the event {τ ∗ < ∞} the solution v of (2.5)
+and the solution u of (1.1) blow up in finite time τ, and τ ≤ τ ∗ P-a.s.
+Proof. Using the hypothesis on g and Jensen’s inequality we get for the terms in (2.6):
+⟨v(·, s), L∗φ0⟩D
+=
+−λ0⟨v(·, s), φ0⟩D,
+exp(−Ns) ⟨g(exp(Ns)v(·, s)), φ0⟩D
+≧
+C exp(βNs)
+�
+v1+β(·, s), φ0
+�
+D ,
+≧
+C exp(βNs)⟨v(·, s), φ0⟩1+β
+D
+.
+Applying these lower bounds to (⟨v(·, t + ε), φ0⟩D − ⟨v(·, t), φ0⟩D)/ε and letting ε → 0 we get
+d
+dt⟨v(·, t), φ0⟩D ≧ −1
+2(λ0k2(t) + a2(t))⟨v(·, t), φ0⟩D + C exp(βNt)⟨v(·, t), φ0⟩1+β
+D
+.
+(3.21)
+The corresponding differential equality reads
+d
+dtI(t) = −1
+2(λ0k2(t) + a2(t))I(t) + C exp(βNt)I(t)1+β,
+and I(t) is a subsolution of (3.21), i.e. ⟨v(·, t), φ0⟩D ≧ I(t). Then
+I(t) = exp[−(λ0K(t) + A(t))]
+�
+⟨ϕ, φ0⟩−β
+D − βC
+� t
+0
+exp [−β(λ0K(s) + A(s)) + βNs] ds
+�−1/β
+13
+
+for all t ∈ [0, τ ∗), where τ ∗ is given by (3.20). Therefore τ ∗ is an upper bound for the blowup time
+of ⟨v(·, t), φ0⟩D, and the function t �→ ∥v(·, t)∥∞ = exp(−Nt)∥u(·, t)∥∞ can not stay finite on [0, τ ∗] if
+τ ∗ < ∞. Therefore u and v blow up before τ ∗ if τ ∗ < ∞.
+Remark 4. Notice that τ ∗ depends on L only by the positive eigenvalue λ0 and the associated eigen-
+function φ0. Moreover, τ ∗ is a decreasing function of ϕ, φ0 and C, and an increasing function of λ0K.
+Therefore small functions ϕ, φ0 and a small constant C, as well as high values of λ0K postpone the
+blowup of I and have, in this sense, the tendency to postpone the blowup of v and u.
+3.2
+A tail probability estimate for the upper bound of the blowup time
+In the following theorem we apply a tail probability estimate for exponential functionals of fBm studied
+by N.T. Dung [8] to estimate the probability that τ ∗ occurs before a fixed time T. Here we assume
+that the process BH is given by the formula
+BH
+t =
+� t
+0
+KH(t, s) dBs,
+(3.22)
+where the kernel KH is given for H > 1/2 by
+KH(t, s) =
+
+
+
+CHs1/2−H � t
+s (σ − s)H−3/2σH−1/2dσ
+if t > s,
+0
+if t ≦ s,
+(3.23)
+where CH = [
+H(2H−1)
+B(2−2H,H−1/2)]
+1
+2 and B is the usual beta function (see Section 5.1.3 in [21] for a general
+representation formula of fBm with H > 1/2). Notice that BH and B are dependent in this case.
+Theorem 4. Under assumptions (3.19) and (3.22), let g(z) ≥ Cz1+β for all z > 0, where C > 0,
+β > 0, are given constants, and let µ(T) =
+� T
+0 exp[−β(λ0K(t) + A(t))]E [exp(βNt)] dt. Then, for any
+T > 0 such that
+1
+Cβ⟨ϕ, φ0⟩−β
+D > µ(T),
+P {τ ∗ ≤ T} ≤ 2 exp
+
+−
+ln2 �
+Cβ⟨ϕ, φ0⟩β
+D µ(T)
+�
+2M(T)
+
+ ,
+where
+M(T) = 2β2
+� T
+0
+a2(r) dr + 4β2HT 2H−1
+� T
+0
+b2(u) du.
+Proof. For t ≥ 0, using (3.22), we have the following representation:
+Xt
+:=
+−β(λ0K(t) + A(t)) + βNt
+(3.24)
+=
+−β(λ0K(t) + A(t)) + β
+�� t
+0
+a(s) dBs +
+� t
+0
+� t
+s
+b(r) ∂
+∂rKH(r, s) dr dBs
+�
+.
+14
+
+From [8, Theorem 3.1] it follows that for any T ≥ 0 and any x > µ(T), there holds
+P
+�� T
+0
+eXtdt ≥ x
+�
+≤ 2 exp
+�
+−(ln x − ln µ(T))2
+2M(T)
+�
+,
+(3.25)
+where µ(T) =
+� T
+0 E
+�
+eXt�
+dt and M(T) is such that
+sup
+t∈[0,T]
+� T
+0
+|DrXt|2 dr ≤ M(T)
+P-a.s.
+(3.26)
+Here DrXt denotes the Malliavin derivative of Xt. In the following we will find an upper bound M(T)
+such that (3.26) holds. For r < t we have, using the representation (3.25),
+DrXt = β
+�
+a(r) +
+� t
+r
+b(s) ∂
+∂sK(s, r) ds
+�
+.
+Hence
+� t
+0 |DrXt|2 dr ≤ 2β2 � t
+0 a2(r) dr + 2β2 � t
+0(
+� t
+r b(s) ∂
+∂sK(s, r) ds)2 dr and
+� t
+0
+�� t
+r
+b(s) ∂
+∂sK(s, r) ds
+�2
+dr
+=
+� t
+0
+�� t
+r
+b(s) ∂
+∂sK(s, r) ds
+��� �� t
+r
+b(s′) ∂
+∂s′ K(s′, r) ds′
+�
+dr
+=
+� t
+0
+b(s) ds
+� s
+0
+∂
+∂sK(s, r) dr
+� t
+r
+b(s′) ∂
+∂s′ K(s′, r) ds′
+=
+� t
+0
+ds b(s)
+� t
+0
+dr1[0,s](r) ∂
+∂sK(s, r)
+� t
+r
+b(s′) ∂
+∂s′ K(s′, r) ds′
+=
+� t
+0
+ds b(s)
+� t
+0
+ds′b(s′)
+� s′
+0
+1[0,s](r) ∂
+∂sK(s, r) ∂
+∂s′ K(s′, r) dr
+=
+� t
+0
+ds
+� t
+0
+ds′ b(s)b(s′)
+� s∧s′
+0
+∂
+∂sK(s, r) ∂
+∂s′ K(s′, r) dr
+=
+� t
+0
+ds
+� t
+0
+ds′ b(s)b(s′)Φ(s, s′)
+=
+� t
+0
+ds
+� s
+0
+ds′ b(s)b(s′)Φ(s, s′) +
+� t
+0
+ds
+� t
+s
+ds′ b(s)b(s′)Φ(s, s′)
+=
+2
+� t
+0
+ds
+� s
+0
+ds′ b(s)b(s′)Φ(s, s′),
+where Φ(s, s′) =
+� s∧s′
+0
+∂
+∂sK(s, r) ∂
+∂s′ K(s′, r) dr. Since
+∂
+∂sK(s, r) = CHr1/2−H(s − r)H−3/2sH−1/2, using
+(5.7) in [21] we obtain
+Φ(s, s′) = C2
+H(ss′)H−1/2
+� s∧s′
+0
+r1−2H(s − r)H−3/2(s′ − r)H−3/2 dr = H(2H − 1)(s − s′)2H−2
+15
+
+for s′ < s, hence
+� t
+0
+�� t
+r
+b(s) ∂
+∂sK(s, r) ds
+�2
+dr
+≤ 2H(2H − 1)
+� t
+0
+ds
+� s
+0
+|b(s)b(s′)|(s − s′)2H−2 ds′
+≤ H(2H − 1)
+�� t
+0
+b(s)2
+� s
+0
+(s − s′)2H−2 ds′ ds +
+� t
+0
+� s
+0
+b(s′)2(s − s′)2H−2 ds′ ds
+�
+= H
+� t
+0
+b(s)2s2H−1 ds + H(2H − 1)
+� t
+0
+b(s′)2
+� t
+s′ (s − s′)2H−2 ds ds′
+= H
+� t
+0
+b(s)2(s2H−1 + (t − s)2H−1) ds
+≤ 2Ht2H−1
+� t
+0
+b(s)2 ds.
+(3.27)
+From the above inequalities we obtain
+sup
+t∈[0,T]
+� T
+0
+|DrXt|2dr ≤ 2β2
+� T
+0
+a2(r)dr + 4β2HT 2H−1
+� T
+0
+b2(u)du := M(T).
+(3.28)
+Now, from (3.20)
+P(τ ∗ ≦ T)
+=
+P
+�� T
+0
+exp[−β(λ0K(t) + A(t)) + βNt] dt ≧
+1
+Cβ ⟨ϕ, φ0⟩−β
+D
+�
+=
+P
+�� T
+0
+exp[X(t)] dt ≥ x
+�
+,
+(3.29)
+where x =
+1
+Cβ⟨ϕ, φ0⟩−β
+D . The result follows from (3.25) and (3.28).
+In the following theorem we obtain upper bounds for the tail of τ ∗ in the case when the Brownian
+motion B and the fractional Brownian motion BH have general dependence structure.
+Theorem 5. Assume (3.19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given
+constants.
+1. Assume that BH
+t
+=
+� t
+0 KH(t, s) dWs, where W is a Brownian motion defined in the same proba-
+bility space, and adapted to the same filtration as the Brownian motion B. Then
+P(τ ∗ ≤ T)
+≤
+Cβ⟨ϕ, φ0⟩β
+D
+� T
+0
+�
+e−βλ0
+� t
+0 k2(s) ds+2β2 � t
+0 a2(s) ds + e−β � t
+0 a2(s) ds+4β2Ht2H−1 � t
+0 b2(s) ds�
+dt.
+16
+
+2. If B and BH are independent, then
+P(τ ∗ ≤ T) ≤ Cβ⟨ϕ, φ0⟩β
+D
+� T
+0
+e−βλ0K(t)+ β2−β
+2
+� t
+0 a2(s) ds+β2Ht2H−1 � t
+0 b2(s) ds.
+Proof.
+1. Using H¨older’s and Chebishev’s inequalities we obtain
+P(τ ∗ ≤ T)
+=
+P
+�� T
+0
+e−βλ0K(t)+β � t
+0 a(s) dBs−βA(t)+β � t
+0 b(s) dBH
+s dt ≥
+1
+Cβ⟨ϕ, φ0⟩−β
+D
+�
+≤
+P
+
+
+�� T
+0
+e−2βλ0K(t)+2β � t
+0 a(s) dBs dt
+� 1
+2
+×
+�� T
+0
+e−2βA(t)+2β � t
+0 b(s) dBH
+s dt
+� 1
+2
+≥
+1
+Cβ ⟨ϕ, φ0⟩−β
+D
+
+
+≤
+P
+�� T
+0
+e−2βλ0K(t)+2β � t
+0 a(s) dBs dt ≥
+1
+Cβ ⟨ϕ, φ0⟩−β
+D
+�
++P
+�� T
+0
+e−2βA(t)+2β � t
+0 b(s) dBH
+s dt ≥
+1
+Cβ⟨ϕ, φ0⟩−β
+D
+�
+≤
+E
+�� T
+0 e−2βλ0K(t)+2β � t
+0 a(s) dBs dt
+�
++ E
+�� T
+0 e−2βA(t)+2β � t
+0 b(s) dBH
+s dt
+�
+1
+Cβ⟨ϕ, φ0⟩−β
+D
+≤
+� T
+0
+�
+e−2βλ0K(t)+2β2 � t
+0 a2(s) ds�
+dt +
+� T
+0 e−2βA(t)E
+�
+e2β � t
+0 b(s) dBH
+s
+�
+dt
+1
+Cβ⟨ϕ, φ0⟩−β
+D
+,
+(3.30)
+where we have used the fact that E
+�
+exp
+�� t
+0 f(s) dB(s)
+��
+= exp
+�
+1
+2
+� t
+0 f 2(s) ds
+�
+to obtain the
+last inequality. In addition,
+E
+�
+e2β
+� t
+0 b(s) dBH
+s
+�
+= E
+�
+e2β
+� t
+0
+� t
+s b(r) ∂
+∂r KH(r,s) dr dWs�
+= e2β2 � t
+0[
+� t
+s b(r) ∂
+∂r KH(r,s) dr]
+2 ds,
+where the last equality follows from [13, Theorem 4.12]. Therefore, using (3.27) we get
+E
+�
+e2β
+� t
+0 b(s) dBH
+s
+�
+≤ exp
+�
+4β2Ht2H−1
+� t
+0
+b2(s) ds
+�
+.
+(3.31)
+Substituting (3.31) into (3.30) we obtain the desired bound.
+17
+
+2. Using Chebishev’s inequality, the independence of B and BH and the proof of (3.31),
+P(τ ∗ ≤ T)
+=
+P
+�� T
+0
+e−βλ0K(t)+β � t
+0 a(s) dBs−βA(t)+β � t
+0 b(s) dBH
+s ) dt ≥
+1
+Cβ ⟨ϕ, φ0⟩−β
+D
+�
+≤
+Cβ⟨ϕ, φ0⟩β
+D
+� T
+0
+E
+�
+e−βλ0K(t)+β � t
+0 a(s) dBs�
+E
+�
+e−βA(t)+β � t
+0 b(s) dBH
+s
+�
+dt
+≤
+Cβ⟨ϕ, φ0⟩β
+D
+� T
+0
+exp
+�
+−βλ0K(t) + β2 − β
+2
+� t
+0
+a2(s) ds + β2Ht2H−1
+� t
+0
+b2(s) ds
+�
+dt.
+4
+Lower bounds for the blowup time and for the probability of finite
+time blowup
+4.1
+A lower bound for the probability of finite time blowup
+In the following theorem we give a lower bound for the probability of finite time blow up of the weak
+solution of (1.1). If f, g are nonnegative functions and c is a constant, we write f(t) ∼ cg(t) as t → ∞
+if limt→∞ f(t)/g(t) = c.
+Theorem 6. Assume (3.19) and (3.22). Let g(z) ≥ Cz1+β and
+� t
+0
+a2(r) dr ∼ C1t2l,
+� t
+0
+b2(r) dr ∼ C2t2m,
+� t
+0
+k2(r) dr ∼ C3t2p
+as t → ∞ for some nonnegative constants l, m, p and positive constants C, β, C1, C2 and C3. Suppose
+additionally that
+1. if β ∈ (0, 1/2), then max{p, l} > H + m − 1
+2,
+2. if β = 1/2, then H+m − 1
+2 < p,
+3. if β > 1/2, then p > max{l, H + m − 1
+2}.
+Under these assumptions the solution of (1.1) blows up in finite time with positive probability. Moreover,
+P(τ < ∞) ≧ P(τ ∗ < ∞) ≧ 1 − exp
+�
+−(mξ − 1)2
+2Lξ
+�
+,
+(4.32)
+where
+ξ =
+1
+Cβ ⟨ϕ, φ0⟩−β
+D ,
+Lξ = sup
+t≧0
+M(t)
+(ln(ξ + 1) + f(t))2 ,
+(4.33)
+18
+
+with f(t) = tmax{H+m−1/2, l} and
+mξ = E
+
+sup
+t≧0
+ln
+�� t
+0 exp (−β(λ0K(s) + A(s)) + βNs) ds + 1
+�
++ f(t)
+ln(ξ + 1) + f(t)
+
+ .
+(4.34)
+Proof. From (3.29) it follows that P(τ ∗ < ∞) = P(
+� ∞
+0 eXt dt ≥ ξ). In order to estimate P(
+� ∞
+0
+eXt dt ≥ ξ)
+we use [9, Theorem 3.1], with a = 0 and σ = 1 :
+Proposition 3 ([9]). Assume that the stochastic process X is adapted and satisfies
+a)
+� ∞
+0
+EeXs ds < ∞,
+b) For each t ≥ 0, Xt ∈ D1,2,
+c) There exists a function f : R+ → R+ such that limt→∞ f(t) = ∞ and for each x > 0,
+sup
+t≧0
+sups∈[0,t]
+� t
+0 |DrXs|2dr
+(ln(x + 1) + f(t))2
+≤ Lx < ∞
+a.s.
+(4.35)
+Then
+P
+�� ∞
+0
+eXt dt < x
+�
+≤ exp
+�
+−(mx − 1)2
+2Lx
+�
+,
+where
+mx = E
+�
+sup
+t≥0
+ln(
+� t
+0 eXs ds + 1) + f(t)
+ln(x + 1) + f(t)
+�
+.
+We now verify that conditions a) - c) of the above proposition hold.
+For condition a) we have from (3.25),
+� ∞
+0
+E exp[Xt] dt
+=
+� ∞
+0
+E exp
+�
+−βλ0
+2
+� t
+0
+k2(s) ds − β
+2
+� t
+0
+a2(s) ds
++ β
+�� t
+0
+a(s) dBs +
+� t
+0
+� t
+s
+b(r) ∂
+∂rKH(r, s) dr dBs
+��
+dt
+=
+� ∞
+0
+E exp
+�
+−βλ0
+2
+� t
+0
+k2(s) ds − β
+2
+� t
+0
+a2(s) ds + β
+� t
+0
+�
+a(s) +
+� t
+s
+b(r) ∂
+∂rKH(r, s) dr
+�
+dBs
+�
+dt
+=
+� ∞
+0
+exp
+�
+−βλ0
+2
+� t
+0
+k2(s) ds − β
+2
+� t
+0
+a2(s) ds + β2
+2
+� t
+0
+�
+a(s) +
+� t
+s
+b(r) ∂
+∂rKH(r, s) dr
+�2
+ds
+�
+dt,
+where, again, we have used [13, Theorem 4.12] to obtain the last equality. Therefore, using (3.27),
+� ∞
+0
+E exp[Xt] dt
+≤
+� ∞
+0
+exp
+�
+−βλ0
+2
+� t
+0
+k2(s) ds − β
+2
+� t
+0
+a2(s) ds + β2
+2
+� t
+0
+2a2(s) ds
++ 2β2Ht2H−1
+� t
+0
+b2(s) ds
+�
+dt.
+(4.36)
+19
+
+The integral (4.36) is finite if and only if the leading power of t in the term
+−βλ0
+2
+� t
+0
+k2(s) ds + 2β2 − β
+2
+� t
+0
+a2(s) ds + 2β2Ht2H−1
+� t
+0
+b2(s) ds
+has negative coefficient, which follows from our assumptions.
+Condition b) is a consequence of (3.28).
+For condition c) we use the inequality (3.28), which implies that for any x > 0 and any fixed
+function f,
+sup
+t≧0
+sups∈[0,t]
+� t
+0 |DrXs|2dr
+(ln(x + 1) + f(t))2
+≤ sup
+t≥0
+M(t)
+(ln(x + 1) + f(t))2 .
+(4.37)
+Due to our assumptions, for big t, the leading power of t in the numerator is max{2l, 2H + 2m − 1}.
+It follows that
+lim
+t→∞
+M(t)
+�
+ln(x + 1) + tmax{l,H+m−1/2}�2 < ∞,
+and therefore the supremum in (4.37) is finite. The result follows from Proposition 3.
+The cases when a = 0 (presence only of fractional Brownian motion) or b = 0 (presence only of
+Brownian motion), are simpler:
+Corollary 2. Under the assumptions in Theorem 6,
+1. When a(t) ≡ 0 and p > H + m − 1/2 the solution of (1.1) explodes in finite time with positive
+probability for all β > 0.
+2. If a(t) ≡ 0 and p = H + m − 1/2, the solution of (1.1) explodes in finite time with positive
+probability for all β > 0 satisfying β < C3λ0
+4C2H .
+3. When b(t) ≡ 0 and 0 < β ≤ 1
+2 the solution of (1.1) exhibits explosion in finite time with positive
+probability for all values of p and l.
+4. If b(t) ≡ 0 and β > 1/2, the solution of (1.1) exhibits explosion in finite time with positive
+probability if p > l or if p = l and C3λ0 > C1(2β − 1).
+Notice that mξ given in (4.34) satisfies mξ > 1 due to Theorem 3.1 in [9]. The formula for mξ
+shows interactions between ϕ and K that have an influence on the lower bound in (4.32). Increasing
+values of K decrease the lower bound in (4.32). In this sense high values of K are in favour of absence
+of finite time blowup.
+20
+
+4.2
+The case H > 3/4 and independent B and BH
+In order to find more explicit lower bounds for P(τ < +∞), we consider in this subsection the case
+H ∈ (3/4, 1) and suppose that B and BH are independent and b(s) = ca(s) for all s ≧ 0, where c
+is a constant. Then Nt =
+� t
+0 a(s)dMs with Ms = Bs + cBH
+s . By [3] M is equivalent to a Brownian
+motion �B, and therefore Nt is equivalent to ˜Nt :=
+� t
+0 a(s) d �Bs. Here equivalence means equality of the
+laws of the processes on (C[0, T], B), the space of continous functions defined on [0, T] endowed with
+the σ−algebra generated by the cylinder sets. Furthermore, ( ˜Nt)t≧0 is a continous martingale and
+therefore a time-changed Brownian motion: ˜Nt = �B2A(t).
+Theorem 7. Assume (3.19). Let H ∈ (3/4, 1), B and BH be independent and b(s) = ca(s) for all
+s ≧ 0, where c is a constant.We assume also that g(z) ≥ Cz1+β, that the functions k and a are positive
+continuous on R+ and that there exist constants η ∈ (0, +∞] and c1 > 0 such that
+1
+a2(t) exp(−βλ0K(t)) ≥ c1 exp
+�
+−2β A(t)
+η
+�
+,
+t ∈ R+.
+(4.38)
+Then
+P(τ < +∞) ≥ P(Zµ ≤ θ),
+(4.39)
+where τ is the blowup time of (1.1), Zµ is a gamma-distributed random variable with parameter µ :=
+2
+β( 1
+η + 1
+2), θ := 2c1
+β2ξ and ξ :=
+1
+Cβ⟨ϕ, φ0⟩−β
+D .
+Proof. From Theorem 3,
+P(τ ∗ = +∞)
+=
+P
+�� t
+0
+dr exp
+�
+−β(λ0K(r) + A(r)) + β ˜Nr
+�
+< ξ for all t > 0
+�
+=
+P
+�� ∞
+0
+dr exp
+�
+−β(λ0K(r) + A(r)) + β ˜Nr
+�
+≤ ξ
+�
+.
+By the change of variable q = 2A(r) we get
+P(τ ∗ = +∞) = P
+�� ∞
+0
+dr exp
+�
+−β(λ0K(r) + A(r)) + β ˜B2A(r)
+�
+≤ ξ
+�
+= P
+�� ∞
+0
+dq
+a2(A−1(q/2)) exp
+�
+−β(λ0K(A−1(q/2)) + 1
+2q) + β ˜Bq
+�
+≤ ξ
+�
+.
+Applying (4.38) to t = A−1(q/2) yields
+1
+a2(A−1(q/2)) exp
+�
+−β(λ0K(A−1(q/2))
+�
+≥ c1 exp
+�
+−β
+η q
+�
+,
+q ∈ R+.
+21
+
+Therefore
+P(τ ∗ = +∞)
+≤
+P
+�
+c1
+� ∞
+0
+dq exp
+�
+−βq
+�1
+η + 1
+2
+�
++ β ˜Bq
+�
+≤ ξ
+�
+=
+P
+�� ∞
+0
+dq exp
+�
+β( ˜Bq − ˜µq)
+�
+≤ ξ
+c1
+�
+,
+where ˜µ := 1
+η + 1
+2. A second change of variable q = 4s
+β2 yields
+P(τ ∗ = +∞) ≤ P
+�� ∞
+0
+ds exp
+�
+2( ˜Bs − µs)
+�
+≤ β2ξ
+4c1
+�
+,
+where µ := ˜µ 2
+β. Due to [27, Corollary 1.2, page 95],
+� ∞
+0
+e2( ˜
+Bs−µs) ds L=
+1
+2Zµ
+,
+where Zµ is a gamma-distributed random variable with parameter µ. Therefore
+P(τ = +∞) ≤ P(τ ∗ = +∞) ≤ P
+� 1
+2Zµ
+≤ β2ξ
+4c1
+�
+= P
+�
+Zµ ≥ 2c1
+β2ξ
+�
+.
+This implies the statement of the theorem.
+Remark 5. If k, a and b are constants, a more explicit lower bound for P(τ < +∞) is available
+without the assumption (4.38). Indeed, starting with (3.20), a straightforward calculation gives a lower
+bound in terms of a gamma-distributed random variable Z again, but this time with parameter �µ :=
+(λ0k2 + a2)/(a2β). More precisely,
+P(τ < ∞) ≧ P(τ ∗ < ∞) = P
+�
+Z�µ ≦ 2C
+a2β ⟨ϕ, φ0⟩β
+D
+�
+.
+4.3
+A lower bound for the blowup time
+Our next goal is to obtain a lower bound for the blowup time τ. Since the proofs of the following results
+are close to those in [1] (where b = 0), we omit them here.
+Theorem 8. Let the function g be such that g(0) = 0, z → g(z)/z is increasing, and g(z) ≤ Λz1+β for
+some positive constant Λ. Then τ ≥ τ∗, where
+τ∗ = inf
+�
+t > 0 :
+� t
+0
+exp(β(Nr − A(r)))
+��U D(r, 0)ϕ
+��β
+∞ dr ≧ 1
+Λβ
+�
+.
+(4.40)
+Let us define for 0 ≦ t < τ∗,
+J(t) =
+�
+1 − Λβ
+� t
+0
+exp(β(Nr − A(r)))
+��U D(r, 0)ϕ
+��β
+∞ dr
+�−1/β
+.
+22
+
+Then the solution u of (1.1) satisfies, for x ∈ D, 0 ≦ t < τ∗, P-a.s.
+0 ≦ u(x, t) ≦ J(t) exp(Nt − A(t))U D(t, 0)ϕ(x).
+(4.41)
+Remark 6. More precisely, the proof of this theorem shows that the mild solution v of (2.5) satisfies
+(4.41) without the factor exp(Nt). By Theorem 2, v is also the weak solution of (2.5), hence the weak
+solution u(·, t) = exp(Nt)v(·, t) of (1.1) satisfies (4.41).
+Corollary 3. Assume that
+Λβ
+� ∞
+0
+exp[β(Nr − A(r))]
+��U D(r, 0)ϕ
+��β
+∞ dr < 1.
+Then the solution u of (1.1) satisfies (4.41) P-a.s. for all t.
+Remark 7. For the special choice of ϕ = pψ0, p > 0, the integrals appearing in (3.20) and (4.40) are
+the same exponential functionals of N. In fact, U D(r, 0)ψ0 = exp(−λ0K(r))ψ0, and τ∗ becomes
+τ∗ = inf
+�
+t > 0 :
+� t
+0
+exp
+�
+β(Nr − λ0K(r) − A(r))
+�
+dr ≧ p−β
+Λβ ∥ψ0∥−β
+∞
+�
+,
+(4.42)
+whereas
+τ ∗ = inf
+�
+t > 0 :
+� t
+0
+exp
+�
+β(Nr − λ0K(r) − A(r))
+�
+dr ≥ p−β
+Cβ ⟨ψ0, φ0⟩−β
+D
+�
+.
+(4.43)
+In fact τ∗ ≦ τ ∗ if C ≦ Λ, since ⟨ψ0, φ0⟩D ≦ ∥ψ0∥∞
+�
+D φ0(x)dx = ∥ψ0∥∞. In order to apply both bounds
+simultaneously, we have to suppose Cz1+β ≦ g(z) ≦ Λz1+β, z > 0. It is therefore of interest to know
+the law of the integral appearing in (4.42) and (4.43). This seems possible only for bH = 0, since, to
+our best knowledge, the law of exponential functionals of fractional Brownian motion is still unknown.
+For the moment it seems that only estimates of the type of those in Section 3.2 are available. See also
+Theorem 7 for H > 3/4.
+5
+A sufficient condition for finite time blowup
+We consider now the mild form of (2.5) obtained in Proposition 2, and obtain a sufficient condition for
+finite time blowup.
+Theorem 9. Suppose that g(z) ≥ Cz1+β and that there exists w∗ > 0 such that
+exp(βA(w∗)) ∥ U D(w∗, 0)ϕ ∥−β
+∞ < βC
+� w∗
+0
+exp(βNs) ds .
+(5.44)
+Then for the explosion time τ of (1.1) there holds τ ≤ w∗.
+23
+
+Remark 8. Inequality (5.44) is understood trajectorywise. Therefore w∗ is random. (5.44) is harder to
+satisfy with a small initial condition ϕ and with a small value of C. Due to the different interpretations
+of the integrals in N, the effects on blowup of B and BH are different.
+If N = 0, (5.44) reads ∥
+U D(w∗, 0)ϕ ∥−β
+∞ < βCw∗ and in this case w∗ is deterministic; if in addition ϕ = ψ0, (5.44) reads
+exp(λ0βK(w∗)) ∥ ψ0 ∥−β
+∞ < βCw∗.
+Proof. We use the approach in [25, Lemma 15.6]; see also [15]. Suppose that v(x, t), x ∈ D, t ≥ 0, is
+a global solution of (2.5), and let 0 < t < t′. Using the semigroup property of the evolution system
+(U D(t, r))0≦r<t we obtain
+exp
+�
+− A(t′, t)
+�
+U D(t′, t)v(·, t)(x)
+=
+exp
+�
+− A(t′, t)
+�
+U D(t′, t)
+�
+exp
+�
+− A(t)
+�
+U D(t, 0)ϕ(·)
+�
+(x)
++ exp
+�
+− A(t′, t)
+�
+U D(t′, t)
+�� t
+0
+exp(−Nr) exp
+�
+− A(t, r)
+�
+U D(t, r)g(exp(Nr)v(·, r))(x) dr
+�
+(x)
+=
+exp
+�
+− A(t′)
+�
+U D(t′, 0)ϕ(·)(x)
++
+� t
+0
+exp(−Nr) exp
+�
+− A(t′, r)
+�
+U D(t′, r)g(exp(Nr)v(·, r))(x) dr
+≧
+exp
+�
+− A(t′)
+�
+U D(t′, 0)ϕ(·)(x)
++C
+� t
+0
+exp(βNr) exp
+�
+− A(t′, r)
+�
+U D(t′, r)v(·, r)1+β(x) dr.
+By Jensen’s inequality
+U D(t′, r)v(·, r)1+β(x)
+=
+�
+D
+pD(r, x; t′, y)v(y, r)1+β dy
+≧
+��
+D
+pD(r, x; t′, y)v(y, r) dy
+�1+β
+=
+�
+U D(t′, r)v(·, r)(x)
+�1+β
+.
+Therefore
+exp
+�
+− A(t′, t)
+�
+U D(t′, t)v(·, t)(x) ≧ exp
+�
+− A(t′)
+�
+U D(t′, 0)ϕ(x)
++ C
+� t
+0
+exp(βNr)
+�
+exp
+�
+− A(t′, r)
+�
+U D(t′, r)v(·, r)(x)
+�1+β
+dr.
+(5.45)
+Let ψ(t) be the last term in (5.45). Then, from the above inequality,
+ψ′(t) = C exp(βNt)
+�
+exp(−A(t′, t))U D(t′, t)v(·, t)(x)
+�1+β
+≧ C exp(βNt)(ψ(t))1+β
+24
+
+Let now Ψ(t) :=
+� ∞
+t
+dz/z1+β = 1
+βt−β, t > 0. Then
+d
+dtΨ(ψ(t)) = −
+ψ′(t)
+(ψ(t))1+β ≦ −C exp(βNt).
+Hence
+C
+� t′
+0
+exp(βNs) ds ≦ Ψ(ψ(0)) − Ψ(ψ(t′)) =
+� ψ(t′)
+ψ(0)
+dz/z1+β <
+� ∞
+exp(−A(t′))UD(t′,0)ϕ(·)(x)
+dz/z1+β
+for all x ∈ D and all t′ > 0. Therefore βC
+� t′
+0 exp(βNs) ds ≦ exp(βA(t′))∥U D(t′, 0)ϕ∥−β
+∞ for all t′ > 0.
+This contradicts (5.44).
+Acknowledgement The authors are grateful to two anonymous referees for their valuable comments,
+which greatly improved our paper. The second- and third-named authors acknowledge the hospitality
+of Institut ´Elie Cartan de Lorraine, where part of this work was done. The research of the second-
+named author was partially supported by CONACyT (Mexico), Grant No. 652255. The fourth-named
+author would like to express her gratitude to the entire staff of the IECL for their hospitality and
+strong support during the completion of her Ph.D. dissertation there.
+References
+[1] A. Alvarez, J.A. L´opez-Mimbela, N. Privault. Blowup estimates for a family of semilinear SPDEs
+with time-dipendent coefficients. Differential Equations and Applications 2 (2015), 201-219.
+[2] X. Chen, J. Wang. Intrinsic ultracontractivity for general L´evy processes on bounded open sets.
+Illinois J. Math. 58 (2014), 1117-1144.
+[3] P. Cheridito. Mixed fractional Brownian motion. Bernoulli 7 (2001), 913-934.
+[4] P.L. Chow. Explosive solutions of stochastic reaction-diffusion equations in mean Lp-norm. J.
+Diff. Equations 250 (2011), 2567-2580.
+[5] M. Dozzi, E.T. Kolkovska, J.A. L´opez-Mimbela. Finite-time blowup and existence of global positive
+solutions of a semi-linear SPDE with fractional noise. In: Modern Stochastics and Applications,
+V. Korolyuk, N. Limnios, Y. Mishura, L. Sakhno, G. Shevchenko (Eds.), Springer 2014, 95-108.
+[6] M. Dozzi, E.T. Kolkovska, J.A. L´opez-Mimbela. Global and non-global solutions of a fractional
+reaction-diffusion equation perturbed by a fractional noise. Stoch. Anal. Appl. 38 (2020), no. 6,
+959-978.
+[7] M. Dozzi, J.A. L´opez-Mimbela. Finite time blowup and existence of global positive solutions of a
+semi-linear SPDE. Stochastic Processes Appl. 120 (2010), 767-776.
+25
+
+[8] N.T. Dung. Tail estimates for exponential functionals and applications to SDEs. Stochastic Pro-
+cesses Appl. 128, Issue 12, (2018), 4154-4170.
+[9] N.T. Dung. The probability of finite-time blowup of a semi-linear SPDE with fractional noise.
+Statist. Probab. Lett. 149 (2019), 86-92.
+[10] H. Fujita. On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations,
+in Nonlinear Functional Analysis, Providence, R.I., 1970, Proc. Symp. Pure Math. 18(1) (1968)
+105-113.
+[11] M.J. Garrido-Atienza, B. Maslowski, J. ˇSnup´arkov´a. Semilinear stochastic equations with bilinear
+fractional noise. Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 9, 3075-3094.
+[12] A. Friedman. Partial differential equations of parabolic type. Prentice-Hall 1964.
+[13] F.C. Klebaner. Introduction to stochastic calculus with applications. Second edition. Imperial
+College Press, London, 2005.
+[14] P. Kim, R. Song, Intrinsic ultracontractivity of non-symmetric L´evy processes. Forum Math. 21
+(2009), 43-66.
+[15] M. Loayza, C.S. Da Paix˜ao. Existence and non-existence of global solutions for a semilinear heat
+equation on a general domain. Electron. J. Differential Equations 2014 (2014), No. 168, 1-9.
+[16] J.A. L´opez-Mimbela, A. P´erez. Global and nonglobal solutions of a system of nonautonomous
+semilinear equations with ultracontractive L´evy generators. J. Math. Anal. Appl. 423 (2015),
+720-733.
+[17] S.V. Lototsky, B.L. Rozovsky, Stochastic partial differential equations. Springer 2017.
+[18] Y. Mishura. Stochastic calculus for fractional Brownian motion and related processes. Springer
+Lecture Notes in Mathematics 1929 2008.
+[19] Y. Mishura, G. M. Shevchenko. Existence and uniqueness of the solution of stochastic differential
+equation involving Wiener process and fractional Brownian motion with Hurst index H > 1/2.
+Comm. Stat. - Theory and Methods 40 (2011), 3492-3508.
+[20] Y. Mishura, K. Ralchenko, G. Shevchenko. Existence and uniqueness of mild solution to stochastic
+heat equation with white and fractional noises. Theory Probab. Math. Statist. No. 98 (2019), 149-
+170.
+[21] D. Nualart. The Malliavin calculus and related topics. Springer Verlag 2006.
+[22] D. Nualart, N. R˘a¸scanu. Differential equations driven by fractional Brownian motion.
+Collect.
+Math. 53 (2002) 55-81.
+26
+
+[23] D. Nualart, P.-A. Vuillermot. Variational solutions for partial differential equations driven by a
+fractional noise. J. Funct. Anal. 232 (2006), 390-454.
+[24] S. Peszat, J. Zabczyk.
+Stochastic partial differential equations with L´evy noise.
+Cambridge
+University Press 2007.
+[25] P. Quittner; P. Souplet. Superlinear parabolic problems. Blow-up, global existence and steady
+states. Birkh¨auser Verlag, Basel, 2007.
+[26] K. Ralchenko, G. Shevchenko, Existence and uniqueness of mild solutions to fractional stochastic
+heat equation. Mod. Stoch. Theory Appl. 6 (2019) 57-79.
+[27] M. Yor. Exponential functionals of Brownian motion and related processes. Springer Verlag 2001.
+[28] M. Z¨ahle, Integration with respect to fractional functions and stochastic calculus I. Prob. Theory
+Rel. Fields 111 (1998) 333-374.
+27
+

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+Robust statistical properties of T1 transitions in
+confluent cell tissues
+Harish P Jain1,*, Axel Voigt2,3,4, and Luiza Angheluta1
+1Njord Centre, Department of Physics, University of Oslo, Oslo, 0371, Norway
+2Institute of Scientific Computing, Technische Universit¨at Dresden, Dresden, 01062, Germany
+3Center of Systems Biology Dresden, Pfotenhauerstr. 108, 01307 Dresden, Germany
+4Cluster of Excellence - Physics of Life, TU Dresden, 01062 Dresden, Germany
+*harishpj@fys.uio.no
+ABSTRACT
+Large-scale tissue deformation which is fundamental to tissue development hinges on local cellular rearrangements, such as
+T1 transitions. In the realm of the multi-phase field model, we analyse the statistical and dynamical properties of T1 transitions
+in a confluent tissue. We identify an energy profile that is robust to changes in several model parameters. It is characterized by
+an asymmetric profile with a fast increase in energy before the T1 transition and a sudden drop after the T1 transition, followed
+by a slow relaxation. The latter being a signature of the fluidity of the cell tissue. We show that T1 transitions are sources of
+localised large deformation of the cells undergoing the neighbour exchange and induce other T1 transitions in the nearby cells
+through a chaining of events that propagate local cell deformation to large scale tissue flows.
+1 Introduction
+Collective motion of cells is essential to several processes including development of an embryo, tissue morphogenesis, wound
+healing, homeostasis and cancer metastasis1–3. These biological processes are highly complex and orchestrate mechanical,
+chemical and biochemical interactions across multiple scales4–7. Through the interplay between directed motion, neighbour
+alignment and mechanical interactions, cell tissues exhibit emergent structures and dynamics that are crucial for their biological
+function. A fundamental underlying process for emergent large-scale behavior is the topological rearrangement of neighbouring
+cells, also known as T1 transition. It is a local, dissipative event that leads to remodelling of the tissue architecture and
+influences the large-scale flow properties of cell tissues that affects tissue homeostasis and epithelial morphogenesis8–10.
+In confluent tissues, the tissue architecture can change in several ways. To isolate the tissue dynamics driven by spontaneous
+T1 transitions, we consider an idealised situation where apoptosis and cell division are neglected, cells have a constant volume,
+identical mechanical properties, and their total number is fixed. During a T1 transition, typically two neighbouring cells move
+apart, while two of their neighbours come towards each other and make contact as illustrated in Figure 1. The average number
+of neighbours before and after the T1 transition is invariant. Through T1 transitions, cells undergo large deformations and
+shape changes, and encounter an energy barrier that they have to overcome through their activity11,12. Albeit, there are several
+competing scenarios of the mechanical-chemical-biological feedback involved in a T1 transition, our understanding of these
+coupled processes is still elusive13.
+T1 transitions are common features also in granular matter and foams under external forcing14–16. The energy relaxation
+after a T1 transition has been studied in foams by measuring the length of T1 junctions17. This concept was adapted for active
+tissues 18, where the length of the T1 junction before and after a T1 transition has been measured. During a T1 transition in
+dry foams15, the cells form a rosette where either four or more edges meet. It has been shown that a junction is energetically
+stable for three edges incident at 120 degrees. So, while undergoing a T1 transition, the cells pass from one metastable state
+to another via an unstable state comprising of a rosette. In confluent tissue extracellular spaces (gaps) change this process19.
+Rosettes and tri-junctions can no longer be defined by the number of edges meeting but are placed where gaps are formed.
+Also, various mathematical models have been used to study different facets of T1 transitions in foams and tissues11,19–23.
+They are mainly based on vertex models, which approximate cells by polygonal shapes. However, cell shape plays a crucial role
+in T1 transitions and the ability to accurately describe complex cell shapes, beyond polygons, might be advantageous19,24,25.
+We consider a multi-phase field model that allows for spontaneous T1 transitions while capturing the cell shape at a high
+resolution and allowing for large shape deformations. Multi-phase field models have been used to probe several questions
+pertaining collective motion of cells26–34. These models consider cells as active incompressible droplets and unlike vertex
+models, T1 transitions emerge spontaneously as a result of shape deformations. In vertex models, also extracellular spaces
+(gaps) need explicit modelling with ad-hoc assumptions19, whereas in multi-phase field models they are emergent.
+arXiv:2301.11758v1  [physics.bio-ph]  27 Jan 2023
+
+In this paper, we focus on characterising the energy profile preceding and succeeding T1 transitions. We show that this
+energy profile is statistically robust to changes in several model parameters. It is characterized by an asymmetric profile with
+a fast increase in energy before the T1 transition, a sudden drop after the T1 transition, followed by a slow relaxation. The
+relaxation profile provides insights into the flow properties of the tissue. Previously, the relaxation has been indirectly studied
+in tissues by examining the relaxation of an ellipsoid droplet immersed in a tissue17,19,35. The relaxation profile was attributed
+to yield stress due to limitations in the measurement timescales35, and also associated with the fluidization of the tissue19,36.
+We further consider the duration of T1 transitions and find that the average duration scales inversely to the maximum average
+energy attained during the T1 transition. Also, we show that T1 transitions may trigger the creation of other T1 transitions
+nearby and the chaining of T1 transitions leads to large-scale deformation and fluid like behaviour.
+We introduce the multi-phase field model in Section 2 and discuss results on local statistical properties of T1 transitions in
+Section 3. We further analyse the dependency of these statistical properties on various model parameters. The effect of cell
+deformability and activity is considered in detail. We study the impact of chaining of T1 transitions on flow at larger scales. In
+Section 4 we relate these finding to mechanical and rheological properties of the tissue and postulate that they can be used to
+characterize fluidization. Details on the numerical methods, the initialization and characterization of T1 transitions are provided
+in Section 5.
+2 Multi-phase field model
+We represent a two-dimensional confluent cell tissue within a multi-phase field model following formulations28,32–34. We
+consider a system of N cells of equal area occupying a square domain of size [0,L]×[0,L] and use periodic boundary conditions.
+Each cell is represented by a scalar phase field φi(x,t) as an indicator function of the domain occupied by each cell labeled by
+i = 1,2,··· ,N. Namely, the bulk phase values φi ≈ 1 and φi ≈ −1 indicate the interior and exterior of the cell, respectively.
+The cell boundary is defined by the localised transition region between the two bulk values. The time evolution of the i-th phase
+field follows a conservative dynamics which preserves the cell areas and is given by
+∂tφi +vi ·∇φi = ∆δF
+δφi
+,
+(1)
+where ∆ is the two-dimensional Laplacian applied to the variational derivative of a free energy functional F with respect to the
+phase field φi. The free energy F = FCH +FINT contains the Cahn-Hilliard energy
+FCH = 1
+Ca
+N
+∑
+i=1
+�
+Ω
+�ε
+2||∇φi||2 + 1
+4ε (φ 2
+i −1)2
+�
+dx,
+(2)
+and the interaction energy28,34
+FINT = 1
+In
+N
+∑
+i=1
+�
+Ω B(φi)∑
+j̸=i
+w(φj)dx.
+(3)
+The capillary number Ca and interaction number In are tuning parameters for the cell deformability and the strength of mutual
+repulsion/attraction interactions, respectively. In equation 2, the Cahn-Hilliard energy has a local free energy density given by
+the double well potential with the minima corresponding to the two bulk values and a gradient energy. The parameter ε controls
+the width of the diffuse interface. The Cahn-Hilliard energy ensures phase separation into two bulk regions which are separated
+by a thin, diffusive interface. This energy alone is minimised by cells with circular shapes. In equation 3, each cell’s interior
+and interface (B(φi) = (φi +1)/2) is coupled with every other cell through a local interaction potential,
+w(φj) = 1−(a+1)
+�φ j −1
+2
+�2
++a
+�φj −1
+2
+�4
+,
+where the parameter a = 1 models repulsion, while a > 1 models attraction and repulsion (see34 for a detailed analysis of role
+of a).
+Cell activity is introduced through the advection velocity vi(x,t) in equation 1 and is given by
+vi(x,t) = v0B(φi)ei(t),
+(4)
+where v0 is a constant parameter that controls the magnitude of the activity, ei = [cosθi(t),sinθi(t)] where θi is the orientation
+of the self-propulsion which evolves as
+dθi =
+�
+2DrdWi(t)+α(βi(t)−θi(t))dt.
+(5)
+2/14
+
+Figure 1. Successive time snapshots of tissue section undergoing a T1 transition, a finite-time neighbour exchange process
+between cells A, B, C and D. The transition starts when cells B and D lose contact and is completed when cells A and C make
+contact. During the T1 transition an extracellular space (gap) is formed between cells A, B, C and D. Also see Supplementary
+Movie 1
+The first term on the right side of equation 5 is a rotational diffusion term with a Wiener process Wi. The second term is a
+relaxation to the orientation of the cell’s shape elongation. The cell elongation is identified by the principal eigenvector of the
+shape deformation tensor29,32
+Si =
+�
+Si,0
+Si,1
+Si,1
+−Si,0
+�
+(6)
+which is symmetric and traceless and has the two components
+Si,0 = 1
+8
+�
+Ω
+��∂φi
+∂y
+�2
+−
+�∂φi
+∂x
+�2�
+dx
+and
+Si,1 = −1
+4
+�
+Ω
+�∂φi
+∂x
+∂φi
+∂y
+�
+dx.
+Its corresponding eigenvalues are λ ±
+i = ±
+�
+S2
+i,0 +S2
+i,1 and eigenvectors are η±
+i = ( Si,0+λ ±
+i
+Si,1
+,1). The vector η+
+i is parallel to the
+elongation axis of the cell and determines the preferred self-propulsion direction as
+βi(t) =
+�
+arg(η+
+i (t))
+: ei(t)·η+
+i (t) > 0
+−arg(η+
+i (t))
+: ei(t)·η+
+i (t) < 0
+(7)
+Therefore, the second term on the right hand side of equation (5) aligns θi(t) with βi(t). The parameter α controls the time
+scale of this alignment of the self-propulsion direction with the elongation axis of the cell. There are different possibilities
+to define the advection velocity vi(x,t) (see Ref.32 for an overview and comparison). The current form includes approaches
+of Ref.29 and, as the elongation is a result of the interaction with neighbouring cells, it accounts for contact inhibition of
+locomotion37,38. The model leads to properties appropriate to describe, e.g., Madin-Darby canine kidney (MDCK) cells32,39.
+(a)
+(b)
+(c)
+Figure 2. (a). Free energy density, in a region surrounding a T1 transition. (b) and (c) Coarse-grained energy density in a
+linear and log-scale, respectively. The white dot represents the epicenter of the T1 transition while the green dotted circle
+represents the coarse graining radius ravg, the estimated core of the T1 transition.
+3/14
+
+A
+A
+A
+A
+B
+D
+D
+D
+B
+B
+D
+B
+c
+c
+c
+c36
+32
+28
+24
+20
+16
+12
+8
+4
+05.4
+4.8
+4.2
+e grained)
+3.6
+3.0
+(coarse
+2.4
+energy
+1.8
+1.2
+0.6
+0.05.4
+4.8
+4.2
+3.6
+3.0
+2.4
+energy (
+1.8
+1.2
+0.6
+0.03 Results
+3.1 Energy profile of T1 transitions
+Within our multi-phase field approach, T1 transitions are neighbour exchange processes with a finite duration. A prototypical
+time sequence of a T1 transition is illustrated in Figure 1. Four cells A, B, C and D are involved. Before the T1 transition, the
+cell junction shared by cells B and D shrink. The T1 transition starts when the cells B and D break contact and move apart.
+This results in the formation of an extracellular space which we call ’gap’. Cells A and C move towards each other, close the
+gap, and form a new contact concluding the T1 transition. After the T1 transition, this new junction between cells A and D
+expands. The junctions that shrink and expand are called T1 junctions. We refer to Section 5 for the procedure to detect T1
+transitions and their durations. A T1 transition not only leads to topological rearrangements of the four neighbouring cells, it
+also involves deformation of the cells. While details, such as the specific shape of the cells and their deformation, the duration
+of the T1 transition and the relaxation process differ between T1 transitions, we will demonstrate that robust statistical features
+of T1 transitions exist.
+Figure 3. (a) Evolution of energy (averaged for 158 T1 transitions) at the epicenter of the T1 transitions. Negative time
+corresponds to time before a T1 transition and positive time corresponds to time after a T1 transition. The shaded region
+denotes a width of 1 standard deviation. The gray dashed line is the average energy across the whole domain. (b) Average
+energy profile during a T1 transition as function of percentage of T1 duration. The standard deviation is also indicated. (c), (d)
+and (e) Montages of deformed cells involved in a T1 transition. Each montage is made up of 5 images, that capture the cells at
+equidistant times, stacked over each other. The darkest colored overlay represents the latest time. (c) Cell shapes before the
+start of the T1 transition, (d) during the T1 transition, and (e) after the end of the T1 transition. Also see Supplementary Movie
+2 for corresponding simulation
+We define the epicenter of a T1 transition as the point with the minimum total distance from the centers of the involved cells
+in the neighbour exchange process midway through the T1 transition. We define the immediate region around the epicenter as
+the core of a T1 transition, which is of essence because it is the region where T1 junctions shrink and expand, and the gap
+appears and disappears.
+Figure 2a shows the total free energy density midway through a T1 transition. The epicentre is shown by the white dot and
+4/14
+
+4.85
+energy before T1
+(a)
+energy during T1
+0.36
+5
+(b)
+energy after T1
+maximum
+ standard deviation of energy
+mean global energy
+4.80
+4
+T
+~75% of maximum !
+~75% of maximum
+4.75
+buunp
+0.32
+rgy standar
+2
+0.30
+ener
+4.65
+1
+0.28
+4.60
+0
+-30
+-20
+-10
+0(T1)
+10
+20
+30
+0%(start)
+20%
+40%
+60%
+80%
+100%(end)
+time
+percent of T1 duration
+(d)
+e
+tT1
+5 to 0
+c
+0 to 100%
+tt1=0the estimated core is highlighted by the green circle. It has a radius ravg = 0.02L, where L is the side length of the computational
+domain. We compute a coarse-grained energy whose value at any point in the domain is the average of the energy density in a
+circular region centered at that point with radius ravg. Figure 2b shows this coarse grained energy field fravg, which we will call
+’energy’ field in the following. The signature of triple-junctions and T1 transitions already becomes appealing due to their
+higher energy. The difference between both is enhanced by using a log scale, see Figure 2c. Considering this energy field in the
+epicenter over time provides a spatial-temporal description of T1 transitions. For discussions on the sensitivity of this procedure
+on ravg we refer to Section 5.
+Figure 3a shows the time evolution of this energy averaged over 158 T1 transitions. The time is negative before a T1
+transition and is positive after a T1 transition, and is denoted by tT1. The energy during the T1 transitions is excluded, which
+leads to a discontinuity at tT1 = 0. The two values at tT1 = 0 correspond to the averaged energies at the start and the end of
+the T1 transitions. As the duration of T1 transitions differs, an averaged energy as a function of time during the T1 transition
+does not provide any meaningful information. Details on the energy during the T1 transition are shown in Figure 3b using a
+normalized time. The energy profile in Figure 3a, 3b has a peak at the T1 transition. The profile is asymmetric with a strong
+increase in energy before the T1 transition and a sudden decrease after the T1 transition followed by a slow relaxation. The
+asymmetry can be quantified by considering the 75% of the maximum value, which is marked in Figure 3a. Figures 3c-3e
+illustrate the evolution for one T1 transition, the one depicted in Figure 1. These figures contain overlays of several snapshots
+as per the time marked in the figures. The darkest of these snapshots pertains to the latest time. The yellow region marks the
+estimated core of the T1 transition. The asymmetry before and after the T1 transition, Figure 3c, 3e, respectively, is clearly
+visible. The T1 junctions are longer at tT1 = 5 compared with tT1 = −5. During the T1 transition, Figure 3d, the asymmetry is
+less pronounced. Most of the deformations are concentrated in the core. These deformations arise as a result of the formation of
+the gap, and subsequently its disappearance. The shrinking and formation of T1 junctions and the deformations within the core
+are a signature of the T1 transition. However, they also influence the deformation of the four cells outside of the core, and their
+neighbours, which can be perceived by the overlayed cell shapes. Interestingly in the depicted T1 transition, the deformations
+of each of the four cells seems to be persistent before, during and after the T1 transition (see the arrows indicating the direction
+of deformations). We will elaborate on this and other coarse grained effects in Section 3.4. The energy profile indicates an
+accumulation of energy to reach the energy barrier at the T1 transition. This is due to probing several possibilities in local
+movement and cell shape deformation, which are coupled by the definition of activity, taking into account cell elongation and
+contact inhibition of locomotion. After the energy barrier has been overcome the fast relaxation of the energy can be associated
+with a steep gradient in the energy landscape in one direction.
+The asymmetric shape of the energy profile is robust to changes in most model parameters, as demonstrated in Figure 4
+where α, Ca, a, D and v0 are varied and the energy profile associated with passive sheared foams is included for comparison.
+Figure 4b shows the energy rescaled by the maximum energy as changes in Ca directly affect the free energy, see equation
+(2). Within the range of parameters explored, the changes in the values of alignment parameter α, interaction coefficient a,
+and diffusivity D have minimal effects on the energy profile. We see that the profile is robust even in absence of noise (D = 0)
+(Figure 4d). On the other hand, the profile deviates from Figure 3a for low values of v0 and Ca. Figure 4e shows that the cell
+activity v0 affects the rate at which the cells approach a T1 transition which is indicated by the slower accumulation of energy
+for low v0. However, change in v0 has a minor effect on the energy relaxation immediately after a T1 transition. The slow
+relaxation afterwards is largest for large values of v0. This can be associated with the definition of activity, which is related to
+cell elongation and at least on average cells elongate in the direction of movement after the T1 transition. The characteristic
+profile of the accumulation of energy before the T1 transition and the fast relaxation of energy after the T1 transition is also
+present for low values of Ca, see Figure 4b. However, as Figure 4b considers a rescaled energy the actual rates depend on Ca.
+The slow relaxation after the sudden decrease only slightly depends on Ca. We would like to point out that the results for low
+values of v0 and Ca should be considered with care, as the number of T1 transitions considered in these cases is much lower.
+While the system is still in the fluid phase, the extreme values for v0 = 0.1 and Ca = 0.05 already approach the transition to the
+solid phase.
+In passive foams T1 transitions can be induced by applying shear. This is considered by an advection velocity field
+vi(x,t) = 0.5|x1 −L/2| and the resulting energy profile is compared with the profile from Figure 3a, see Figure 4f. The profiles
+differ before the T1 transition and within the slow relaxation, but are similar in the sudden drop of energy right after the T1
+transition. The latter reiterates that the energy relaxation right after a T1 transition is independent on activity. The differences
+in the accumulation of the energy can be associated with the persistent orientation of advection velocity due to shear, which
+results in collective deformation and a more deterministic approach of the T1 transition. Also the termination of the decay in
+the passive case results from the restricted possibilities of relaxation due to the applied shear.
+5/14
+
+Figure 4. Evolution of energy for different parameter values. The pink and cyan shaded region are used to denote time before
+and after the T1 transitions, respectively. The number of T1 transition used to obtain these results in indicated. (a) The aligning
+parameter α is varied. (b) The parameter to control cell deformability, Ca is varied. As Ca is a parameter that influences the
+overall total energy, for better comparison the energy is rescaled by division with the maximum energy. (c) Adhesion and
+repulsion corresponds to a = 1.5 and repulsion corresponds to a = 1. (d) The diffusivity D is varied. (e) The magnitude of the
+activity v0 is varied. (f) The passive shear corresponds to advection field vi(x,t) = 0.5|x1 − L
+2| while the active case
+corresponds to parameters in Table 1.
+3.2 Duration and other properties of T1 transitions
+As mentioned earlier, the duration of T1 transitions strongly depends on the specific cell arrangements. We now discuss the
+statistical properties of the duration. Figure 5a shows the probability distributions of the duration of T1 transitions. The
+distributions peak at smaller values and have a long tail for larger values. The profiles corresponds to repulsive and adhesive
+(a > 1), and only repulsive interactions (a = 1), and are fitted by Gamma distributions. The average duration of T1 transitions
+for repulsive interactions (3.418 measured for 539 T1 transitions across 3 simulations) is smaller compared to that for repulsive
+and adhesive interactions (3.826 for 631 T1 transitions across 4 simulations). Keeping other parameters fixed, the average
+number of T1 transitions in the repulsive and adhesive case was 157.75 while for the repulsive case was 179.66, respectively.
+Therefore, in the repulsive case, cells undergo neighbour exchanges faster and more often. Figure 5b shows the duration of T1
+6/14
+
+α: 0.0, Total Tl: 173
+(b)
+1.0
+5.0
+(a)
+α: 0.001, Total T1: 194
+α: 0.01, Total T1: 159
+0.9
+4.5
+α: 0.1, Total T1: 158
+::
+:
+α: 1.0, Total T1: 140
+0
+::
+4.0
+::
+8
+:
+8:
+:
+:
+:
+.
+:::
+ner
+led
+1:
+.
+ 3.0-
+0.6
+:
+::
+Ca: 0.05, Total T1: 25
+2.5
+::::
+Ca: 0.1, Total T1: 96
+:
+.
+Ca: 0.15, Total T1: 160
+0.4
+2.0
+:
+Ca: 0.2, Total T1: 151
+Ca: 0.25, Total T1: 193
+0
+0.3
+1.5
+Ca: 0.3, Total T1: 182
+-30
+-20
+20
+-10
+0(T1)
+10
+30
+-30
+-10
+0(T1)
+-20
+10
+20
+30
+time
+time
+(c)
+D: 0.0 , Total T1: 182
+5.0
+(d)
+5.0-
+D: 0.01 , Total T1: 158
+..
+D: 0.02 , Total T1: 177
+4.5
+4.5
+D: 0.03, Total T1: 145
+8
+:
+:
+4.0.
+:
+0
+4.0
+8
+::
+:
+:
+ner
+ 3.0
+8
+ 3.0
+:.
+2.5
+2.5.
+1...:
+2.0
+2.0
+8888888
+adhesion & repulsion, Total T1: 158
+repulsion, Total T1: 188
+1.5
+1.5
+-30
+-20
+-10
+0(T1)
+-30
+-20
+10
+20
+30
+-10
+0(T1)
+10
+20
+30
+time
+time
+Vo: 0.1 , Total T1: 8
+active, Total T1: 158
+(4)
+5.0
+5.0
+(e)
+Vo: 0.2 , Total T1: 27
+passive shear, Total T1: 64
+Vo: 0.3 , Total T1: 72
+.
+4.5
+Vo: 0.4 , Total T1: 98
+4.5
+.
+Vo: 0.5 , Total T1: 158
+0
+.
+o:
+Vo: 0.6 , Total Tl: 268
+4.0
+4.0
+.
+Vo: 0.7 , Total T1: 266
+:
+8
+:
+:
+.
+:
+..
+ 3.0
+..
+.......
+ 3.0
+:
+2.5-
+2.5
+2.0
+8.8
+2.0
+1.5
+1.5
+-30
+-20
+-10
+0(T1)
+10
+20
+30
+-30
+-20
+-10
+0(T1)
+10
+20
+30
+time
+time(a)
+(b)
+(c)
+(d)
+Figure 5. (a) Probability distributions of the duration of T1 transitions for only repulsion interactions (magenta dots) and for
+both repulsion and adhesion (cyan dots). Both data sets are fitted by Gamma distributions highlighting the exponential tails. (b)
+Scatter plot of duration of T1 transition as function of the maximum energy reached during a T1 transition. (c) Evolution of
+average shape index and (d) Evolution of the average velocity of center of mass of the cells involved in the T1 transitions as
+function of time relative to a T1 transition. The shaded regions mark the standard deviations of both quantities.
+transitions as a function of the maximum energy reached during a T1 transition. While the data is scattered, it qualitatively
+shows that high energy T1 transitions are faster. This qualitative result holds for both cases and can be explained by a larger
+accumulation of energy in the core, which increases the spatial energy gradients and in turn speeds up the relaxation of the
+energy which leads to the shorter duration.
+Figure 5c shows the averaged shape index (perimeter/√area) of the four cells involved in a T1 transition as function of
+time relative to a T1 transition. The asymmetry found for the energy profile and the discontinuity at tT1 = 0 is also present
+for this quantity. The cells deform and elongate as they approach a T1 transition and relax afterwards. This increases and
+decreases their shape index, respectively. The faster relaxation leads to the asymmetry in the evolution of the shape index.
+The asymmetry around a T1 transition is also seen in the average velocity of the center of mass of the cells involved in a T1
+transition as shown in Figure 5d. While the velocity is almost constant before the T1 transition, the velocity peaks at the
+T1 transition and slows down afterwards until it reaches the average value before the T1 transition. The peak in the average
+velocity of the center of mass is due to the large deformations of the portions of cells within the core and their fast relaxation
+after the T1 transition. Both quantities, the shape index and the cell velocity of the four cells involved in a T1 transition are also
+experimentally accessible. These quantities can be related to the energy considered above.
+3.3 Effect of cell deformability, activity and gaps on T1 transitions
+The asymmetric energy profile in Figure 3a is robust to tuning of most of the model parameters. Significant variations only
+occur for low values of Ca and v0, see Figure 4b, 4e. We now analyse the effect of cell deformability and activity on T1
+transitions in more detail. This requires a detailed analysis of the influence of gaps. The gap fraction is related to the confluency
+as confluency = 100(1−gap fraction). It essentially is a fixed quantity set by the initial data. We fix all parameters as per table
+1 and compare two different initial cell sizes, denoted by ’low gap’ with gap fraction 0.00048 and ’high gap’ with gap fraction
+0.00212. Both can be considered as confluent. The number of T1 transitions within the considered time frame is not influenced
+by this variation. The total numbers of T1 transitions are 162 and 158 for low and high gap cases, respectively. However, the
+7/14
+
+0.22
+before Tl
+after T1
+0.20
+S
+cell
+0.18
+0.16
+f
+velocity
+0.14
+0.12
+0.10
+0.08
+-30
+-20
+-10
+0(T1)
+10
+20
+30
+timerepulsion gamma fit
+adhesion & repulsion gamma fit
+repulsion
+0.3
+adhesion & repulsion
+probability
+0.2
+0.1
+0.0
+0
+2
+4
+6
+8
+10
+Tl duration10
+adhesion & repulsion
+repulsion
+8
+ duration
+6
+4
+2
+4.0
+4.5
+5.0
+5.5
+max energy4.15
+before Tl
+after T1
+shape index of Tl cells
+4.10
+4.05
+O
+4.00
+000:
+0000
+3.95
+3.90
+-30
+-20
+-10
+0(T1)
+10
+20
+30
+timeFigure 6. Dependency of various properties on deformability Ca ((a) - (f)) and activity v0 ((g) - (l)). Total T1 considers the
+total number of T1 transitions within the considered time frame, T1 duration is the averaged time from start to end of all T1
+transitions, Gap fraction is the extracellular space, considered as ∑i B(φi) below a fixed threshold, again averaged over time,
+Shape index considers the averaged shape index of the four cells involved in the T1 transitions. Time between T1 is the average
+time a cell spends between successive T1 transitions, Max energy is the maximum energy reached at a T1 transition and vavg is
+the average velocity of center of mass of all cells.
+average duration of T1 transitions is reduced by reducing the gap fraction. The values are 2.559 and 3.794 for low and high gap
+cases, respectively. We measure the gap fraction as the fraction of domain where ∑i B(φi) is less than a fixed threshold which is
+set to 0.2. This essentially excludes possible partial overlap of the diffuse interface region of cells and only accounts for gaps at
+tri-junctions and rosettes. This makes the measured gap fraction to depend on deformability and activity. For the considered
+cases low Ca leads to rounder cells with stronger overlap of the diffuse interfaces of the cells, which are in contact. This leads
+8/14
+
+300
+8
+0.012
+(a)
+(b)
+(c)
+250
+duration
+6
+0.009
+4
+2
+% 0.003
+50
+0
+0
+0.0
+0.05 0.10 0.15 0.20 0.25 0.30
+0.05 0.10 0.15 0.20 0.25 0.30
+0.05 0.10 0.15 0.20 0.25 0.30
+Ca
+Ca
+Ca
+4.2
+125
+(d)
+(f)
+(e)
+Regular pentagon
+15
+4.1
+Regular hexagon
+energy
+index
+100
+between
+4.0
+75
+10
+1/Ca fit
+pe
+3.9
+ xeu
+50
+hal
+time
+5
+S
+3.8
+25
+3.7
+0.05 0.10 0.15 0.20 0.25 0.30
+0.05 0.10 0.15 0.20 0.25 0.30
+0.05 0.10 0.15 0.20 0.25 0.30
+Ca
+Ca
+Ca
+300
+8
+0.012
+.(g).
+(h)
+(i)
+250
+6
+150
+4
+b
+2
+50
+0
+0.0
+0
+0.2 0.3 0.4 0.5 0.6 0.7
+0.2
+¥0.7
+0.4 0.5 0.6 0.7
+0.1
+0.1 (
+0.3 0.4 0.5 0.6
+0.1
+0.2
+0.3
+Vo
+Vo
+Vo
+4.2
+125
+(k)
+·(I)
+Regular pentagon
+0.20
+4.1
+index
+Regular hexagon
+100
+between
+0.15
+4.0
+75
+Vavg
+0.10
+50
+leus
+time
+S
+3.8
+25
+0.05
+3.7
+0
+0.00
+0.2 0.3 0.4 0.5
+0.60.7
+0.1 0.2 0.3 
+0.40.50.60.7
+0.1
+0.2
+0.1
+0.3 0.4 0.50.60.7
+Vo
+Vo
+Voto an increase in the measured gap fraction, see Figure 6c. A similar dependency, but smaller in magnitude, is found for activity.
+Larger v0 lead to stronger interactions between cells and thus more overlap of the diffuse interface region of cells in contact
+which again leads to an increase in measured gap fraction, see 6i. The gap fraction in both figures is the average quantity over
+the considered time frame. Both results and the dependencies discussed below are considered for the ’high gap’ setting.
+As shown in Figure 6a, the number of T1 transitions increases with increasing cell deformability parameter Ca. Cells that
+are more deformable can more easily acquire the shape deformations associated with T1 transitions. When Ca is low, these
+deformations are energetically more expensive resulting in fewer T1 transitions. Also the duration of T1 transitions depends on
+Ca, as shown in Figure 6b. T1 transitions are shorter when cells are more deformable. We suspect that this might be due to the
+presence of smaller gaps at T1 transitions, as this requires less shape deformation. Figure 6d shows the average cell shape index
+of the four involved cells in a T1 transition as function of cell deformability Ca. The shape index increases as deformability
+increases. The shape index of Ca = 0.05 is less than that of a regular pentagon. The shape index of regular pentagon (3.813)
+was attributed as the critical shape index for jamming transition in classical vertex models40 without gaps. It has been argued
+that gaps influence the mechanical properties and solid-liquid transition17, which might explain this discrepancy, as our system
+is still within the fluid phase. Further details, which are related to the previous dependencies are shown in Figures 6e and 6f.
+Figure 6e shows the average time a cell spends between successive T1 transitions as function of Ca. This quantity is large for
+low Ca but decreases and plateaus to low values upon increasing Ca. Figure 6f shows the maximum energy reached during a T1
+transition against Ca. We see from the dotted curve that the maximum energy is proportional to 1/Ca. Recall that 1/Ca scales
+the Cahn-Hilliard energy as per equation (2). This means that Fravg is primarily affected by the Cahn-Hilliard energy, which
+explains the correspondence of our results with the length of T1 junctions discussed earlier and considered in17,18.
+The dependency on v0 shows qualitatively similar behaviour for the number of T1 transitions, the duration of T1 transitions,
+the shape index of the cells involved in T1 transitions and the time a cell spends between successive T1 transitions, see Figures
+6g, 6h, 6j, 6k, respectively. The increase in T1 transitions and decrease in the time between T1 transitions with activity is a
+property of active systems, which are driven out of equilibrium. T1 transitions are topological defects and thus an indication of
+out of equilibrium. The decrease in duration with increasing activity can again be associated with the decrease in measured
+gap fraction, see 6i, and also the increasing shape index with activity is a direct consequence of the form of active forcing
+considered. Figure 6l shows the average velocity of center of mass of all cells as a function of v0. As expected, activity is
+primarily converted into motion with an almost linear dependency.
+3.4 Chaining of T1 transitions
+So far, we have analysed robust statistical properties of T1 transitions within their cores. However, we have also seen that these
+local features influence the position and shape of the four cells involved in a T1 transition, and their neighbours. This can
+induce new T1 transitions and lead to the formation of chains of T1 transitions as illustrated in Figure 7. Each of these images
+consists of 10 tissue states captured at equally-spaced time instants and overlaid on top of each other. The cell shapes outlined
+in the darkest colors correspond to the latest time. The yellow circles mark the cores of the T1 transitions at those time instants.
+The chaining of T1 transitions is a result of the assumptions on constant cell area and a confluent tissue. Any cell deformation
+associated with a T1 transition induces deformation of the neighbouring cells and thereby increases the possibility of new
+T1 transitions. This is further enhanced by activity and the considered propulsion mechanism which favours the direction of
+elongation.
+This chaining of T1 transitions is also observed experimentally in sheared foams41 and in our simulations of passive foams
+which are sheared with a constant shear velocity profile. For v0 = 0, typically one or two T1 transitions occur due to the initial
+non-equilibrium configuration of the tissue. As cells relax toward an equilibrium state, their motility is reduced which prevents
+any further T1 transitions. The situation for small v0 is similar. The tissue becomes jammed by cells being caged amongst
+their neighbours and no T1 transitions occur32. Furthermore, when cell deformability (Ca) is low, the energetic cost for cell
+deformations that are necessary to undergo T1 transitions is high, which prevents or at least reduces T1 transitions and the
+tissue also becomes jammed32. This corresponds to the low number of T1 transitions in Figure 4b, 4e for low Ca and low v0,
+respectively.
+However, in the considered case in Figure 7 we are far away from jamming and the chaining of T1 transitions leads to
+cell deformation propagating to larger scales. This is highlighted in Figure 8a, which shows the evolution of the cell tissue in
+the whole time window considered in Figure 7 together with the trajectory of the center of mass of the colored cells, which
+highlights the movement on larger spatial scales. The chaining of T1 transitions is also a source of large-scale flows as evidenced
+in Figure 8b. We consider the velocities of the centers of mass of all cells, average this quantity with the neighboring cells and
+construct a continuous velocity field by interpolating in space. The velocity field is shown together with the cell boundaries at
+t = 52. The mean direction corresponds with the direction of the black path shown in Figure 8a. However, as the variations in
+magnitude and direction of the flow field in Figure 8b indicate, T1 transitions can also induce fluctuations and could play an
+important role in sustaining chaotic flows (active turbulence) in cell tissues42–44.
+9/14
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 7. Chaining of T1 transitions. Each panel is a montage of 10 snapshots of tissue configurations taken successively at
+constant times intervals. Latest time is represented by the cell shapes marked in the darkest color shades. The cores of the T1
+transitions are highlighted in yellow. Also see Supplementary Movie 3.
+Figure 8. (a) Montage of tissue snapshots from time t = 25 to t = 79 (see figure 7). The black path is the trajectory of the
+center of mass of the 11 coloured cells. (b) LIC visualization of streamlines, magnitude (color) and direction (black arrows) of
+the flow velocity. The velocity and the cell boundaries correspond to time t = 52.
+10/14
+
+time: 25 to 34time: 34 to 43time: 43 to 52time: 52 to 61time: 61 to 70time: 70 to 794 Discussion
+Large-scale tissue deformation requires cellular rearrangements. The simplest rearrangement in confluent cell tissue is a T1
+transition. We have analysed these neighbour exchanges among cells in detail using a multi-phase field model and identified
+a characteristic asymmetric energy profile, see Figure 3. The energy profile has a peak at the T1 transition. The profile is
+asymmetric with a strong increase in energy before the T1 transition and a sudden decrease after the T1 transition which is
+followed by a slow relaxation. Detailed studies on the dependency of this profile on model parameters show robustness to
+variations in most parameters. They also allowed to associate the strong energy increase before the T1 transition with the
+strength in activity. This region is characterized by an accumulation of energy to reach the energy barrier at the T1 transition.
+This is achieved by probing several possibilities of direction of movement and shape deformation. This process is enhanced
+by activity, which is quantified by Figure 4e. In contrast to this the sudden relaxation after the T1 transition can clearly be
+associated with energy relaxation. It is almost independent of activity, see Figure 4e, and cell deformability, see Figure 4b, and
+also present in sheared foams, see Figure 4f. We would like to remark that the behaviour is independent but the actual slope and
+duration of this regime depends on deformability, as the energy is scaled in Figure 4b. The sudden decrease is associated with a
+steep gradient in the energy landscape in one direction set by the deformation of the cells in the core of the T1 transition. The
+third characteristic region, the slow relaxation, depends on activity and cell deformability. This relaxation profile provides
+insight in the mechanical properties of the tissue. Similar energy profiles have been obtained by actuation and relaxation of
+magnetic microdroplets which are injected into the tissue17,19,35. In these experiments a slow relaxation is associated with the
+fluidization of the tissue19,35, while stagnation of the relaxation indicates more solid-like behaviour17 and is associated with
+irreversible (plastic) tissue rearrangements. We postulate that these mechanical characterizations can also be obtained from the
+energy decay of the T1 transitions.
+In the considered confluent tissue the type of interaction between the cells, if repulsive or repulsive and attractive, seems to
+play a minor role on the characteristic energy profile of a T1 transition, see Figure 4c. However, the degree of confluency is
+known to influence the solid-fluid phase transition35. Increasing the extracellular space enhances fluidization. While we only
+consider the fluid phase, we observe an increased duration of T1 transitions for larger extracellular space. A finite duration
+of T1 transitions in cell tissues has been associated with molecular processes and is considered in an adhoc manner in vertex
+models22. Within the multi-phase field model a finite duration is a result of the mechanical properties of the cells and the their
+interactions. An increased duration of T1 transitions is observed for low deformability and low activity, see Figures 6b and 6h,
+respectively. Both indicating more solid-like behaviour, which is consistent with22, where increased duration of T1 transitions
+leads to decreasing the overall number of T1 transitions and a possible stiffening of the global tissue mechanics. However,
+these results don’t take extracellular space into account.
+Even if characterized locally, due to the confluent cell tissue, large enough deformations induced by T1 transitions lead to
+permanent cell deformations in the neighbourhood, which can trigger other T1 transitions, leading to a chaining effect. This
+behaviour is associated with the foam-like architecture and consistent with previously reported nonlinear tissue mechanics35. It
+is this chaining of T1 transitions which allows for large-scale tissue deformations and flow patterns which can be associated
+with sustaining chaotic flows, see Figure 8b.
+We believe these results also to hold in more general situations, e.g. for varying cell sizes and varying mechanical cell
+properties.
+5 Numerical Methods
+Model Parameters
+Unless otherwise specified, we use the model parameters as per Table 1
+τ
+τsave
+T
+L
+ε
+v0
+a
+Ca
+In
+Dr
+α
+0.005
+0.5
+150
+100
+0.15
+0.5
+1.5
+0.2
+0.1
+0.1
+0.1
+Table 1. Default values of the model parameters.
+Finite element simulations
+The simulations are run for time interval [0,T] discretised into Nt units with a uniform timestep size τ, i.e. T = Ntτ. We employ
+a semi-implicit discretization in time. Discretization in space follows the finite element method. We adaptively refine the diffuse
+interface and employ a parallelization approach which scales with the number of cells. For details we refer to28,32–34,45,46. The
+algorithm is implemented in the open-source library AMDiS47,48.
+11/14
+
+Detecting T1 transitions
+The T1 transitions are detected by tracking the neighbour relations of all cells. If two cells A and B are in contact, their neighbour
+relation is denoted by (A,B) or (B,A), both of which are equivalent. Suppose, there are four cells as in the Figure 1. The set of
+neighbour relations between these four cells before, during and after a T1 transition are {(A,B),(B,C),(C,D),(D,A),(B,D)},
+{(A,B),(B,C),(C,D),(D,A)} and {(A,B),(B,C),(C,D),(D,A),(A,C)} respectively. Before and after a T1 transition, there
+are 5 distinct neighbour relations between the four cells. The sets of relations before and after a T1 transition have four elements
+in common. These common elements make up the set of relations during a T1 transition. The duration of a T1 transition is time
+difference when the number of neighbour relations between the four cells change from 5 to 4 and back to 5.
+Sensitivity of fravg on ravg
+The coarse graining region of a point p is the region with all points x such that |p−x| < ravg. As the free energy is high at the
+cell edges, the points which include the edges within its coarse graining region around it would have high fravg. Moreover,
+points with triple junctions (where 3 edges meet) within its coarse graining region would have a higher fravg due to the presence
+of longer total length of cell edges. Usually at a given time, fravg has peaks near the T1 epicenter. This is because, the region
+around it would have either two triple junctions along with a gap as seen in the snapshots of Figure 1. Also, it is clear that points
+that do not have any cell edges within its coarse graining region, would have zero fravg. We have found that increasing the ravg
+loses information about the T1 transition in the value of fravg at the epicenter. A larger coarse graining region would entail a
+larger contribution from the bulk of the interior of the cell and would reduce fravg at the epicenters such that fravg at epicenters
+would not be uniquely discerned as a signature of a T1 transition. On the other hand, reducing ravg would mean that we might
+not encompass the information of the two triple junctions and the gap formed during the T1 transition. It also increases the
+deviations in the statistics that we describe. Moreover, if the energy along the length of the edge is uniform then the energy
+field fravg at a point gives an approximate measure of length of edges within the coarse graining region around that point.
+Data availability
+All data are available from the corresponding author upon reasonable request. The AMDiS implementation and additional
+codes for pre- and postprocessing are available from the corresponding author upon reasonable request
+Supplementary Information
+Supplementary Movie 1
+Supplementary Movie 2
+Supplementary Movie 3
+References
+1. Ladoux, B. & Mège, R. M. Mechanobiology of collective cell behaviours. Nat. Rev. Mol. Cell Biol. 18, 743–757, DOI:
+10.1038/nrm.2017.98 (2017).
+2. Brugués, A. et al. Forces driving epithelial wound healing. Nat. Phys. 10, 683–690, DOI: 10.1038/nphys3040 (2014).
+3. Friedl, P., Locker, J., Sahai, E. & Segall, J. E. Classifying collective cancer cell invasion. Nat. Cell Biol. 14, 777–783, DOI:
+10.1038/ncb2548 (2012).
+4. Guirao, B. et al. Unified quantitative characterization of epithelial tissue development. eLife 4, e08519, DOI: 10.7554/
+eLife.08519 (2015).
+5. Etournay, R. et al. Interplay of cell dynamics and epithelial tension during morphogenesis of the Drosophila pupal wing.
+eLife 4, e07090, DOI: 10.7554/eLife.07090 (2015).
+6. Iyer, K. V., Piscitello-Gómez, R., Paijmans, J., Jülicher, F. & Eaton, S. Epithelial viscoelasticity is regulated by mechanosen-
+sitive E-cadherin turnover. Curr. Biol. 29, 578–591.e5, DOI: 10.1016/j.cub.2019.01.021 (2019).
+7. Dye, N. A. et al. Self-organized patterning of cell morphology via mechanosensitive feedback. eLife 10, e57964, DOI:
+10.7554/eLife.57964 (2021).
+8. Keller, R. et al. Mechanisms of convergence and extension by cell intercalation. Philos. Transactions Royal Soc. B: Biol.
+Sci. 355, 897–922 (2000).
+9. Walck-Shannon, E. & Hardin, J. Cell intercalation from top to bottom. Nat. Rev. Mol. Cell Biol. 15, 34–48, DOI:
+10.1038/nrm3723 (2014).
+12/14
+
+10. Rauzi, M. Cell intercalation in a simple epithelium. Philos. Transactions Royal Soc. B: Biol. Sci. 375, 20190552, DOI:
+10.1098/rstb.2019.0552 (2020).
+11. Bi, D., Lopez, J. H., Schwarz, J. M. & Manning, M. L. A density-independent rigidity transition in biological tissues. Nat.
+Phys. 11, 1074–1079, DOI: 10.1038/nphys3471 (2015).
+12. Oswald, L., Grosser, S., Smith, D. M. & Käs, J. A. Jamming transitions in cancer. J. Phys. D: Appl. Phys. 50, 483001,
+DOI: 10.1088/1361-6463/aa8e83 (2017).
+13. Rauzi, M., Verant, P., Lecuit, T. & Lenne, P.-F. Nature and anisotropy of cortical forces orienting Drosophila tissue
+morphogenesis. Nat. Cell Biol. 10, 1401–1410, DOI: 10.1038/ncb1798 (2008).
+14. Schall, P., Weitz, D. A. & Spaepen, F. Structural rearrangements that govern flow in colloidal glasses. Science 318,
+1895–1899, DOI: 10.1126/science.1149308 (2007).
+15. Weaire, D. & Hutzler, S. The Physics of Foams (Oxford University Press, Oxford, New York, 2001).
+16. Stavans, J. The evolution of cellular structures. Reports on Prog. Phys. 56, 733–789, DOI: 10.1088/0034-4885/56/6/002
+(1993).
+17. Durand, M. & Stone, H. A. Relaxation time of the topological T1 process in a two-dimensional foam. Phys. Rev. Lett. 97,
+226101, DOI: 10.1103/PhysRevLett.97.226101 (2006).
+18. Curran, S. et al. Myosin II controls junction fluctuations to guide epithelial tissue ordering. Dev. Cell 43, 480–492.e6,
+DOI: 10.1016/j.devcel.2017.09.018 (2017).
+19. Kim, S., Pochitaloff, M., Stooke-Vaughan, G. A. & Campàs, O. Embryonic tissues as active foams. Nat. Phys. 17, 859–866,
+DOI: 10.1038/s41567-021-01215-1 (2021).
+20. Barton, D. L., Henkes, S., Weijer, C. J. & Sknepnek, R. Active Vertex Model for cell-resolution description of epithelial
+tissue mechanics. PLOS Comput. Biol. 13, e1005569, DOI: 10.1371/journal.pcbi.1005569 (2017).
+21. Sknepnek, R., Djafer-Cherif, I., Chuai, M., Weijer, C. J. & Henkes, S.
+Generating active T1 transitions through
+mechanochemical feedback (2022). arXiv:2106.12394.
+22. Erdemci-Tandogan, G. & Manning, M. L. Effect of cellular rearrangement time delays on the rheology of vertex models
+for confluent tissues. PLOS Comput. Biol. 17, e1009049, DOI: 10.1371/journal.pcbi.1009049 (2021).
+23. Drenckhan, W. et al. Rheology of ordered foams—on the way to Discrete Microfluidics. Colloids Surfaces A: Physicochem.
+Eng. Aspects 263, 52–64, DOI: 10.1016/j.colsurfa.2005.01.005 (2005).
+24. Boromand, A., Signoriello, A., Ye, F., O’Hern, C. S. & Shattuck, M. D. Jamming of deformable polygons. Phys. Rev. Lett.
+121, 248003, DOI: 10.1103/PhysRevLett.121.248003 (2018).
+25. Perrone, M. C., Veldhuis, J. H. & Brodland, G. W. Non-straight cell edges are important to invasion and engulfment as
+demonstrated by cell mechanics model. Biomech. Model. Mechanobiol. 15, 405–418, DOI: 10.1007/s10237-015-0697-6
+(2016).
+26. Nonomura, M. Study on multicellular systems using a phase field model. PLoS ONE 7, 1–9, DOI: 10.1371/journal.pone.
+0033501 (2012). 1109.5246.
+27. Palmieri, B., Bresler, Y., Wirtz, D. & Grant, M. Multiple scale model for cell migration in monolayers: Elastic mismatch
+between cells enhances motility. Sci. Reports 5, 1–13, DOI: 10.1038/srep11745 (2015).
+28. Marth, W. & Voigt, A. Collective migration under hydrodynamic interactions: A computational approach. Interface Focus.
+6, DOI: 10.1098/rsfs.2016.0037 (2016). 1605.06108.
+29. Mueller, R., Yeomans, J. M. & Doostmohammadi, A. Emergence of active nematic behavior in monolayers of isotropic
+cells. Phys. Rev. Lett. 122, 048004, DOI: 10.1103/PhysRevLett.122.048004 (2019).
+30. Loewe, B., Chiang, M., Marenduzzo, D. & Marchetti, M. C. Solid-liquid transition of deformable and overlapping active
+particles. Phys. Rev. Lett. 125, 038003, DOI: 10.1103/PhysRevLett.125.038003 (2020).
+31. Camley, B. A. et al. Polarity mechanisms such as contact inhibition of locomotion regulate persistent rotational motion
+of mammalian cells on micropatterns. Proc. Natl. Acad. Sci. United States Am. 111, 14770–14775, DOI: 10.1073/pnas.
+1414498111 (2014).
+32. Wenzel, D. & Voigt, A. Multiphase field models for collective cell migration. Phys. Rev. E 104, 054410, DOI: 10.1103/
+PhysRevE.104.054410 (2021).
+13/14
+
+33. Wenzel, D., Praetorius, S. & Voigt, A. Topological and geometrical quantities in active cellular structures. J. Chem. Phys.
+150, DOI: 10.1063/1.5085766 (2019). 1812.10416.
+34. Jain, H. P., Wenzel, D. & Voigt, A. Impact of contact inhibition on collective cell migration and proliferation. Phys. Rev. E
+105, 034402, DOI: 10.1103/PhysRevE.105.034402 (2022).
+35. Mongera, A. et al. A fluid-to-solid jamming transition underlies vertebrate body axis elongation. Nature 561, 401–405,
+DOI: 10.1038/s41586-018-0479-2 (2018).
+36. Mongera, A. et al. Mechanics of the cellular microenvironment as probed by cells in vivo during zebrafish presomitic
+mesoderm differentiation. Nat. Mater. 22, 135–143, DOI: 10.1038/s41563-022-01433-9 (2023).
+37. Smeets, B. et al. Emergent structures and dynamics of cell colonies by contact inhibition of locomotion. Proc. Natl. Acad.
+Sci. 113, 14621–14626, DOI: 10.1073/pnas.1521151113 (2016).
+38. Stramer, B. & Mayor, R. Mechanisms and in vivo functions of contact inhibition of locomotion. Nat. Rev. Mol. Cell Biol.
+18, 43–55, DOI: 10.1038/nrm.2016.118 (2017).
+39. Peyret, G. et al. Sustained oscillations of epithelial cell sheets. Biophys. J. 117, 464–478, DOI: 10.1016/j.bpj.2019.06.013
+(2019).
+40. Park, J. A. et al. Unjamming and cell shape in the asthmatic airway epithelium. Nat. Mater. 2014 14:10 14, 1040–1048,
+DOI: 10.1038/nmat4357 (2015).
+41. Rosa, M. E. & Fortes, M. A. Nucleation and glide of dislocations in a monodisperse two-dimensional foam under uniaxial
+deformation. Philos. Mag. A 77, 1423–1446, DOI: 10.1080/01418619808214261 (1998).
+42. Doostmohammadi, A., Ignés-Mullol, J., Yeomans, J. M. & Sagués, F. Active nematics. Nat. Commun. 9, DOI: 10.1038/
+s41467-018-05666-8 (2018).
+43. Wenzel, D., Nestler, M., Reuther, S., Simon, M. & Voigt, A. Defects in active nematics – algorithms for identification and
+tracking. Comput. Methods Appl. Math. 21, 683–692, DOI: 10.1515/cmam-2020-0021 (2021).
+44. Alert, R., Casademunt, J. & Joanny, J.-F. Active turbulence. Annu. Rev. Condens. Matter Phys. 13, 143–170, DOI:
+10.1146/annurev-conmatphys-082321-035957 (2022).
+45. Marth, W., Aland, S. & Voigt, A. Margination of white blood cells: A computational approach by a hydrodynamic phase
+field model. J. Fluid Mech. 790, 389–406, DOI: 10.1017/jfm.2016.15 (2016). 1507.01544.
+46. Praetorius, S. & Voigt, A. Collective cell behaviour – a cell-based parallelisation approach for a phase field active polar gel
+model. In NIC Series, 49, 369–376 (Forschungszentrum Jülich GmbH, Zentralbibliothek, Jülich, 2018).
+47. Vey, S. & Voigt, A. AMDiS: Adaptive multidimensional simulations. Comput. Vis. Sci. 10, 57–67, DOI: 10.1007/
+s00791-006-0048-3 (2007).
+48. Witkowski, T., Ling, S., Praetorius, S. & Voigt, A. Software concepts and numerical algorithms for a scalable adaptive
+parallel finite element method. Adv. Comput. Math. 41, 1145–1177, DOI: 10.1007/s10444-015-9405-4 (2015).
+Acknowledgements
+This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the
+Marie Skłodowska-Curie grant agreement No 945371. We acknowledge computing resources provided within project WIR at
+ZIH at TU Dresden.
+Author contributions statements
+H.P.J. implemented the codes, performed all simulations, analysed data and contributed to conceptual development and
+manuscript writing. A.V. and L.A. contributed to supervision, conceptual development, data analysis and manuscript writing.
+Additional information
+Competing interests The authors declare no competing interests.
+14/14
+

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+arXiv:2301.02599v1  [math.GM]  2 Jan 2023
+Wigner–Yanase–Dyson function and logarithmic mean
+Shigeru Furuichi1∗
+1Department of Information Science,
+College of Humanities and Sciences, Nihon University,
+3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan
+Abstract.
+The ordering between Wigner–Yanase–Dyson function and logarithmic mean
+is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse
+inequalities for Wigner–Yanase–Dyson function and logarithmic mean. We also compare the
+obtained results with the known bounds of the logarithmic mean.
+Finally we give operator
+inequalities based on the obtained results.
+Keywords :
+Wigner–Yanase–Dyson function, logarithmic mean, Kantorovich constant,
+Specht ratio and reverse inequalities
+2020 Mathematics Subject Classification :
+Primary 26E60, Secondary 26D07.
+1
+Introduction
+In this paper, we study the ordering of the symmetric homogeneous means N(x, y) for x, y > 0.
+The mean N(x, y) is called the symmetric homogeneous mean if the following conditions are
+satisfied ([8]):
+(i) N(x, y) = N(y, x).
+(ii) N(kx, ky) = kN(x, y) for k > 0.
+(iii) min{x, y} ≤ N(x, y) ≤ max{x, y}.
+(iv) N(x, y) is non–decreasing in x and y.
+Since we do not treat the weighted means, a symmetric homogeneous mean is often called a
+mean simply in this paper. In order to determine the ordering of two means such as N1(x, y) ≤
+N2(x, y) for x, y > 0, it is sufficient to show the ordering N1(x, 1) ≤ N2(x, 1) for x > 0 by
+homogeneity such that yN (x/y, 1) = N(x, y) for a symmetric homogeneous mean N(·, ·) and
+x, y > 0. Throughout this paper, we use the standard symbol A(x, y) := x + y
+2
+, L(x, y) :=
+x − y
+log x − log y, (x ̸= y > 0) with L(x, x) := x, G(x, y) := √xy and H(x, y) :=
+2xy
+x + y as the
+arithmetic mean, logarithmic mean, geometric mean and harmonic mean, respectively.
+The
+Wigner–Yanase–Dyson function is given by
+Wp (x, y) :=
+p (1 − p) (x − y)2
+(xp − yp) (x1−p − y1−p), (x ̸= y > 0, p ∈ R) , with Wp(x, x) = x
+∗E-mail:furuichi.shigeru@nihon-u.ac.jp
+1
+
+which was firstly appeared in [13]. Since Wp(x, 1) is matrix monotone function on x ∈ (0, ∞)
+when −1 ≤ p ≤ 2 [16], the parameter p is often considered to be −1 ≤ p ≤ 2. We mainly
+consider the case of 0 ≤ p ≤ 1 in this paper, as it was done so in [4, 5, 6] to study the
+Wigner–Yanase–Dyson metric with Morozova–Chentsov function or the Wigner–Yanase–Dyson
+skew information. It is easily seen that W1−p(x, y) = Wp(x, y) and W1/2(x, y) =
+�√x + √y
+2
+�2
+which is called the Wigner–Yanase function or the binomial mean Bp(x, y) :=
+�xp + yp
+2
+�1/p
+with p = 1/2. It is also known that
+H(x, y) ≤ G(x, y) ≤ L(x, y) ≤ Wp(x, y) ≤ W1/2(x, y) ≤ A(x, y), (x, y > 0, 0 ≤ p ≤ 1).
+The set M(n, C) represents all n×n matrices on complex field. The set M+(n, C) represents
+all positive semi–definite matrices in M(n, C). The stronger ordering N1(x, y) ⪯ N2(x, y) for
+means N1 and N2 have been studied in [3, 7, 8, 11, 14] for the study of the unitarily invariant
+norm inequalities and recent advances on the related topics.
+It is known [8, 11] that the ordering N1(x, y) ⪯ N2(x, y) is equivalent to the unitarily
+invariant norm inequality |||N1(S, T)X||| ≤ |||N1(S, T)X||| for S, T ∈ M+(n, C) and arbitrary
+X ∈ M(n, C), implies the usual ordering N1(x, y) ≤ N2(x, y) which is equivalent to the Hilbert–
+Schmidt (Frobenius) norm inequality ∥N1(S, T)X∥2 ≤ ∥N2(S, T)X∥2. See [8, 11] the precise
+definition and equivalent conditions on the stronger ordering N1(x, y) ⪯ N2(x, y). We study the
+usual ordering for some means in this paper. The following propositions are known.
+Proposition 1.1. ([9]) For S, T ∈ M+(n, C) and any X ∈ M(n, C), if 1/2 ≤ p ≤ 1 ≤ q ≤ 2 or
+−1 ≤ q ≤ 0 ≤ p ≤ 1/2, then we have
+|||H(S, T)X||| ≤ |||Wq(S, T)X||| ≤ |||L(S, T)X||| ≤ |||Wp(S, T)X||| ≤
+������B1/2(S, T)X
+������.
+In particular, p ∈ [0, 1] =⇒ |||L(S, T)X||| ≤ |||Wp(S, T)X|||.
+Proposition 1.2. ([1]) For S, T ∈ M+(n, C) and any X ∈ M(n, C),if |p| ≤ 1, then
+���
+���
+��� ˆGp(S, T)X
+���
+���
+��� ≤ |||L(S, T)X||| ≤
+���
+���
+��� ˆAp(S, T)X
+���
+���
+���,
+where ˆGp(x, y) := p(xy)p/2(x − y)
+xp − yp
+and ˆAp(x, y) := p(xp + yp)(x − y)
+2(xp − xp)
+for |p| ≤ 1 and x ̸= y.
+See [1] for the details on ˆGp(x, y) and ˆAp(x, y). From Proposition 1.1, we see L(x, y) ≤
+Wp(x, y) for 0 ≤ p ≤ 1. In Section 2, we study the reverse inequalities of L(x, y) ≤ Wp(x, y). In
+addition, we compare the obtained results in Section 2 with the bounds in Proposition 1.2, in
+Section 3.
+2
+Reverse inequalities
+For x > 0, t > 0,we have ln−t x ≤ log x ≤ lnt x, where lnt x := xt − 1
+t
+, (x > 0, t ̸= 0). Thus we
+have the simple bounds of Wp, (0 ≤ p ≤ 1) as
+Wp(x, 1) ≤ L(x, 1)2, (x ≥ 1),
+Wp(x, 1) ≥ L(x, 1)2, (0 < x ≤ 1).
+2
+
+Since fp(t) := xpt log x is convex in t when x ≥ 1, 0 ≤ p ≤ 1, taking an account for
+� 1
+0 fp(t)dt = lnp x, we have xp/2 log x ≤ lnp x ≤
+�xp + 1
+2
+�
+log x from Hermite–Hadamard in-
+equality. Thus the slightly improved upper bound was obtained under the condition x ≥ 1:
+4
+(xp + 1)(x1−p + 1)L(x, 1)2 ≤ Wp(x, 1) ≤
+1
+√xL(x, 1)2, (x ≥ 1).
+Also,we have the reverse inequality of the above for 0 < x ≤ 1 since fp(t) concave in t when
+0 < x ≤ 1.In this section, we study the reverse inequalities of L(x, y) ≤ Wp(x, y) for all x > 0
+not restricted as x ≥ 1 or 0 < x ≤ 1.
+We firstly consider the difference type reverse inequality of L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤
+p ≤ 1). From the simple calculations, we have
+Wp(x, 1) ≤
+�√x + 1
+2
+�2
+≤ r
+�√x − 1
+�2+√x ≤ r
+�√x − 1
+�2+L(x, 1), (x > 0, 0 ≤ p ≤ 1, r ≥ 1/4)
+(1)
+Considering the parameter p, we can obtain the first inequality in the following as a general
+result.
+Theorem 2.1. Let x > 0. For 0 ≤ p ≤ 1, we have
+Wp(x, 1) ≤ p(1 − p)
+�√x − 1
+�2 + L(x, 1) ≤ A(x, 1).
+(2)
+Proof. If the inequality (2) holds for x ≥ 1, then the inequality (2) holds for 0 < x ≤ 1. It is
+easily seen by putting x := 1/y ≥ 1 in the proven inequality (2) for x ≥ 1. Thus it is sufficient
+to prove inequality (2) for x ≥ 1 to show the inequality (2) for x > 0.
+In (2), put x instead of √x. Then the denominator is
+2p(1 − p)(x − 1)(x2p − 1)(x2(1−p) − 1) log x ≥ 0
+for x ≥ 1 when we reduce the difference right hand side minus the left hand side to a common
+denominator. Also we set the numerator as f(x, p), namely
+f(x, p) := (x + 1)(x2p − 1)(x2(1−p) − 1) + 2p(1 − p)
+�
+(x2p − 1)(x2(1−p) − 1) − (x + 1)2�
+log x.
+Since f(x, 1 − p) = f(x, p),we have only to prove f(x, p) ≥ 0 for x ≥ 1 and 0 ≤ p ≤ 1/2. We
+calculate
+df(x, p)
+dp
+= 4(x1−2p + 1)(log x)g(x, p),
+g(x, p) := h(x, p) + p(1 − p)(x − 1)(x − x2p) log x
+h(x, p) := px2 − (1 − p)x2p+1 − px + x − px2p,
+dh(x, p)
+dx
+= 2px − (1 − p)(1 + 2p)x2p − 2p2x2p−1 + 1 − p,
+d2h(x, p)
+dx2
+= 2p
+�
+−(1 − p)(1 + 2p)x2p−1 + p(1 − 2p)x2p−2 + 1
+�
+,
+d3h(x, p)
+dx3
+= 2p(1 − p)(1 − 2p)x2p−3 {2p(x − 1) + x} ≥ 0, (x ≥ 1, 0 ≤ p ≤ 1/2),
+so that we have
+d2h(x, p)
+dx2
+≥ d2h(1, p)
+dx2
+= 0 =⇒ dh(x, p)
+dx
+≥ dh(1, p)
+dx
+= 0 =⇒ h(x, p) ≥ h(1, p) = 0.
+3
+
+From p(1 − p)(x − 1)(x − x2p) log x ≥ 0, (x ≥ 1, 0 ≤ p ≤ 1/2) with the above results, we have
+df(x, p)
+dp
+≥ 0 which implies f(x, p) ≥ f(x, 0) = 0.
+To prove the second inequality, we set
+k(x, p) := x + 1
+2
+− p(1 − p)
+�√x − 1
+�2 − x − 1
+log x , (x > 1).
+Then we have
+k(x, p) ≥ k(x, 1/2) = 4 − 4x + (√x + 1)2 log x
+4 log x
+≥ 0.
+Indeed, we have x − 1
+log x ≤
+�√x + 1
+2
+�2
+which implies 4−4x+(√x +1)2 log x ≥ 0 for x > 1. This
+completes the proof with k(1, p) = 0.
+For the special case p = 1/2 in Theorem 2.1, the inequalities in (2) are reduced to G(x, 1) ≤
+L(x, 1) ≤ B1/2(x, 1). Note that the right hand side of the second inequality in (2) can not be
+replaced by W1/2(x, 1) which is less than or equal to A(x, 1).
+Secondly we consider the ratio type reverse inequality of L(x, y) ≤ Wp(x, y). From the known
+wesults, we have
+Wp(x, 1) ≤
+�√x + 1
+2
+�2
+≤ A(x, 1) ≤ S(x)G(x, 1) ≤ S(x)L(x, 1), (x > 0, 0 ≤ p ≤ 1).
+(3)
+Where S(x) :=
+x
+1
+x−1
+e log x
+1
+x−1
+is Specht ratio [15]. From the relation K(x) := (x + 1)2
+4x
+≥ S(x),
+Specht ratio in (3) can be replaced by Kantorovich constant K(x) [10]. See [2, Chapter 2] and
+references therein for the recent results on the inequalities with Specht ratio and Kantorovich
+constant. Moreover we have the following inequality if we use Kantorovich constant K(x).
+Wp(x, 1) ≤
+�√x + 1
+2
+�2
+= K(√x)√x ≤ K(√x)L(x, 1), (x > 0, 0 ≤ p ≤ 1)
+(4)
+From (3) and (4), it may be expected that
+�√x + 1
+2
+�2
+≤ S(√x)√x. However, this fails.
+Indeed we have the following proposition. In this point, we see that the ordering K(x) ≥ S(x)
+is effective.
+Proposition 2.2. For x > 0,
+�√x + 1
+2
+�2
+≥ S(√x)√x.
+(5)
+Proof. When x = 1, we have equality of (5) since S(1) = 1. The inequality (5) is equivalent to
+the following inequality:
+(x − 1)x
+x
+x−1
+e log x
+≤
+�x + 1
+2
+�2
+.
+(6)
+By the similar reason as we stated in the beginning of the proof in Theorem 2.1, it is sufficient
+to prove (6) for x > 1. Taking a logarithm of both sides in (6) and considering its difference:
+f(x) := 2 log
+�x + 1
+2
+�
+− log(x − 1) −
+x
+x − 1 log x + 1 + log (log x) .
+4
+
+Since L(x, 1) ≥ H(x, 1) and L(x, 1)−1 ≥ A(x, 1)−1 for x > 0, we have
+f ′(x) =
+1
+x(x − 1)
+�x − 1
+log x + x log x
+x − 1 −
+4x
+x + 1
+�
+≥ 0, (x > 1).
+Thus we have f(x) ≥ f(1) = 0.
+It is notable that the inequality (5) can be also obtaind by putting v = 1/2 in [2, Theorem
+2.10.1], taking a square the both sides and then replacing x by √x.
+The following result is the ratio type reverse inequality of L(x, y) ≤ Wp(x, y) for 0 ≤ p ≤ 1.
+Theorem 2.3. For x > 0, 0 ≤ p ≤ 1, we have
+Wp(x, 1) ≤ K(x)p(1−p)L(x, 1).
+(7)
+Proof. For x = 1, we have equality in (7). So it is sufficient to prove (7) for x > 1. Take a
+logarithm of the both sides in (7) and put the function f(x, p) as its difference, namely
+f(x, p)
+:=
+− log(x − 1) − log (log x) + 2p(1 − p) log(x + 1) − p(1 − p) log 4x
+− log p − log(1 − p) + log(xp − 1) + log(x1−p − 1).
+We calculate
+df(x, p)
+dx
+=
+−
+1
+x − 1 − p(1 − p)
+x
++ 2p(1 − p)
+x + 1
++ pxp−1
+xp − 1 + p − 1
+xp − x +
+1
+x log x
+=
+−
+1
+x − 1 + p(1 − p) (x − 1)
+x(x + 1) +
+1
+x1−p lnp x +
+1
+xp ln1−p x +
+1
+x log x
+≥
+−
+1
+x − 1 +
+1
+x1−p lnp x +
+1
+xp ln1−p x =: g(x, p)
+and
+g(x, p) =
+h(x, p)
+(x − 1)(xp − 1)(x − xp) ≥ 0, (x > 1, 0 ≤ p ≤ 1).
+Indeed,
+h(x, p) :
+=
+−(xp − 1)(x − xp) + pxp−1(x − 1)(x − xp) + (1 − p)(x − 1)(x − xp)
+=
+(1 − p) + px − 2xp + (1 − p)x2p + px2p−1
+=
+(1 − p)(1 + x2p) + p(x + x2p−1) − 2xp
+≥
+(1 − p) × 2xp + p × 2xp − 2xp = 0.
+Therefore we have f(x, p) ≥ f(1, p) = 0.
+Remark 2.4. It is natural to consider the replacement K(x) by S(x) in (7). However we have
+not prove
+Wp(x, 1) ≤ S(x)p(1−p)L(x, 1),
+(x > 0, 0 ≤ p ≤ 1).
+We also have not found any counter-example of the above inequality.
+It is known that S(x) ≤ K(x), S(xr) ≤ S(x)r and K(xr) ≤ K(x)r for x > 0 and 0 ≤ r ≤ 1
+[2, Section 2.10]. Also it is known that both K(x) and S(x) are decreasing for 0 < x < 1 and
+increasing x > 1 with S(1) = K(1) = 1.
+We are interested to find the smaller constant depending p ∈ [0, 1] than K(x)p(1−p). By the
+numerical computations we found the counter-example for
+Wp(x, 1) ≤ K(xp)L(x, 1),
+(x > 0, 0 ≤ p ≤ 1).
+Thus the inequalities Wp(x, 1) ≤ S(xp)L(x, 1) and Wp(x, 1) ≤ K(xp(1−p))L(x, 1) do not hold in
+general.
+5
+
+3
+Comparisons of the bounds for logarithmic mean
+From Proposition 1.1 and Theorem 2.3, we have
+K(x)−p(1−p)Wp(x, 1) ≤ L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤ p ≤ 1).
+(8)
+On the other hand, from Proposition 1.2, we have
+ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1), (x > 0, 0 ≤ p ≤ 1).
+(9)
+In this section, we compare the bounds of L(x, 1) in (8) and (9). The first result is the
+comparison on the upper bounds of L(x, 1).
+Theorem 3.1. Let x > 0 and p ∈ R.
+(i) If p ∈ [0, 1/2], then Wp(x, 1) ≥ ˆAp(x, 1).
+(ii) If p /∈ (0, 1/2), then Wp(x, 1) ≤ ˆAp(x, 1).
+Proof. It is sufficient to prove for x ≥ 1. Taking account of x1−p − 1
+1 − p
+≥ 0 for x ≥ 1, we set
+fp(x) := 2(x − 1) − (xp + 1)(x1−p − 1)
+(1 − p)
+=
+�
+2 −
+1
+1 − p
+�
+(x − 1) + xp − x1−p
+1 − p
+.
+Since we have
+f ′
+p(x) =
+�
+2 −
+1
+1 − p
+�
++ pxp−1 − (1 − p)x−p
+1 − p
+, f ′′
+p (x) = px−p−1(1 − x2p−1),
+we have f ′′
+p (x) ≥ 0 for p ∈ [0, 1/2].Thus we have f ′
+p(x) ≥ f ′
+p(1) = 0 which implies fp(x) ≥
+fp(1) = 0. Therefore we obtain (i). Similarly we have f ′′
+p (x) ≤ 0 for p /∈ (0, 1/2). Thus we have
+f ′
+p(x) ≤ f ′
+p(1) = 0 which implies fp(x) ≤ fp(1) = 0. Therefore we obtain (ii).
+The second result is the comparison on the lower bounds of L(x, 1). To prove it, we prepare
+the following lemma which is interesting itself.
+Lemma 3.2. For x > 0, we have
+L(x, 1)2 − G(x, 1)2
+2L(x, 1)2
+≤ A(x, 1) − L(x, 1)
+L(x, 1)
+≤ log K(x).
+(10)
+Proof. It is sufficient to prove (10) for x ≥ 1. We firstly prove the second inequality. To this
+end, we set
+u(x) := 2(x − 1) − (x + 1) log x + 4(x − 1) log(x + 1) − 2(x − 1) log(4x), (x ≥ 1).
+We calculate
+u′(x) =
+4x
+x + 1 −
+4
+x + 1 + 1
+x − 1 − 3 log x + 4 log(x + 1) − 4 log 2
+u′′(x) = (x − 1)(x2 + 6x + 1)
+x2(x + 1)2
+≥ 0.
+Thus we have
+u′(x) ≥ u′(1) = 0 =⇒ u(x) ≥ u(1) = 0.
+6
+
+We secondly prove the first inequality. To this end, we set
+v(x) := (x2 − 1) log x + x(log x)2 − 3(x − 1)2,
+(x ≥ 1).
+We calculate
+v′(x) = 2(x + 1) log x + (log x)2 − 5x + 6 − 1
+x,
+v′′(x) = 1
+x2
+�
+2x(x + 1) log x − 3x2 + 2x + 1
+�
+v(3)(x) = 2
+x3 w(x),
+w(x) := x2 − 1 − x log x,
+w′(x) = 2x − 1 − log x ≥ 0.
+Thus we have
+w(x) ≥ w(1) = 0 =⇒ v(3)(x) ≥ 0 =⇒ v′′(x) ≥ v′′(1) = 0 =⇒ v′(x) ≥ v′(1) = 0 =⇒ v(x) ≥ v(1) = 0.
+Therefore we obtain 3L(x, 1) ≤ 2A(x, 1) + G(x, 1)2/L(x, 1), (x > 0) which is equivalent to the
+first inequality of (10).
+Here, the second inequality of (10) is equivalent to the inequality:
+1 − (x + 1) log x
+2(x − 1)
++ log
+�(x + 1)2
+4x
+�
+≥ 0
+(11)
+Also the inequality L(x, 1)2 − G(x, 1)2
+2L(x, 1)2
+≤ log K(x) is equivalent to the inequality:
+1 − x(log x)2
+(x − 1)2 + 2 log
+�
+4x
+(x + 1)2
+�
+≤ 0.
+(12)
+The inequalities (11) and (12) will be used in the proof of Theorem 3.3 below.
+Theorem 3.3. Let x > 0 and p ∈ R.
+(i) If 0 ≤ p ≤ 1
+2, then ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1).
+(ii) If p ≤ 0 or p ≥ 1, then ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1).
+Proof. It is sufficient to prove for x ≥ 1. Since K(x) ≥ 1, in order to prove (i),
+ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1) ⇐⇒ ˆGp(x, 1)K(x)p(1−p) ≥ Wp(x, 1)
+⇐⇒ x
+p
+2
+�(x + 1)2
+4x
+�p(1−p)
+≥ (1 − p)(x − 1)
+x1−p − 1
+we set
+fp(x) := p
+2 log x + p(1 − p) {2 log(x + 1) − log(4x)} − log(1 − p) − log(x − 1) + log(x1−p − 1).
+Then we have
+f ′
+p(x)
+=
+−
+1
+x − 1 + p
+2x + p(1 − p)(x − 1)
+x(x + 1)
++ 1 − p
+x − xp
+=
+gp(x)
+2(x − 1)(x + 1)(x − xp)
+7
+
+where
+gp(x) := 2(x + 1)(xp − 1 − p(x − 1)) + p(x − 1)(1 − xp−1) ((3 − 2p)x + (2p − 1)) .
+When x ≥ 1 and 0 ≤ p ≤ 1/2, we have 1 ≥ xp−1, −xp−3 ≥ −xp−2 and (1−p)(1−2p)(p−2) ≤ 0.
+Also when x ≥ 1 and 0 ≤ p ≤ 1/2, we have xp−1 ≥ xp−2. Thus we calculate
+g′
+p(x) = (p + 1)(2p2 − 3p + 2)xp − 2p(2p2 − 2p − 1)xp−1 + p(1 − p)(1 − 2p)xp−2
++2p(1 − 2p)x + 2(2p2 − 2p − 1)
+g′′
+p(x) = p
+�
+(p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2
++(1 − p)(1 − 2p)(p − 2)xp−3 + 2(1 − 2p)
+�
+≥ p
+�
+(p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − 2p)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2
++(1 − p)(1 − 2p)(p − 2)xp−2�
+= (2p3 − p2 − 5p + 4)(xp−1 − xp−2) = 2(1 − p) {(1 − p)(1 + p) + 1 − p/2} (xp−1 − xp−2) ≥ 0.
+Thus we have g′
+p(x) ≥ g′
+p(1) = 0 so that gp(x) ≥ gp(1) = 0. Therefore we have f ′
+p(x) ≥ 0 for
+x ≥ 1 and taking an accout for lim
+x→1
+(1 − p)(x − 1)
+x1−p − 1
+= 1, we havefp(x) ≥ fp(1) = 0 which proves
+(i).
+It is also sufficent to prove (ii) for x ≥ 1. For the special cases p = 0 or p = 1 we have
+equality. Since
+ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1) ⇐⇒ x
+p
+2
+�(x + 1)2
+4x
+�p(1−p)
+≤ (1 − p)(x − 1)
+x1−p − 1
+,
+we have only to prove fp(x) ≤ 0 for x ≥ 1.
+(a) We consider the case p > 1. We set g′′
+p(x) = p · hp(x), namely
+hp(x) := (p+1)(2p2−3p+2)xp−1+2(1−p)(2p2−2p−1)xp−2+(1−p)(1−2p)(p−2)xp−3+2(1−2p).
+Then
+h′
+p(x) = (p − 1)xp−4kp(x),
+where
+kp(x) := 2p3(x − 1)2 − p2(x − 1)(x − 11) − p(x2 + 6x − 17) + 2(x − 3)(x + 1)
+and we have
+k′
+p(x) = 4p3(x−1)−2p2(x−6)−2p(x+3)+4(x−1), k′′
+p(x) = 2(p+1)(2p2−3p+2) > 0, (p > 1).
+Thus we have k′
+p(x) ≥ k′
+p(1) = 10p2 − 8p > 0, (p > 1) so that kp(x) ≥ kp(1) = 10p − 8 >
+0, (p > 1). Therefore we have h′
+p(x) ≥ 0, (p > 1) which implies hp(x) ≥ hp(1) = 0. Thus
+we have g′′
+p(x) ≥ 0 so that we have g′
+p(x) ≥ g′
+p(1) = 0 which implies gp(x) ≥ gp(1) = 0.
+Taking account of x − xp ≤ 0 when x ≥ 1, p > 1, we have f ′
+p(x) ≤ 0 Therefore we have
+fp(x) ≤ fp(1) = 0 which proves (ii) for the case p > 1.
+(b) We consider the case p < 0. We calculate
+dfp(x)
+dp
+=
+1
+1 − p +
+�1
+2 +
+x
+xp − x
+�
+log x + (2p − 1) log(4x) + (2 − 4p) log(x + 1),
+d2fp(x)
+dp2
+= −xp+1(log x)2
+(x − xp)2
++
+1
+(p − 1)2 + 2 log(4x) − 4 log(x + 1),
+d3fp(x)
+dp3
+= xp+1 (xp + x) (log x)3
+(xp − x)3
+−
+2
+(p − 1)3 .
+8
+
+We further calculate
+d
+dx
+�d3fp(x)
+dp3
+�
+= −xp(log x)2
+(x − xp)4 s(x, p),
+s(x, p) := 3(x2 − x2p) + (p − 1)
+�
+x2 + x2p + 4x1+p�
+log x,
+ds(x, p)
+dp
+=
+�
+x2 − 5x2p + 4xp+1 + 2(p − 1)
+�
+x2p + 2xp+1�
+log(x)
+�
+log x,
+d2s(x, p)
+dp2
+= 4xp(log x)2 {2(x − xp) + (p − 1)(x + xp) log x} ≤ 0 (p ≤ 1, x ≥ 1).
+Indeed, putting a := x, b := xp in the inequality
+a − b
+log a − log b ≤ a + b
+2
+, (a, b > 0), we have
+2(x−xp)+(p−1)(x+xp) log x ≤ 0 for p < 1 and x ≥ 1. (The equality holds when p = 1.)
+From d2s(x, p)
+dp2
+≤ 0, we have ds(x, p)
+dp
+≥ ds(x, p)
+dp
+����
+p=1
+= 0 which implies s(x, p) ≤ s(x, 1) =
+0 so that we have d
+dx
+�d3fp(x)
+dp3
+�
+≥ 0. From this, we have
+d3fp(x)
+dp3
+≥ d3fp(1)
+dp3
+= −
+2
+(p − 1)3 > 0, (p < 1).
+By the inequality (12), we have
+p ≤ 0 =⇒ d2fp(x)
+dp2
+≤ d2fp(x)
+dp2
+����
+p=0
+= 1 − x(log x)2
+(x − 1)2 + 2 log
+�
+4x
+(x + 1)2
+�
+≤ 0.
+(13)
+Thus we have by the inequality (11),
+p ≤ 0 =⇒ dfp(x)
+dp
+≥ dfp(x)
+dp
+����
+p=0
+= 1 − (x + 1) log x
+2(x − 1)
++ log
+�(x + 1)2
+4x
+�
+≥ 0.
+Therefore p ≤ 0 =⇒ fp(x) ≤ f0(x) = 0 which proves (ii) for the case p < 0.
+Remark 3.4.
+(i) Since fp(x) = log
+��(x + 1)2
+4x
+�p(1−p)
+× xp/2(x1−p − 1)
+(1 − p)(x − 1)
+�
+appeared in the
+proof of Theorem 3.3 and comparing the maximum degree of the numerator and the
+denominator insides of the logarithmic function, we found that if p < 0 or p > 1/2, then
+we have
+lim
+x→∞
+��(x + 1)2
+4x
+�p(1−p)
+× xp/2(x1−p − 1)
+(1 − p)(x − 1)
+�
+= 0.
+Thus we have lim
+x→∞ fp(x) = −∞ if p < 0 or p > 1/2. On the other hand, by the muner-
+ical computations, we have f3/4(e) ≃ 0.0063209. Therefore there is no ordering between
+K(x)−p(1−p)Wp(x, 1) and ˆGp(x, 1) for x > 0 and 1/2 < p < 1.
+(ii) From Theorem 3.1 (i) and Theorem 3.3 (i), we have for x > 0 and 0 ≤ p ≤ 1/2,
+K(x)−p(1−p)Wp(x, 1) ≤ ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1) ≤ Wp(x, 1).
+(iii) From the proof (b) in Theorem 3.3, we obtained d3fp(x)
+dp3
+≥ 0, (p < 1). From this with
+simple calculations, we have
+H(x, xp)3 ≤ xp+1A(x, xp) ≤ L(x, xp)3, (p ≤ 1, x > 0).
+9
+
+4
+Conclusion
+As we have seen, we studied the inequalities on the relations between the Wigner–Yanase–
+Dyson function Wp(·, ·) and the logarithmic mean L(·, ·). As one of main results, we obtained
+two kinds of the reverse inequalities for L(x, y) ≤ Wp(x, y) for x, y > 0 and 0 ≤ p ≤ 1. That
+is, the inequalities (2) and (7) shown in Theorem 2.1 and 2.3 are respectively equivalent to the
+following inequalities for x, y > 0 and 0 ≤ p ≤ 1:
+Wp(x, y) ≤ p(1 − p)
+�√x − √y
+�2 + L(x, y),
+(14)
+Wp(x, y) ≤ K (x/y)p(1−p) L(x, y).
+(15)
+The inequality (14) and (15) are the difference type reverse inequality and the ratio type reverse
+inequality for L(x, y) ≤ Wp(x, y), (0 ≤ p ≤ 1), respectively.
+In addition, we compared the obtained inequality (15) with the known result in Section 3.
+It is summerized in the following. The inequalities given in Remark 3.4 (ii) are equivalent to
+the following inequalities for x, y > 0 and 0 ≤ p ≤ 1/2:
+K (x/y)−p(1−p) Wp(x, y) ≤ ˆGp(x, y) ≤ L(x, y) ≤ ˆAp(x, y) ≤ Wp(x, y).
+(16)
+We conclude this paper by giving operator inequalities baesd on Theorem 2.1 and 2.3. For
+positive operators S, T and 0 ≤ p ≤ 1, we define the operator version of the logarithmic mean
+and the Wigner–Yanase–Dyson function as
+L(S, T) :=
+� 1
+0
+S♯tTdt,
+Wp(S, T) := p(1 − p)
+2
+(S − T) (S∇T − Hzp(S, T))−1 (S − T), (S ̸= T),
+Wp(S, S) := S.
+where
+S♯pT := S1/2 �
+S−1/2TS−1/2�p
+S1/2, S∇T := S + T
+2
+, Hzp(S, T) := 1
+2 (S♯pT + S♯1−pT) .
+We often use the symbol S♯T := S♯1/2T for short. Hzp(S, T) is often called the Heinz mean. It
+is notable that we have the relation for the validity in the case p := 1/2,
+�S − T
+2
+�
+(S∇T + S♯T)−1
+�S − T
+2
+�
+= S∇T − S♯T,
+which can be confirmed by multiplying S−1/2 to both sides.
+From Theorem 2.1, we have the following corollary.
+Corollary 4.1. Let S and T be positive operators and let 0 ≤ p ≤ 1. Then we have
+Wp(S, T) ≤ 2p(1 − p) (S∇T − S♯T) + L(S, T).
+From Theorem 2.3, we also have the following corollary.
+Corollary 4.2. Let S and T be positive operators with αS ≤ T ≤ βS for 0 < α ≤ β and let
+0 ≤ p ≤ 1. Then we have
+Wp(S, T) ≤ kp · L(S, T),
+kp := max
+α≤x≤β K(x)p(1−p).
+10
+
+Acknowledgement
+The author (S.F.) was partially supported by JSPS KAKENHI Grant Number 21K03341.
+References
+[1] S.Furuichi,
+Unitarily
+invariant
+norm
+inequalities
+for
+some
+means,
+J.Inequal.Appl.,2014(2014), Art.158.
+[2] S.Furuichi and H.R.Moradi, Advances in mathematical inequalities, De Gruyter, 2020.
+[3] S.Furuichi and M.E.Amlashi, On bounds of logarithmic mean and mean inequality chain,
+arXiv:2203.01134.
+[4] S.Furuichi and K.Yanagi, Schr¨odinger uncertainty relation, Wigner–Yanase–Dyson skew
+information and metric adjusted correlation measure, J. Math. Anal. Appl.,388(2)(2012),
+1147–1156.
+[5] P. Gibilisco, F.Hansen and T. Isola, On a correspondence between regular and non-regular
+operator monotone functions, Linear Alg. Appl., 430 (2009) 2225–2232.
+[6] F.Hansen, Metric adjusted skew information, Proc. Nat.Acad. Sci.,105(2008), 9909–9916.
+[7] F. Hiai and H. Kosaki, Means for matrices and comparison of their norms, Indiana Univ.
+Math. J. 48 (1999) 899–936.
+[8] F.Hiai and H.Kosaki, Means of Hilbert space operators, Springer–Verlag, 2003.
+[9] F.Hiai, H.Kosaki,D.Petz and B.Ruskai Families of completely positive maps associated with
+monotone metrics, Linear Alg. Appl., 48(439)(2013), 1749–1791.
+[10] L. V. Kantorovich, Functional analysis and applied mathematics, Uspekhi Mat. Nauk,
+3:6(28) (1948), 89–185. http://mi.mathnet.ru/eng/umn/v3/i6/p89.
+[11] H. Kosaki, Positive definiteness of functions with applications to operator norm inequalities,
+Mem. Amer. Math. Soc., 212(997), 2011.
+[12] H.Kosaki, Strong monotonicity for various means, J.Func.Anal.,267(2014),1917–1958.
+[13] D.Petz and H.Hasegawa, On the Riemannian metric of α–entropies of density matrices,
+Lett.Math.Phys.,38(1996), 221-225.
+[14] H.Kosaki, Positive definiteness and infinite divisibility of certain functions of hyperbolic
+cosine function, Int. J. Math.,33(7) (2022), 2250050.
+[15] W.Specht,
+Zur
+Theorie
+der
+elementaren
+Mittel,
+Math.Z,
+74
+(1960),
+91–98.
+10.1007/BF01180475.
+[16] V.E.S.Szab´o, A class of matrix monotone functions, Linear Alg. Appl.,420(2007), 79–85.
+11
+

0dE0T4oBgHgl3EQfuAFm/content/tmp_files/load_file.txt

@@ -0,0 +1,354 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf,len=353
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='02599v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='GM]  2 Jan 2023  Wigner–Yanase–Dyson function and logarithmic mean  Shigeru Furuichi1∗  1Department of Information Science,  College of Humanities and Sciences, Nihon University,  3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  The ordering between Wigner–Yanase–Dyson function and logarithmic mean  is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Also bounds for logarithmic mean are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' In this paper, we give two reverse  inequalities for Wigner–Yanase–Dyson function and logarithmic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' We also compare the  obtained results with the known bounds of the logarithmic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Finally we give operator  inequalities based on the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Keywords :  Wigner–Yanase–Dyson function, logarithmic mean, Kantorovich constant,  Specht ratio and reverse inequalities  2020 Mathematics Subject Classification :  Primary 26E60, Secondary 26D07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  1  Introduction  In this paper, we study the ordering of the symmetric homogeneous means N(x, y) for x, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  The mean N(x, y) is called the symmetric homogeneous mean if the following conditions are  satisfied ([8]):  (i) N(x, y) = N(y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (ii) N(kx, ky) = kN(x, y) for k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (iii) min{x, y} ≤ N(x, y) ≤ max{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (iv) N(x, y) is non–decreasing in x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Since we do not treat the weighted means, a symmetric homogeneous mean is often called a  mean simply in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' In order to determine the ordering of two means such as N1(x, y) ≤  N2(x, y) for x, y > 0, it is sufficient to show the ordering N1(x, 1) ≤ N2(x, 1) for x > 0 by  homogeneity such that yN (x/y, 1) = N(x, y) for a symmetric homogeneous mean N(·, ·) and  x, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Throughout this paper, we use the standard symbol A(x, y) := x + y  2  , L(x, y) :=  x − y  log x − log y, (x ̸= y > 0) with L(x, x) := x, G(x, y) := √xy and H(x, y) :=  2xy  x + y as the  arithmetic mean, logarithmic mean, geometric mean and harmonic mean, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  The  Wigner–Yanase–Dyson function is given by  Wp (x, y) :=  p (1 − p) (x − y)2  (xp − yp) (x1−p − y1−p), (x ̸= y > 0, p ∈ R) , with Wp(x, x) = x  ∗E-mail:furuichi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='shigeru@nihon-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='jp  1  which was firstly appeared in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Since Wp(x, 1) is matrix monotone function on x ∈ (0, ∞)  when −1 ≤ p ≤ 2 [16], the parameter p is often considered to be −1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' We mainly  consider the case of 0 ≤ p ≤ 1 in this paper, as it was done so in [4, 5, 6] to study the  Wigner–Yanase–Dyson metric with Morozova–Chentsov function or the Wigner–Yanase–Dyson  skew information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It is easily seen that W1−p(x, y) = Wp(x, y) and W1/2(x, y) =  �√x + √y  2  �2  which is called the Wigner–Yanase function or the binomial mean Bp(x, y) :=  �xp + yp  2  �1/p  with p = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It is also known that  H(x, y) ≤ G(x, y) ≤ L(x, y) ≤ Wp(x, y) ≤ W1/2(x, y) ≤ A(x, y), (x, y > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  The set M(n, C) represents all n×n matrices on complex field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' The set M+(n, C) represents  all positive semi–definite matrices in M(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' The stronger ordering N1(x, y) ⪯ N2(x, y) for  means N1 and N2 have been studied in [3, 7, 8, 11, 14] for the study of the unitarily invariant  norm inequalities and recent advances on the related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  It is known [8, 11] that the ordering N1(x, y) ⪯ N2(x, y) is equivalent to the unitarily  invariant norm inequality |||N1(S, T)X||| ≤ |||N1(S, T)X||| for S, T ∈ M+(n, C) and arbitrary  X ∈ M(n, C), implies the usual ordering N1(x, y) ≤ N2(x, y) which is equivalent to the Hilbert–  Schmidt (Frobenius) norm inequality ∥N1(S, T)X∥2 ≤ ∥N2(S, T)X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' See [8, 11] the precise  definition and equivalent conditions on the stronger ordering N1(x, y) ⪯ N2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' We study the  usual ordering for some means in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' The following propositions are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' ([9]) For S, T ∈ M+(n, C) and any X ∈ M(n, C), if 1/2 ≤ p ≤ 1 ≤ q ≤ 2 or  −1 ≤ q ≤ 0 ≤ p ≤ 1/2, then we have  |||H(S, T)X||| ≤ |||Wq(S, T)X||| ≤ |||L(S, T)X||| ≤ |||Wp(S, T)X||| ≤  ������B1/2(S, T)X  ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  In particular, p ∈ [0, 1] =⇒ |||L(S, T)X||| ≤ |||Wp(S, T)X|||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' ([1]) For S, T ∈ M+(n, C) and any X ∈ M(n, C),if |p| ≤ 1, then  ���  ���  ��� ˆGp(S, T)X  ���  ���  ��� ≤ |||L(S, T)X||| ≤  ���  ���  ��� ˆAp(S, T)X  ���  ���  ���,  where ˆGp(x, y) := p(xy)p/2(x − y)  xp − yp  and ˆAp(x, y) := p(xp + yp)(x − y)  2(xp − xp)  for |p| ≤ 1 and x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  See [1] for the details on ˆGp(x, y) and ˆAp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' From Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1, we see L(x, y) ≤  Wp(x, y) for 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' In Section 2, we study the reverse inequalities of L(x, y) ≤ Wp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' In  addition, we compare the obtained results in Section 2 with the bounds in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='2, in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  2  Reverse inequalities  For x > 0, t > 0,we have ln−t x ≤ log x ≤ lnt x, where lnt x := xt − 1  t  , (x > 0, t ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Thus we  have the simple bounds of Wp, (0 ≤ p ≤ 1) as  Wp(x, 1) ≤ L(x, 1)2, (x ≥ 1),  Wp(x, 1) ≥ L(x, 1)2, (0 < x ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  2  Since fp(t) := xpt log x is convex in t when x ≥ 1, 0 ≤ p ≤ 1, taking an account for  � 1  0 fp(t)dt = lnp x, we have xp/2 log x ≤ lnp x ≤  �xp + 1  2  �  log x from Hermite–Hadamard in-  equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Thus the slightly improved upper bound was obtained under the condition x ≥ 1:  4  (xp + 1)(x1−p + 1)L(x, 1)2 ≤ Wp(x, 1) ≤  1  √xL(x, 1)2, (x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Also,we have the reverse inequality of the above for 0 < x ≤ 1 since fp(t) concave in t when  0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='In this section, we study the reverse inequalities of L(x, y) ≤ Wp(x, y) for all x > 0  not restricted as x ≥ 1 or 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  We firstly consider the difference type reverse inequality of L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤  p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' From the simple calculations, we have  Wp(x, 1) ≤  �√x + 1  2  �2  ≤ r  �√x − 1  �2+√x ≤ r  �√x − 1  �2+L(x, 1), (x > 0, 0 ≤ p ≤ 1, r ≥ 1/4)  (1)  Considering the parameter p, we can obtain the first inequality in the following as a general  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Let x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' For 0 ≤ p ≤ 1, we have  Wp(x, 1) ≤ p(1 − p)  �√x − 1  �2 + L(x, 1) ≤ A(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' If the inequality (2) holds for x ≥ 1, then the inequality (2) holds for 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It is  easily seen by putting x := 1/y ≥ 1 in the proven inequality (2) for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Thus it is sufficient  to prove inequality (2) for x ≥ 1 to show the inequality (2) for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  In (2), put x instead of √x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Then the denominator is  2p(1 − p)(x − 1)(x2p − 1)(x2(1−p) − 1) log x ≥ 0  for x ≥ 1 when we reduce the difference right hand side minus the left hand side to a common  denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Also we set the numerator as f(x, p), namely  f(x, p) := (x + 1)(x2p − 1)(x2(1−p) − 1) + 2p(1 − p)  �  (x2p − 1)(x2(1−p) − 1) − (x + 1)2�  log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Since f(x, 1 − p) = f(x, p),we have only to prove f(x, p) ≥ 0 for x ≥ 1 and 0 ≤ p ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' We  calculate  df(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dp  = 4(x1−2p + 1)(log x)g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p) := h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p) + p(1 − p)(x − 1)(x − x2p) log x  h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p) := px2 − (1 − p)x2p+1 − px + x − px2p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  dh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dx  = 2px − (1 − p)(1 + 2p)x2p − 2p2x2p−1 + 1 − p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  d2h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dx2  = 2p  �  −(1 − p)(1 + 2p)x2p−1 + p(1 − 2p)x2p−2 + 1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  d3h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dx3  = 2p(1 − p)(1 − 2p)x2p−3 {2p(x − 1) + x} ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' (x ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' 0 ≤ p ≤ 1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  so that we have  d2h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dx2  ≥ d2h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dx2  = 0 =⇒ dh(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dx  ≥ dh(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p)  dx  = 0 =⇒ h(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p) ≥ h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  3  From p(1 − p)(x − 1)(x − x2p) log x ≥ 0, (x ≥ 1, 0 ≤ p ≤ 1/2) with the above results, we have  df(x, p)  dp  ≥ 0 which implies f(x, p) ≥ f(x, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  To prove the second inequality, we set  k(x, p) := x + 1  2  − p(1 − p)  �√x − 1  �2 − x − 1  log x , (x > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Then we have  k(x, p) ≥ k(x, 1/2) = 4 − 4x + (√x + 1)2 log x  4 log x  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Indeed, we have x − 1  log x ≤  �√x + 1  2  �2  which implies 4−4x+(√x +1)2 log x ≥ 0 for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' This  completes the proof with k(1, p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  For the special case p = 1/2 in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1, the inequalities in (2) are reduced to G(x, 1) ≤  L(x, 1) ≤ B1/2(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Note that the right hand side of the second inequality in (2) can not be  replaced by W1/2(x, 1) which is less than or equal to A(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Secondly we consider the ratio type reverse inequality of L(x, y) ≤ Wp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' From the known  wesults, we have  Wp(x, 1) ≤  �√x + 1  2  �2  ≤ A(x, 1) ≤ S(x)G(x, 1) ≤ S(x)L(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (3)  Where S(x) :=  x  1  x−1  e log x  1  x−1  is Specht ratio [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' From the relation K(x) := (x + 1)2  4x  ≥ S(x),  Specht ratio in (3) can be replaced by Kantorovich constant K(x) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' See [2, Chapter 2] and  references therein for the recent results on the inequalities with Specht ratio and Kantorovich  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Moreover we have the following inequality if we use Kantorovich constant K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Wp(x, 1) ≤  �√x + 1  2  �2  = K(√x)√x ≤ K(√x)L(x, 1), (x > 0, 0 ≤ p ≤ 1)  (4)  From (3) and (4), it may be expected that  �√x + 1  2  �2  ≤ S(√x)√x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' However, this fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Indeed we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' In this point, we see that the ordering K(x) ≥ S(x)  is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' For x > 0,  �√x + 1  2  �2  ≥ S(√x)√x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' When x = 1, we have equality of (5) since S(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' The inequality (5) is equivalent to  the following inequality:  (x − 1)x  x  x−1  e log x  ≤  �x + 1  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (6)  By the similar reason as we stated in the beginning of the proof in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1, it is sufficient  to prove (6) for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Taking a logarithm of both sides in (6) and considering its difference:  f(x) := 2 log  �x + 1  2  �  − log(x − 1) −  x  x − 1 log x + 1 + log (log x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  4  Since L(x, 1) ≥ H(x, 1) and L(x, 1)−1 ≥ A(x, 1)−1 for x > 0, we have  f ′(x) =  1  x(x − 1)  �x − 1  log x + x log x  x − 1 −  4x  x + 1  �  ≥ 0, (x > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Thus we have f(x) ≥ f(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  It is notable that the inequality (5) can be also obtaind by putting v = 1/2 in [2, Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1], taking a square the both sides and then replacing x by √x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  The following result is the ratio type reverse inequality of L(x, y) ≤ Wp(x, y) for 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' For x > 0, 0 ≤ p ≤ 1, we have  Wp(x, 1) ≤ K(x)p(1−p)L(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' For x = 1, we have equality in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' So it is sufficient to prove (7) for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Take a  logarithm of the both sides in (7) and put the function f(x, p) as its difference, namely  f(x, p)  :=  − log(x − 1) − log (log x) + 2p(1 − p) log(x + 1) − p(1 − p) log 4x  − log p − log(1 − p) + log(xp − 1) + log(x1−p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  We calculate  df(x, p)  dx  =  −  1  x − 1 − p(1 − p)  x  + 2p(1 − p)  x + 1  + pxp−1  xp − 1 + p − 1  xp − x +  1  x log x  =  −  1  x − 1 + p(1 − p) (x − 1)  x(x + 1) +  1  x1−p lnp x +  1  xp ln1−p x +  1  x log x  ≥  −  1  x − 1 +  1  x1−p lnp x +  1  xp ln1−p x =: g(x, p)  and  g(x, p) =  h(x, p)  (x − 1)(xp − 1)(x − xp) ≥ 0, (x > 1, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Indeed,  h(x, p) :  =  −(xp − 1)(x − xp) + pxp−1(x − 1)(x − xp) + (1 − p)(x − 1)(x − xp)  =  (1 − p) + px − 2xp + (1 − p)x2p + px2p−1  =  (1 − p)(1 + x2p) + p(x + x2p−1) − 2xp  ≥  (1 − p) × 2xp + p × 2xp − 2xp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Therefore we have f(x, p) ≥ f(1, p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It is natural to consider the replacement K(x) by S(x) in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' However we have  not prove  Wp(x, 1) ≤ S(x)p(1−p)L(x, 1),  (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  We also have not found any counter-example of the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  It is known that S(x) ≤ K(x), S(xr) ≤ S(x)r and K(xr) ≤ K(x)r for x > 0 and 0 ≤ r ≤ 1  [2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Also it is known that both K(x) and S(x) are decreasing for 0 < x < 1 and  increasing x > 1 with S(1) = K(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  We are interested to find the smaller constant depending p ∈ [0, 1] than K(x)p(1−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' By the  numerical computations we found the counter-example for  Wp(x, 1) ≤ K(xp)L(x, 1),  (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Thus the inequalities Wp(x, 1) ≤ S(xp)L(x, 1) and Wp(x, 1) ≤ K(xp(1−p))L(x, 1) do not hold in  general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  5  3  Comparisons of the bounds for logarithmic mean  From Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3, we have  K(x)−p(1−p)Wp(x, 1) ≤ L(x, 1) ≤ Wp(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (8)  On the other hand, from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='2, we have  ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1), (x > 0, 0 ≤ p ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (9)  In this section, we compare the bounds of L(x, 1) in (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' The first result is the  comparison on the upper bounds of L(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Let x > 0 and p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (i) If p ∈ [0, 1/2], then Wp(x, 1) ≥ ˆAp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (ii) If p /∈ (0, 1/2), then Wp(x, 1) ≤ ˆAp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It is sufficient to prove for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Taking account of x1−p − 1  1 − p  ≥ 0 for x ≥ 1, we set  fp(x) := 2(x − 1) − (xp + 1)(x1−p − 1)  (1 − p)  =  �  2 −  1  1 − p  �  (x − 1) + xp − x1−p  1 − p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Since we have  f ′  p(x) =  �  2 −  1  1 − p  �  + pxp−1 − (1 − p)x−p  1 − p  , f ′′  p (x) = px−p−1(1 − x2p−1),  we have f ′′  p (x) ≥ 0 for p ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Thus we have f ′  p(x) ≥ f ′  p(1) = 0 which implies fp(x) ≥  fp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Therefore we obtain (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Similarly we have f ′′  p (x) ≤ 0 for p /∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Thus we have  f ′  p(x) ≤ f ′  p(1) = 0 which implies fp(x) ≤ fp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Therefore we obtain (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  The second result is the comparison on the lower bounds of L(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' To prove it, we prepare  the following lemma which is interesting itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' For x > 0, we have  L(x, 1)2 − G(x, 1)2  2L(x, 1)2  ≤ A(x, 1) − L(x, 1)  L(x, 1)  ≤ log K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (10)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It is sufficient to prove (10) for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' We firstly prove the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' To this  end, we set  u(x) := 2(x − 1) − (x + 1) log x + 4(x − 1) log(x + 1) − 2(x − 1) log(4x), (x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  We calculate  u′(x) =  4x  x + 1 −  4  x + 1 + 1  x − 1 − 3 log x + 4 log(x + 1) − 4 log 2  u′′(x) = (x − 1)(x2 + 6x + 1)  x2(x + 1)2  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Thus we have  u′(x) ≥ u′(1) = 0 =⇒ u(x) ≥ u(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  6  We secondly prove the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' To this end, we set  v(x) := (x2 − 1) log x + x(log x)2 − 3(x − 1)2,  (x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  We calculate  v′(x) = 2(x + 1) log x + (log x)2 − 5x + 6 − 1  x,  v′′(x) = 1  x2  �  2x(x + 1) log x − 3x2 + 2x + 1  �  v(3)(x) = 2  x3 w(x),  w(x) := x2 − 1 − x log x,  w′(x) = 2x − 1 − log x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Thus we have  w(x) ≥ w(1) = 0 =⇒ v(3)(x) ≥ 0 =⇒ v′′(x) ≥ v′′(1) = 0 =⇒ v′(x) ≥ v′(1) = 0 =⇒ v(x) ≥ v(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Therefore we obtain 3L(x, 1) ≤ 2A(x, 1) + G(x, 1)2/L(x, 1), (x > 0) which is equivalent to the  first inequality of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Here, the second inequality of (10) is equivalent to the inequality:  1 − (x + 1) log x  2(x − 1)  + log  �(x + 1)2  4x  �  ≥ 0  (11)  Also the inequality L(x, 1)2 − G(x, 1)2  2L(x, 1)2  ≤ log K(x) is equivalent to the inequality:  1 − x(log x)2  (x − 1)2 + 2 log  �  4x  (x + 1)2  �  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (12)  The inequalities (11) and (12) will be used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Let x > 0 and p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (i) If 0 ≤ p ≤ 1  2, then ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (ii) If p ≤ 0 or p ≥ 1, then ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It is sufficient to prove for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Since K(x) ≥ 1, in order to prove (i),  ˆGp(x, 1) ≥ K(x)−p(1−p)Wp(x, 1) ⇐⇒ ˆGp(x, 1)K(x)p(1−p) ≥ Wp(x, 1)  ⇐⇒ x  p  2  �(x + 1)2  4x  �p(1−p)  ≥ (1 − p)(x − 1)  x1−p − 1  we set  fp(x) := p  2 log x + p(1 − p) {2 log(x + 1) − log(4x)} − log(1 − p) − log(x − 1) + log(x1−p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Then we have  f ′  p(x)  =  −  1  x − 1 + p  2x + p(1 − p)(x − 1)  x(x + 1)  + 1 − p  x − xp  =  gp(x)  2(x − 1)(x + 1)(x − xp)  7  where  gp(x) := 2(x + 1)(xp − 1 − p(x − 1)) + p(x − 1)(1 − xp−1) ((3 − 2p)x + (2p − 1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  When x ≥ 1 and 0 ≤ p ≤ 1/2, we have 1 ≥ xp−1, −xp−3 ≥ −xp−2 and (1−p)(1−2p)(p−2) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Also when x ≥ 1 and 0 ≤ p ≤ 1/2, we have xp−1 ≥ xp−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Thus we calculate  g′  p(x) = (p + 1)(2p2 − 3p + 2)xp − 2p(2p2 − 2p − 1)xp−1 + p(1 − p)(1 − 2p)xp−2  +2p(1 − 2p)x + 2(2p2 − 2p − 1)  g′′  p(x) = p  �  (p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2  +(1 − p)(1 − 2p)(p − 2)xp−3 + 2(1 − 2p)  �  ≥ p  �  (p + 1)(2p2 − 3p + 2)xp−1 + 2(1 − 2p)xp−1 + 2(1 − p)(2p2 − 2p − 1)xp−2  +(1 − p)(1 − 2p)(p − 2)xp−2�  = (2p3 − p2 − 5p + 4)(xp−1 − xp−2) = 2(1 − p) {(1 − p)(1 + p) + 1 − p/2} (xp−1 − xp−2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Thus we have g′  p(x) ≥ g′  p(1) = 0 so that gp(x) ≥ gp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Therefore we have f ′  p(x) ≥ 0 for  x ≥ 1 and taking an accout for lim  x→1  (1 − p)(x − 1)  x1−p − 1  = 1, we havefp(x) ≥ fp(1) = 0 which proves  (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  It is also sufficent to prove (ii) for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' For the special cases p = 0 or p = 1 we have  equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Since  ˆGp(x, 1) ≤ K(x)−p(1−p)Wp(x, 1) ⇐⇒ x  p  2  �(x + 1)2  4x  �p(1−p)  ≤ (1 − p)(x − 1)  x1−p − 1  ,  we have only to prove fp(x) ≤ 0 for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (a) We consider the case p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' We set g′′  p(x) = p · hp(x), namely  hp(x) := (p+1)(2p2−3p+2)xp−1+2(1−p)(2p2−2p−1)xp−2+(1−p)(1−2p)(p−2)xp−3+2(1−2p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Then  h′  p(x) = (p − 1)xp−4kp(x),  where  kp(x) := 2p3(x − 1)2 − p2(x − 1)(x − 11) − p(x2 + 6x − 17) + 2(x − 3)(x + 1)  and we have  k′  p(x) = 4p3(x−1)−2p2(x−6)−2p(x+3)+4(x−1), k′′  p(x) = 2(p+1)(2p2−3p+2) > 0, (p > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Thus we have k′  p(x) ≥ k′  p(1) = 10p2 − 8p > 0, (p > 1) so that kp(x) ≥ kp(1) = 10p − 8 >  0, (p > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Therefore we have h′  p(x) ≥ 0, (p > 1) which implies hp(x) ≥ hp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Thus  we have g′′  p(x) ≥ 0 so that we have g′  p(x) ≥ g′  p(1) = 0 which implies gp(x) ≥ gp(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Taking account of x − xp ≤ 0 when x ≥ 1, p > 1, we have f ′  p(x) ≤ 0 Therefore we have  fp(x) ≤ fp(1) = 0 which proves (ii) for the case p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (b) We consider the case p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' We calculate  dfp(x)  dp  =  1  1 − p +  �1  2 +  x  xp − x  �  log x + (2p − 1) log(4x) + (2 − 4p) log(x + 1),  d2fp(x)  dp2  = −xp+1(log x)2  (x − xp)2  +  1  (p − 1)2 + 2 log(4x) − 4 log(x + 1),  d3fp(x)  dp3  = xp+1 (xp + x) (log x)3  (xp − x)3  −  2  (p − 1)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  8  We further calculate  d  dx  �d3fp(x)  dp3  �  = −xp(log x)2  (x − xp)4 s(x, p),  s(x, p) := 3(x2 − x2p) + (p − 1)  �  x2 + x2p + 4x1+p�  log x,  ds(x, p)  dp  =  �  x2 − 5x2p + 4xp+1 + 2(p − 1)  �  x2p + 2xp+1�  log(x)  �  log x,  d2s(x, p)  dp2  = 4xp(log x)2 {2(x − xp) + (p − 1)(x + xp) log x} ≤ 0 (p ≤ 1, x ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Indeed, putting a := x, b := xp in the inequality  a − b  log a − log b ≤ a + b  2  , (a, b > 0), we have  2(x−xp)+(p−1)(x+xp) log x ≤ 0 for p < 1 and x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' (The equality holds when p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=')  From d2s(x, p)  dp2  ≤ 0, we have ds(x, p)  dp  ≥ ds(x, p)  dp  ����  p=1  = 0 which implies s(x, p) ≤ s(x, 1) =  0 so that we have d  dx  �d3fp(x)  dp3  �  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' From this, we have  d3fp(x)  dp3  ≥ d3fp(1)  dp3  = −  2  (p − 1)3 > 0, (p < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  By the inequality (12), we have  p ≤ 0 =⇒ d2fp(x)  dp2  ≤ d2fp(x)  dp2  ����  p=0  = 1 − x(log x)2  (x − 1)2 + 2 log  �  4x  (x + 1)2  �  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (13)  Thus we have by the inequality (11),  p ≤ 0 =⇒ dfp(x)  dp  ≥ dfp(x)  dp  ����  p=0  = 1 − (x + 1) log x  2(x − 1)  + log  �(x + 1)2  4x  �  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Therefore p ≤ 0 =⇒ fp(x) ≤ f0(x) = 0 which proves (ii) for the case p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (i) Since fp(x) = log  ��(x + 1)2  4x  �p(1−p)  × xp/2(x1−p − 1)  (1 − p)(x − 1)  �  appeared in the  proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3 and comparing the maximum degree of the numerator and the  denominator insides of the logarithmic function, we found that if p < 0 or p > 1/2, then  we have  lim  x→∞  ��(x + 1)2  4x  �p(1−p)  × xp/2(x1−p − 1)  (1 − p)(x − 1)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Thus we have lim  x→∞ fp(x) = −∞ if p < 0 or p > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' On the other hand, by the muner-  ical computations, we have f3/4(e) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='0063209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Therefore there is no ordering between  K(x)−p(1−p)Wp(x, 1) and ˆGp(x, 1) for x > 0 and 1/2 < p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (ii) From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1 (i) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3 (i), we have for x > 0 and 0 ≤ p ≤ 1/2,  K(x)−p(1−p)Wp(x, 1) ≤ ˆGp(x, 1) ≤ L(x, 1) ≤ ˆAp(x, 1) ≤ Wp(x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (iii) From the proof (b) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3, we obtained d3fp(x)  dp3  ≥ 0, (p < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' From this with  simple calculations, we have  H(x, xp)3 ≤ xp+1A(x, xp) ≤ L(x, xp)3, (p ≤ 1, x > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  9  4  Conclusion  As we have seen, we studied the inequalities on the relations between the Wigner–Yanase–  Dyson function Wp(·, ·) and the logarithmic mean L(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' As one of main results, we obtained  two kinds of the reverse inequalities for L(x, y) ≤ Wp(x, y) for x, y > 0 and 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' That  is, the inequalities (2) and (7) shown in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3 are respectively equivalent to the  following inequalities for x, y > 0 and 0 ≤ p ≤ 1:  Wp(x, y) ≤ p(1 − p)  �√x − √y  �2 + L(x, y),  (14)  Wp(x, y) ≤ K (x/y)p(1−p) L(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (15)  The inequality (14) and (15) are the difference type reverse inequality and the ratio type reverse  inequality for L(x, y) ≤ Wp(x, y), (0 ≤ p ≤ 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  In addition, we compared the obtained inequality (15) with the known result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  It is summerized in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' The inequalities given in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='4 (ii) are equivalent to  the following inequalities for x, y > 0 and 0 ≤ p ≤ 1/2:  K (x/y)−p(1−p) Wp(x, y) ≤ ˆGp(x, y) ≤ L(x, y) ≤ ˆAp(x, y) ≤ Wp(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  (16)  We conclude this paper by giving operator inequalities baesd on Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' For  positive operators S, T and 0 ≤ p ≤ 1, we define the operator version of the logarithmic mean  and the Wigner–Yanase–Dyson function as  L(S, T) :=  � 1  0  S♯tTdt,  Wp(S, T) := p(1 − p)  2  (S − T) (S∇T − Hzp(S, T))−1 (S − T), (S ̸= T),  Wp(S, S) := S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  where  S♯pT := S1/2 �  S−1/2TS−1/2�p  S1/2, S∇T := S + T  2  , Hzp(S, T) := 1  2 (S♯pT + S♯1−pT) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  We often use the symbol S♯T := S♯1/2T for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Hzp(S, T) is often called the Heinz mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' It  is notable that we have the relation for the validity in the case p := 1/2,  �S − T  2  �  (S∇T + S♯T)−1  �S − T  2  �  = S∇T − S♯T,  which can be confirmed by multiplying S−1/2 to both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Let S and T be positive operators and let 0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Then we have  Wp(S, T) ≤ 2p(1 − p) (S∇T − S♯T) + L(S, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='3, we also have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Let S and T be positive operators with αS ≤ T ≤ βS for 0 < α ≤ β and let  0 ≤ p ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Then we have  Wp(S, T) ≤ kp · L(S, T),  kp := max  α≤x≤β K(x)p(1−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  10  Acknowledgement  The author (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=') was partially supported by JSPS KAKENHI Grant Number 21K03341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Furuichi,  Unitarily  invariant  norm  inequalities  for  some  means,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Inequal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=',2014(2014), Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Furuichi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Moradi, Advances in mathematical inequalities, De Gruyter, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Furuichi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Amlashi, On bounds of logarithmic mean and mean inequality chain,  arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='01134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Furuichi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Yanagi, Schr¨odinger uncertainty relation, Wigner–Yanase–Dyson skew  information and metric adjusted correlation measure, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=',388(2)(2012),  1147–1156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Gibilisco, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Hansen and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Isola, On a correspondence between regular and non-regular  operator monotone functions, Linear Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=', 430 (2009) 2225–2232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Hansen, Metric adjusted skew information, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=',105(2008), 9909–9916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Hiai and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Kosaki, Means for matrices and comparison of their norms, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' 48 (1999) 899–936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Hiai and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Kosaki, Means of Hilbert space operators, Springer–Verlag, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Hiai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Kosaki,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Petz and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Ruskai Families of completely positive maps associated with  monotone metrics, Linear Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=', 48(439)(2013), 1749–1791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Kantorovich, Functional analysis and applied mathematics, Uspekhi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Nauk,  3:6(28) (1948), 89–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' http://mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='mathnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='ru/eng/umn/v3/i6/p89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Kosaki, Positive definiteness of functions with applications to operator norm inequalities,  Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=', 212(997), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Kosaki, Strong monotonicity for various means, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Func.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=',267(2014),1917–1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Petz and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Hasegawa, On the Riemannian metric of α–entropies of density matrices,  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=',38(1996), 221-225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [14] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Kosaki, Positive definiteness and infinite divisibility of certain functions of hyperbolic  cosine function, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=',33(7) (2022), 2250050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Specht,  Zur  Theorie  der  elementaren  Mittel,  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Z,  74  (1960),  91–98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='1007/BF01180475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  [16] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='Szab´o, A class of matrix monotone functions, Linear Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content=',420(2007), 79–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE0T4oBgHgl3EQfuAFm/content/2301.02599v1.pdf'}

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+Tower of quantum scars in a partially many-body localized system
+Michael Iversen and Anne E. B. Nielsen
+Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
+Isolated quantum many-body systems are often well-described by the eigenstate thermalization
+hypothesis. There are, however, mechanisms that cause different behavior: many-body localization
+and quantum many-body scars. Here, we show how one can find disordered Hamiltonians hosting
+a tower of scars by adapting a known method for finding parent Hamiltonians. Using this method,
+we construct a spin-1/2 model which is both partially localized and contains scars. We demonstrate
+that the model is partially localized by studying numerically the level spacing statistics and bipar-
+tite entanglement entropy. As disorder is introduced, the adjacent gap ratio transitions from the
+Gaussian orthogonal ensemble to the Poisson distribution and the entropy shifts from volume-law
+to area-law scaling. We investigate the properties of scars in a partially localized background and
+compare with a thermal background. At strong disorder, states initialized inside or outside the scar
+subspace display different dynamical behavior but have similar entanglement entropy. We demon-
+strate that localization stabilizes scar revivals of initial states with support both inside and outside
+the scar subspace. Finally, we show how strong disorder introduces additional towers of approximate
+scar states.
+I.
+INTRODUCTION
+The eigenstate thermalization hypothesis (ETH) de-
+scribes how isolated quantum systems reach thermal
+equilibrium [1–3]. The hypothesis is a statement about
+generic quantum many-body systems and has been veri-
+fied for a wide variety of physical models [3–13]. Despite
+the effectiveness of ETH, several phenomena are known
+to cause non-thermal behavior.
+One such mechanism is many-body localization (MBL)
+[14–17].
+MBL appears in many-body interacting sys-
+tems with local disorder.
+When the disorder strength
+is sufficiently strong, it causes a change in the structure
+of the energy eigenstates.
+An extensive set of quasi-
+local integrals of motion (LIOM) emerges and the en-
+ergy eigenstates localize [18, 19]. Consequently, all en-
+ergy eigenstates behave non-thermally and MBL repre-
+sents a strong violation of ETH. While this phenomenon
+is well-established for finite systems, the stability of MBL
+in the thermodynamic limit is still an open question [20].
+Another mechanism leading to non-thermal behav-
+ior was discovered in experiments with kinetically con-
+strained Rydberg atoms [21]. The atoms were arranged
+with strong nearest neighbor interactions so the simul-
+taneous excitation of neighboring atoms was prohibited.
+When initializing the system in the N´eel state, observ-
+ables displayed abnormal persistent oscillations – con-
+trary to the predictions by ETH. Subsequent theoreti-
+cal works uncovered that the revivals were caused by a
+small number of non-thermal eigenstates dubbed quan-
+tum many-body scars (QMBS) [22–25]. These scar states
+have approximately equal energy spacing so any initial
+state in the scar subspace displays revivals.
+The scar
+states are uncommon and represent a vanishingly small
+part of an otherwise thermalizing spectrum. Therefore,
+QMBS represent a weak violation of ETH.
+In this work, we realize both ETH-breaking mecha-
+nisms simultaneously. We study a one-dimensional dis-
+ordered spin-1/2 chain hosting a tower of QMBS. As the
+disorder strength is increased, the model transitions from
+the thermal phase to being partially localized while pre-
+serving the scar states.
+In earlier works, a single scar
+state was embedded in an otherwise MBL spectrum [26–
+28]. Our work adds to these studies by considering a full
+tower of QMBS in an MBL spectrum. The presence of
+multiple scar states, enables us to study the effect of lo-
+calization on the dynamical revivals characteristic of scar
+states. Using this model, we demonstrate how scar states
+can be distinguished from a localized background. We
+also find two phenomena originating from the interplay
+between QMBS and localization: disorder stabilization
+of scar revivals and disorder induced approximate scars.
+The paper is structured as follows. In Sec. II A, we
+summarize the model by Iadecola and Schecter which is
+the starting point of our analysis. In Sec. II B, we explain
+how we find Hamiltonians having a set of scar states with
+equal energy spacing. In Sec. II C, we use this method to
+determine all local 1- and 2-body Hamiltonians for the
+tower of scar states in the Iadecola and Schecter model.
+In Sec. III A, we show that a subset of these Hamiltoni-
+ans partially localize as disorder is introduced. We quan-
+tify the partial localization as a special structure in the
+energy eigenstates and compare with results from exact
+diagonalization. We verify the localization by studying
+the level spacing statistics in Sec. III B and the entan-
+glement entropy in Sec. III C. In Sec. IV, we show that
+the fidelity between initial states and the corresponding
+time evolved states can be utilized to distinguish the scar
+states from the partially localized background. We fur-
+ther show that the bipartite entanglement entropy is an
+ineffective tool for distinguishing scar states from a par-
+tially localized background. In Sec. V, we demonstrate
+how scar revivals are stabilized by strong disorder. In
+Sec. VI, we uncover additional towers of approximate scar
+states which emerge as disorder is introduced. Finally, we
+summarize our results in Sec. VII.
+arXiv:2301.01681v1  [cond-mat.dis-nn]  4 Jan 2023
+
+2
+II.
+MODEL
+A.
+Model by Iadecola and Schecter
+We take the model by Iadecola and Schecter as our
+starting point [29]. Consider a one-dimensional spin- 1
+2
+chain of even length L with periodic boundary conditions.
+The local Hilbert space on each site is described by the
+eigenkets |↑⟩ and |↓⟩ of the Pauli z-matrix, i.e. ˆσz |↑⟩ = |↑⟩
+and ˆσz |↓⟩ = − |↓⟩. The model by Iadecola and Schecter
+is given by
+ˆH0 =
+L
+�
+i=1
+�
+λ(ˆσx
+i − ˆσz
+i−1ˆσx
+i ˆσz
+i+1) + ∆ˆσz
+i + J ˆσz
+i ˆσz
+i+1
+�
+, (1)
+with λ, ∆, J ∈ R. All indices are understood as modulo
+L, i.e. the index i+L is identified as i. The operators ˆσx
+i ,
+ˆσy
+i and ˆσz
+i are the Pauli matrices acting on site i. The
+first term in Eq. (1) flips the spin si at site i if its nearest
+neighbors are in different states, i.e. si−1 ̸= si+1. The
+second term is a magnetic field along the z-direction with
+strength ∆. The third term represents nearest neighbor
+interactions with strength J.
+Two adjacent spins in different states represent a do-
+main wall, i.e. ↑↓ or ↓↑. The Hamiltonian conserves the
+number of domain walls Ndw because only spins with dif-
+ferent neighbors are allowed to change their state. Fur-
+thermore, the Hamiltonian is invariant under spatial in-
+version and translation, but these symmetries are broken
+when disorder is introduced in section III and we will not
+consider them any further.
+For nonzero values of λ, ∆ and J, the energy eigen-
+states are thermal except for a small number of ETH-
+violating scar states grouped into two towers. Through-
+out this work, we only focus on one of these towers. This
+tower contains L/2+1 eigenstates and the n-th state |Sn⟩
+is constructed by acting n times with the operator ˆQ† on
+the “all-spin-down” state
+|Sn⟩ ∝
+� ˆQ†�n |↓↓ . . . ↓⟩ .
+(2)
+The operator ˆQ† is given by
+ˆQ† =
+L
+�
+i=1
+(−1)i ˆP ↓
+i−1ˆσ+
+i ˆP ↓
+i+1,
+(3)
+where ˆσ+
+i = (ˆσx
+i +iˆσy
+i )/2 is the raising operator and ˆP ↓
+i =
+(ˆ1 − ˆσz
+i )/2 is the local projection onto spin down. The
+n-th scar state has energy En = 2(∆ − 2J)n + (J − ∆)L,
+number of domain walls Ndw = 2n and generally appears
+central in the spectrum after resolving all symmetries.
+Since the scar states are equally spaced in energy, any
+initial state in the scar subspace displays the dynami-
+cal revivals characteristic of QMBS. Furthermore, it was
+shown in Ref. [29] that the bipartite entanglement en-
+tropy of the scar states displays logarithmic scaling with
+system size.
+B.
+Determining Hamiltonians
+All eigenstates of ˆH0 located near the middle of the
+spectrum are thermal except the scar states. We wish
+to extend the model so the scar states are embedded
+in a MBL background instead of a thermal background.
+MBL is possible in systems with quench disorder, and it
+has been realized in numerous models by e.g. introduc-
+ing a disordered magnetic field [17], bond-disorder [30]
+or disordered nearest-neighbor interactions [31]. Unfor-
+tunately, disorder cannot be introduced naively to the
+Hamiltonian ˆH0. When promoting any parameter to be-
+ing site-dependent λ → λi, ∆ → ∆i or J → Ji, the
+scar states are no longer eigenstates. Therefore, disorder
+must be introduced through new terms. In this section,
+we uncover all local few-body Hamiltonians which share
+the scar states as eigenstates and maintain equal energy
+spacing. In the next section, we show that a subset of
+these Hamiltonians are partially localized.
+We search for local Hamiltonians following Refs. [32,
+33]. The set of 2L×2L Hermitian operators form a vector
+space. Most of these operators are long-ranged, contain
+many-body interactions and are difficult to realize in ex-
+periments. Therefore, we restrict ourselves to Hamiltoni-
+ans containing local 1- and 2-body Hermitian operators.
+This subspace is spanned by the operator basis
+B2 =
+�
+ˆσa
+i
+���a ∈ {x, y, z}, i ∈ ZL
+�
+∪
+�
+ˆσa
+i ˆσb
+i+1
+���a, b ∈ {x, y, z}, i ∈ ZL
+�
+,
+(4)
+where ZL = {1, 2, . . . , L} are the first L integers. This
+subspace is considerably smaller than the full operator
+vector space and has dimension |B2| = 12L where | · |
+denotes the number of elements in a set. Any local 1-
+or 2-body interacting Hamiltonian can be expressed as a
+linear combination of the basis elements
+ˆH =
+|B2|
+�
+i=1
+αiˆhi,
+ˆhi ∈ B2,
+(5)
+where αi ∈ R are free coefficients.
+To simplify no-
+tation, we collect the coefficients in a vector α
+=
+(α1, α2, . . . , α|B2|)T where T is the transpose.
+We search for the vector of parameters α so the re-
+sulting Hamiltonian has |Sn⟩ as eigenstates for n =
+0, 1, . . . , L/2. The scar state |Sn⟩ is an eigenstate of ˆH if
+and only if the energy variance of |Sn⟩ is exactly zero
+⟨Sn| ˆH2|Sn⟩ − ⟨Sn| ˆH|Sn⟩
+2 = 0.
+(6)
+Inserting Eq. (5), the expression becomes
+αT Cnα = 0,
+(7)
+where Cn is the quantum covariance matrix
+[Cn]ij = ⟨Sn|ˆhiˆhj|Sn⟩ − ⟨Sn|ˆhi|Sn⟩ ⟨Sn|ˆhj|Sn⟩ .
+(8)
+
+3
+Equation (7) is satisfied when the vector of coefficients
+lies in the null space of the quantum covariance matrix
+α ∈ Null(Cn), i.e. Cnα = 0. We ensure all scar states
+|Sn⟩ are simultaneously eigenstates of ˆH by demanding
+the vector of coefficients α lies in the null space of ev-
+ery covariance matrix α ∈ Null(C0) ∩ Null(C1) ∩ . . . ∩
+Null(CL/2). While this condition ensures all scar states
+are eigenstates of ˆH, they are not necessarily equally
+spaced in energy.
+Equal energy spacing is established
+by imposing another set of requirements
+⟨Sn+2| ˆH|Sn+2⟩ − ⟨Sn+1| ˆH|Sn+1⟩
+= ⟨Sn+1| ˆH|Sn+1⟩ − ⟨Sn| ˆH|Sn⟩ ,
+(9)
+for all n = 0, 1, . . . , L/2 − 2. Inserting Eq. (5), we find
+Gα = 0,
+(10)
+where we introduce the rectangular matrix of energy gap
+differences
+[G]ij = ⟨Si+2|ˆhj|Si+2⟩ − 2 ⟨Si+1|ˆhj|Si+1⟩ + ⟨Si|ˆhj|Si⟩ .
+(11)
+We observe that the scar states are equally spaced in en-
+ergy when the coefficient vector resides in the null space
+of the gap matrix. In summary, the scar states appear as
+eigenstates of the Hamiltonian with equal energy spacing
+when the vector of coefficients lies in the intersection
+α ∈
+L/2
+�
+n=0
+Null(Cn) ∩ Null(G).
+(12)
+It is straightforward to determine this subspace numeri-
+cally since the scar states are known analytically. Note
+however, that while the matrices Cn and G are com-
+plex, we only search for real vectors α ∈ R|B2| (for com-
+plex vectors α ∈ C|B2|, the linear combination in Eq.
+(5) is not necessarily Hermitian).
+We find real coeffi-
+cient vectors by stacking the real and imaginary parts of
+the matrices (Re(C0), Im(C0), . . . , Re(CL/2), Im(CL/2),
+Re(G), Im(G))T and determining the null space of the
+resulting rectangular matrix by e.g. singular value de-
+composition.
+The vectors αi produced by this numerical method are
+typically dense, i.e. have few nonzero entries. As a con-
+sequence, the corresponding operator �
+i αiˆhi is difficult
+to interpret. We overcome this difficulty by noting that
+if {αi|i = 1, 2, . . .} lies in the null space Eq. (12), then
+any linear combination of these vectors also lies in the
+null space. We apply a heuristic algorithm to determine
+sparse vectors in the subspace [34].
+C.
+Generalized models
+We apply the numerical method for system sizes L = 8,
+10, 12, 14 and for all sizes find L + 4 linearly indepen-
+dent vectors αi satisfying Eq. (12). The corresponding
+(i)
+ˆHz = �L
+i=1 ˆσz
+i
+(ii)
+ˆDi = ˆσz
+i + ˆσz
+i+1 + ˆσz
+i ˆσz
+i+1,
+for i ∈ ZL
+(iii) ˆHodd
+zz
+= �L/2
+i=1 ˆσz
+2i−1ˆσz
+2i
+(iv)
+ˆHalt
+xz = �L
+i=1(−1)i(ˆσx
+i ˆσz
+i+1 + ˆσz
+i ˆσx
+i+1)
+(v)
+ˆHalt
+yz = �L
+i=1(−1)i(ˆσy
+i ˆσz
+i+1 + ˆσz
+i ˆσy
+i+1)
+TABLE I. Local 1- and 2-body operators which have |Sn⟩
+for n = 0, 1, . . . , L/2 as energy eigenstates with equal energy
+spacing. The operators are determined by applying the nu-
+merical method presented in Sec. II B and Appendix A proves
+the statement rigorously.
+operators are summarized in Tab. I. The first operator
+ˆHz was already present in the initial model Eq. (1) and
+adds nothing new. The L operators ˆDi act locally on
+sites i and i+1 and represent good candidates for adding
+quench disorder into the model in Eq. (1). Indeed, in Sec.
+III, we demonstrate the system partially localizes when
+introducing sufficiently strong disorder via these opera-
+tors. The third operator ˆHodd
+zz
+represents an interaction
+between every odd site and its right neighbor with equal
+interaction strength. The fourth and fifth operators ˆHalt
+xz
+and ˆHalt
+yz flip spins with the sign of the term determined
+by the nearest neighbors.
+Using the numerical method, we rediscover the 1- and
+2-body terms of the model in Eq. (1) by starting from
+the scar states. As noted above, the operator ˆHz was
+already present in the original model. Furthermore, the
+third term in Eq. (1) is a linear combination of the oper-
+ators in Tab. I: �L
+i=1 ˆσz
+i ˆσz
+i+1 = �L
+i=1 ˆDi − 2 ˆHz. Hence,
+the operators in Tab. I only represent L + 2 non-trivial
+extensions to the initial model.
+The numerical method presented in Sec. II B finds all
+operators in the operator subspace span(B2) hosting the
+tower of scars for finite L (up to length L = 14 in our
+case). However, in principle, the scar states may not be
+eigenstates of these operators at larger L. Therefore, in
+Appendix A we prove analytically for all even L that the
+scar states remain eigenstates with equal energy spacing
+for all operators in Tab. I.
+The method from Sec. II B can be extended by in-
+cluding all 3-body terms to the basis B3
+=
+B2 ∪
+{ˆσa
+i ˆσb
+i+1ˆσc
+i+2
+���a, b, c ∈ {x, y, z}, i ∈ ZL}.
+This results
+in a myriad of new operators – including the first term
+from Eq. (1). Hence, with a large enough operator basis,
+the numerical method fully recovers the original model.
+Since long-ranged many-body interactions are less rele-
+vant experimentally, we will not explore this possibility
+any further.
+Finally, we remark that the effectiveness of this ap-
+proach is highly non-trivial. For an eigenstate of a generic
+local Hamiltonian, it is unlikely for another local Hamil-
+
+4
+tonian to exist that shares the same eigenstate [35]. Con-
+trary to this, we find a large subspace of local Hamiltoni-
+ans sharing a full tower of scar states. We attribute the
+effectiveness of our study to the analytical structure of
+the scar states, i.e. Eq. (2) and (3). Our methods are not
+expected to be valuable starting from generic eigenstates
+but may be equally effective in other scarred models with
+similar amount of structure.
+III.
+MANY-BODY LOCALIZATION
+In the last section, we determined a subspace of Hamil-
+tonians with the scar states |Sn⟩ as eigenstates equally
+spaced in energy. Now, we study a concrete Hamiltonian
+from this subspace
+ˆH = ˆH0 +
+L
+�
+i=1
+di ˆDi,
+(13)
+with di chosen randomly from the uniform probability
+distribution di ∈ [−W, W] where W > 0 is the disorder
+strength. The action of ˆDi is given by
+ˆDi |s1 . . . sisi+1 . . . sL⟩
+=
+�
+3 |s1 . . . sisi+1 . . . sL⟩ ,
+if si = si+1 = ↑
+− |s1 . . . sisi+1 . . . sL⟩ ,
+otherwise
+(14)
+The operator ˆDi is related to the projection operators
+through ˆDi = 4 ˆP ↑
+i ˆP ↑
+i+1 − ˆ1 with ˆP ↑
+i = (ˆ1 + ˆσz
+i )/2. We
+remark that Ref. [29] also observes that the operator
+ˆP ↑
+i ˆP ↑
+i+1 preserves the scar states.
+The model conserves the number of domain walls. The
+dimension of the symmetry sector containing Ndw do-
+main walls is given by the binomial coefficient 2(
+L
+Ndw ).
+We generally consider the largest symmetry sector with
+Ndw = 2⌊L/4⌋ domain walls where ⌊·⌋ is the function
+rounding down to the nearest integer.
+A.
+Partial many-body localization
+A physical system may transition to the MBL phase
+when disorder is introduced.
+MBL is usually real-
+ized with the disorder term in the Hamiltonian acting
+uniquely on each basis state. Consequently, a complete
+set of LIOMs emerge and all energy eigenstates are fully
+described by their eigenvalues of the LIOMs.
+The situation is slightly different in our model because
+the disorder term �
+i di ˆDi treats some basis states the
+same. The operator ˆDi is only sensitive to whether spins
+i and i+1 are both up (it acts identically on states where
+spins i and i+1 are ↓↓, ↓↑ or ↑↓). Therefore, the operator
+�
+i di ˆDi has the same action on product states with all
+consecutive spin-ups placed identically. We do not expect
+these to localize in the usual sense. Instead, we anticipate
+the spectrum to separate into fully MBL eigenstates and
+partially localized eigenstates.
+This structure is most easily described when the prod-
+uct states |s1s2 . . . sL⟩ are relabeled to reflect the ac-
+tion of �
+i di ˆDi.
+In this spirit, we define |Ndw, D, n⟩
+as a simultaneous eigenstate of the ˆDi’s with eigenvalues
+D = (D1, D2, . . . DL) where Di ∈ {−1, 3}. We will refer
+to D as the disorder indices. As discussed above, the
+state |s1s2 . . . sL⟩ is not fully described by D since mul-
+tiple states can have the same eigenvalues. Therefore, we
+further label the states by their number of domain walls
+Ndw and introduce a dummy index n = 1, 2, . . . , N (Ndw)
+D
+to distinguish states with identical Ndw and D. For in-
+stance, if two states |s1s2 . . . sL⟩ and |s′
+1s′
+2 . . . s′
+L⟩ have the
+same number of domain walls Ndw and disorder indices
+D, then they are relabeled as |Ndw, D, n⟩ for n = 1, 2.
+Note that some labelings are invalid. Consider the vector
+of eigenvalues D = (3, −1, 3, 3) for a small system L = 4.
+The “3”s imply all spins are up, while the “−1” entail at
+least one spin is down. In the following, we study a single
+symmetry sector and hence omit the Ndw index for clar-
+ity but reintroduce it in Secs. V and VI when studying
+multiple symmetry sectors at once.
+Upon introducing strong disorder, we expect LIOMs to
+emerge which are localized on the operators ˆDi and en-
+ergy eigenstates are characterized by their eigenvalues of
+the LIOMs. Therefore, we expect the energy eigenstates
+to be close to linear combinations of product states with
+the same disorder indices
+|ED,m⟩ ≈
+ND
+�
+n=1
+αmn |D, n⟩ .
+(15)
+with αmn ∈ R and m = 1, 2, . . . , ND. This expression
+is an approximation rather than an equality due to an
+exponentially small overlap with states |D′, n⟩ with dif-
+ferent disorder indices D′ ̸= D. The special case ND = 1
+corresponds to the disorder term acting uniquely on the
+basis state |D, 1⟩. We expect the corresponding energy
+eigenstate |ED,1⟩ ≈ |D, 1⟩ to be MBL. For ND > 1,
+the states {|ED,m⟩ |m = 1, 2, . . . , ND} are only partially
+MBL since the LIOMs do not fully describe each state
+and all additional structure is captured by the extra in-
+dex m.
+The above considerations are verified in numerical sim-
+ulations by considering a system of size L = 8 at strong
+disorder W = 10. Figure 1 illustrates the norm squared
+overlap of all energy eigenstates |ED,m⟩ with the prod-
+uct states |D, n⟩. The (i, j)-th pixel displays the norm
+squared overlap between the i-th product state and j-
+th energy eigenstate. The product states on the second
+axis are sorted according to ND. The energy eigenstates
+are reordered to allow the diagonal shape in Fig. 1. In
+the upper left corner of Fig. 1, each eigenstate has high
+overlap with a single product state. Numerical analysis
+reveals that these product states exactly coincide with
+those being fully described by their disorder indices, i.e.
+ND = 1. These results support the claim that such eigen-
+
+5
+|ED,m⟩
+|D, n⟩
+1
+2
+3
+4
+20
+ND
+(a)
+(b)
+(c)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+|⟨D, n|ED,m⟩|2
+FIG. 1.
+The norm squared overlap of the energy eigenstates
+with the product states | ⟨D, n|ED,m⟩ |2 for system size L = 8,
+disorder strength W = 10 and parameters λ = ∆ = J = 1.
+The color of pixel (i, j) displays the overlap between the i’th
+product state and the j’th eigenstate. The product states are
+sorted into ascending order according to ND. The second axis
+on the right hand side groups the product states according to
+ND. The insets show eigenstates with significant weight on
+(a) two, (b) three and (c) four product states.
+The figure
+verifies that all energy eigenstates are approximately linear
+combinations of product states with the same disorder indices.
+states fully localize. The next eigenstates shown in Fig.
+1(a) each has significant overlap with exactly two product
+states of the same disorder indices. The pattern contin-
+ues: we find eigenstates that are linear combinations of
+Fig. 1(b) three, Fig. 1(c) four, and (bottom right corner)
+twenty product states. In each case, the product states
+have the same disorder indices and hence correspond to
+{|D, n⟩ |n = 1, 2, . . . , ND} for ND = 3, 4, 20. These ob-
+servations are not restricted to L = 8, but seem universal
+at all system sizes. For larger system sizes, the number
+and sizes of the blocks increase. Finally, we note that
+the scar state within the considered symmetry sector is
+located in the block ND = 20 in Fig. 1. The scar state
+is generally an equal weight linear combination of prod-
+uct states with the maximum ND. This fact will play an
+important role when we explore the system dynamics in
+Sec. V.
+Next, we discuss how the eigenstates are distributed
+in energy. The magnetization MD = �
+i σz
+i of a prod-
+uct state |D, n⟩ is fixed by the symmetry sector Ndw
+E
+Thermal
+|Sn⟩
+Partial MBL
+|ED1,m1⟩
+|ED2,1⟩
+|ED2,2⟩
+|ED2,3⟩
+|ED3,m3⟩
+|ED4,m4⟩
+|ED5,m5⟩
+FIG. 2.
+Sketch of the spectrum in the thermal phase (left)
+and in the partially localized phase (right). In the thermal
+phase, the energy levels follow the Wigner-Dyson surmise.
+As disorder is introduced, the spectrum experiences partial
+localization. Eigenstates with similar indices D are near de-
+generate and the spectrum forms clusters of such eigenstates.
+The scar state lies in the largest of these clusters.
+and disorder indices D. Likewise, the number N (↑↑,↓↓)
+D
+of adjacent spins pointing in the same direction (↑↑ or
+↓↓) and the number N (↑↓,↓↑)
+D
+of adjacent spins point-
+ing in opposite directions (↑↓ or ↓↑) are also fully de-
+termined.
+Therefore, the terms ∆ �
+i ˆσz
+i , J �
+i ˆσz
+i ˆσz
+i+1
+and �
+i di ˆDi have the same action on all product states
+with the same number of domain walls and disorder in-
+dices: {|D, n⟩ |n = 1, 2, . . . , ND}.
+At strong disorder,
+the energy of an eigenstate is approximately ED,m ≈
+∆MD + J(N (↑↑,↓↓)
+D
+− N (↑↓,↓↑)
+D
+) + �
+i diDi with a small
+correction that depends on the value of the m index. The
+slight additional contribution originates from the term
+�
+i λ(ˆσx
+i − ˆσz
+i−1ˆσx
+i ˆσz
+i+1) and scales with λ. Consequently,
+at large disorder, the set of eigenstates {|ED,m⟩ |m =
+1, 2, . . . , ND} are near degenerate and form clusters. A
+scar state resides in the largest of these clusters in all
+symmetry sectors. Figure 2 illustrates the spectral struc-
+ture. Note that Fig. 2 is highly idealized to highlight the
+structure described above. In practice, it is highly likely
+for two or more clusters to overlap making the structure
+less apparent.
+B.
+Spectral statistics
+The distribution of energy gaps distinguishes the ther-
+mal and MBL phases.
+Let Ei be the energies of the
+Hamiltonian in ascending order and δi = Ei+1 − Ei ≥ 0
+the i-th energy gap. In the thermal phase, the number of
+energy levels in an interval [E, E+∆E] is known to follow
+the Wigner-surmise [36, 37]. In particular, it follows the
+Gaussian orthogonal ensemble (GOE) since the model in
+Eq. (13) is time-reversal invariant. On the other hand,
+the number of energy levels in an interval follows the Pois-
+son distribution in the MBL phase. Since our model only
+partially localizes, we review how the Poisson distribu-
+tion accurately describes the MBL phase and investigate
+
+T6
+the validity of these arguments in our model. Consider
+two adjacent eigenstates with energies Ei and Ei+1. At
+large disorder, the energy of these states are dominated
+by the disorder term �
+i di ˆDi. If the states have differ-
+ent disorder indices |ED,m⟩ and |ED′,m′⟩, then their en-
+ergies originate from different linear combinations of the
+random numbers di: �
+i diDi ≈ �
+i diD′
+i with Di ̸= D′
+i
+for some i’s. Consequently, the eigenstates “arrive” at
+this energy independently of each other and hence fol-
+low the Poisson distribution.
+These arguments are no
+longer valid when two adjacent eigenstates have the same
+disorder indices and different m indices.
+In this case,
+we expect the level spacing distribution to follow GOE.
+Thus, the distribution of energy levels still identifies the
+transition to partial localization if we only consider level
+spacings between eigenstates of different disorder indices.
+Instead of working directly with the level spacing dis-
+tribution, it is convenient to analyze the adjacent gap
+ratio since it removes the need for unfolding the spec-
+trum [37, 38]. The adjacent gap ratio is defined by [16]
+ri = min(δi, δi+1)
+max(δi, δi+1).
+(16)
+This quantity is bounded by the interval ri ∈ [0, 1] and
+follows the distributions below in the thermal and MBL
+phases respectively [39]
+PGOE(r) = 27
+4
+r(1 + r)
+(1 + r + r2)5/2 ,
+(17a)
+PPoisson(r) =
+2
+(1 + r)2 .
+(17b)
+The mean values of the distributions in Eq. (17) are given
+by ⟨r⟩GOE = 2(2 −
+√
+3) ≈ 0.536 and ⟨r⟩Poisson = 2 ln 2 −
+1 ≈ 0.386.
+Figure 3(a) illustrates the mean adjacent gap ratio as
+a function of disorder strength for different system sizes.
+We average the adjacent gap ratio over 2 × 103 disor-
+der realizations for L = 8, 103 disorder realizations for
+L = 10, 12, 14 and 500 disorder realizations for L = 16.
+For each disorder realization, we average over all energies
+in the interval Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the
+q-th quantile of the energy distribution for the current
+disorder realization. For system size L = 16, we average
+over the 103 energies closest to (Emin + Emax)/2 where
+Emin and Emax are the smallest and largest energies in
+the spectrum. The errorbars indicate two standard devi-
+ations of the average when assuming a Gaussian distri-
+bution. As discussed above, the distribution of adjacent
+gap ratios only converges to Eq. (17b) if the analysis is
+restricted to adjacent energy levels with different disor-
+der indices. In practice, however, it is unlikely for two
+neighboring eigenstates to have the same disorder indices.
+Furthermore, the likelihood of neighboring eigenstates
+having the same disorder indices decreases rapidly with
+system size. With this in mind, we study the mean adja-
+cent gap ratio using all eigenstates in the central third of
+the spectrum. We verify the considerations above by also
+computing the mean adjacent gap ratio using only adja-
+cent eigenstates with different disorder indices at large
+disorder. For each energy gap δi = Ei+1 −Ei, we inspect
+the eigenstates |ED,m⟩ and |ED′,m′⟩ corresponding to the
+energies Ei and Ei+1. At large disorder, the disorder in-
+dices D are accurately determined by computing which
+D yields �ND
+m=1 | ⟨D, m|ED,m⟩ |2 ≈ 1. The mean of the
+adjacent gap ratio is then restricted to energy gaps with
+D ̸= D′. For small system sizes, there is a large differ-
+ence between the two methods, but the difference is seen
+to be small for large systems.
+The mean adjacent gap ratio agrees well with the GOE
+value at weak disorder 0 <∼ W <∼ 1.
+As the disorder
+strength is increased, the mean adjacent gap ratio de-
+creases and ultimately approaches the Poisson value at
+5 <∼ W. The agreement of data with the GOE and Pois-
+son values improves with increasing system size and the
+transition between the thermal and localized phase be-
+comes steeper for larger systems.
+Figures 3(b)-(d) illustrate the adjacent gap ratio dis-
+tribution at (b) weak disorder W = 0.46, (c) intermedi-
+ate disorder strength W = 2.27 and (d) strong disorder
+W = 6. The figures display the distributions in Eq. (17)
+for comparison. As expected, the data agrees with Eq.
+(17a) at weak disorder and (17b) at strong disorder. Fig-
+ure 3 indicates the system transitions from the thermal
+phase to being partially localized as disorder is intro-
+duced.
+C.
+Bipartite entanglement entropy
+In this section, we further verify the transition from
+the thermal phase to partial localization by studying the
+bipartite entanglement entropy. We separate the system
+into a left part L containing the first L/2 sites and a
+right part R containing the remaining sites. The reduced
+density matrix of the left part is obtained by tracing out
+the right part
+ρL = TrR(ρ)
+(18)
+where ρ is the density matrix of the full system and
+TrR(·) is the partial trace over R.
+The entanglement
+entropy between the left and right halves is given by,
+S = − TrL
+�
+ρL ln(ρL)
+�
+.
+(19)
+In the thermal phase, we expect eigenstates near the
+center of the spectrum to display volume-law scaling with
+system size. Specifically, the entropy is approximately
+described by the Page value SPage = [L ln(2) − 1]/2 [40].
+On the other hand, the entanglement entropy displays
+area-law scaling for MBL eigenstates [41]. While some
+eigenstates in our model are fully MBL, others are only
+partially localized. Hence, the precise scaling behavior
+of the entanglement entropy is not clear. Nonetheless,
+we expect the entropy of partially localized eigenstates
+to grow slower with system size than thermal eigenstates
+
+7
+P(r)
+(b)
+Poisson
+GOE
+P(r)
+(c)
+0.0
+0.5
+1.0
+r
+P(r)
+(d)
+0
+1
+2
+3
+4
+5
+6
+W
+0.40
+0.45
+0.50
+⟨r⟩
+(a)
+GOE
+Poisson
+L = 8
+10
+12
+14
+16
+FIG. 3.
+(a) Mean adjacent gap ratio ⟨r⟩ (solid line) as a function of disorder strength W for different system sizes L with
+parameters λ = ∆ = J = 1. The shaded areas display two standard deviations on the estimate of ⟨r⟩ when assuming a Gaussian
+distribution of data. For L = 8, the adjacent gap ratio is averaged over 2 × 103 disorder realizations, for L = 10, 12, 14 we
+use 103 disorder realizations and for L = 16 we use 500 disorder realizations. For system sizes L = 8, 10, 12, 14, we average
+over all energies Ei ∈ [E(q=1/3), E(q=2/3)] where E(q) is the q-th quantile. For system size L = 16, we average over the 103
+energies closest to (Emin + Emax)/2 where Emin and Emax are the smallest and largest energies in the spectrum.
+At low
+disorder 0 <∼ W <∼ 1, the system is thermal and ⟨r⟩ coincides with the Gaussian orthogonal ensemble ⟨r⟩GOE ≈ 0.536 (upper
+dashed line). At strong disorder 5 <∼ W, the mean adjacent gap ratio agrees with the Poisson distribution ⟨r⟩Poisson ≈ 0.386
+(lower dotted line). The agreement between data and the GOE and Poisson values improves with system size. Additionally,
+the transition from the thermal phase to partial localization happens more rapidly as a function of disorder strength for larger
+system sizes. The figure also illustrates the mean adjacent gap ratio when only averaging over neighboring energy eigenstates
+with different disorder indices (dots). The errorbars show two standard deviations on the estimate of the mean. This average
+coincides with the naive calculation at large system sizes. The figure also shows the adjacent gap ratio distribution for L = 16
+at (b) weak disorder W = 0.46, (c) intermediate disorder strength W = 2.27 and (d) strong disorder W = 6. These plots
+include the distributions Eq. (17a) (dashed curve) and Eq. (17b) (dotted curve). The data agrees with Eq. (17a) at weak
+disorder and transitions to the distribution (17b) at strong disorder.
+and we use the entropy to identify the onset of partial
+localization.
+Figure 4(a) shows the entropy of the eigenstate with
+energy closest to (Emin + Emax)/2 as a function of disor-
+der strength W for different system sizes L. Each data
+point represents the average entropy over 103 disorder
+realizations with errorbars displaying two standard devi-
+ations of the mean when assuming a Gaussian distribu-
+tion. For low disorder, the entanglement entropy scales
+linearly with the system size and hence agrees with the
+expected volume-law scaling in the thermal phase. Ad-
+ditionally, the entropy approaches the Page value with
+increasing system size.
+At large disorder, the entropy
+seems to be roughly independent of system size. Thus,
+the scaling of entropy is consistent with area-law for par-
+tially localized eigenstates.
+The sudden shift in scaling behavior of the entropy
+verifies the transition from the thermal phase to partial
+localization at strong disorder. The transition point is
+identified by analyzing the variance of entanglement en-
+tropy. Figure 4(b) illustrates the sample variance of the
+entropy over 103 disorder realizations. The variance dis-
+plays a peak when the system transitions from volume-
+law to area-law scaling.
+IV.
+DISTINGUISHABLE FEATURES OF SCAR
+STATES IN A PARTIALLY LOCALIZED
+BACKGROUND
+Scar states are commonly distinguished from a thermal
+background by their low entanglement and oscillatory dy-
+namics. In this section, we show that oscillatory dynam-
+ics can also be utilized to distinguish scar states from
+a partially localized background, while entanglement en-
+tropy turns out not to be an effective tool to identify the
+scar states.
+A.
+Entanglement entropy
+The entanglement entropy of the scar states scales log-
+arithmically with system size [29], while thermal states
+display volume-law scaling. Therefore, the entanglement
+
+8
+0
+1
+2
+3
+4
+5
+⟨S⟩
+(a)
+L = 8
+10
+12
+14
+16
+0
+2
+4
+6
+W
+0
+1
+Var(S)
+(b)
+FIG. 4.
+(a) Average bipartite entanglement entropy of the
+eigenstate closest to the center of the spectrum ⟨S⟩ as a func-
+tion of disorder strength W for different system sizes L. The
+entropy is averaged over 103 disorder realizations with system
+parameters λ = ∆ = J = 1.
+Errorbars display two stan-
+dard deviations on the estimate of average entropy assuming
+a Gaussian distribution. At low disorder, the entropy displays
+volume-law scaling with system size and approaches the Page
+value (dashed lines) as expected in the thermal phase.
+At
+large disorder, the entropy follows area-law scaling with sys-
+tem size. (b) Variance of bipartite entanglement entropy of
+the eigenstate closest to the center of the spectrum.
+The
+variance is computed from 103 disorder realizations. As the
+disorder strength is increased, the variance displays a sudden
+peak. This indicates a transition from the thermal phase to
+partial localization. The peak becomes higher at larger sys-
+tem sizes.
+entropy provides a way to identify the scar states in a
+thermal background. Figure 5(a) illustrates the entropy
+as a function of energy of a thermal system with size
+L = 14 and disorder strength W = 0.5. The thermal
+states form a narrow arc with maximum in the middle
+of the spectrum while the scar state appears as an out-
+lier at much lower entropy. The situation is different in
+a partially localized background. Figure 5(b) illustrates
+the entropy as a function of energy at strong disorder
+W = 6. As discussed above, partially localized eigen-
+states are weakly entangled making it difficult to identify
+the scar state. We conclude that entanglement entropy
+is an ineffective tool for distinguishing scar states from a
+partially localized background.
+0.0
+0.5
+1.0
+ϵ
+0
+2
+4
+S
+(a)
+0.0
+0.5
+1.0
+ϵ
+(b)
+FIG. 5.
+The entanglement entropy S as a function of nor-
+malized energy ϵ = (E − Emin)/(Emax − Emin) where Emin
+and Emax are the smallest and largest energies in the spec-
+trum.
+Lighter (darker) colors indicate lower (higher) den-
+sity of points.
+(a) We consider a thermal system of size
+L = 14, disorder strength W = 0.5 and system parameters
+λ = ∆ = J = 1. In the thermal phase, the energy eigenstates
+form a narrow band with maximum at the center of the spec-
+trum. The scar state (inside the green ring) is easily identified
+since it appears isolated below the curve. (b) We consider a
+partially localized system at strong disorder W = 6. The en-
+ergy eigenstates are spread out at low entropy with the scar
+state embedded among them. The entanglement entropy is
+hence not an effective tool to distinguish the scar state from
+a partially localized background.
+B.
+Fidelity
+States initialized in the scar subspace distinguish them-
+selves from a thermal background by displaying persis-
+tent dynamic revivals. We now show that this behavior
+also enables the identification of scar states from a par-
+tially localized background. We quantify the dynamics of
+quantum systems by the fidelity F(t). Let |ψ(0)⟩ be the
+initial state and |ψ(t)⟩ = e−i ˆ
+Ht |ψ(0)⟩ the time evolved
+state. The fidelity is given by
+F(t) = | ⟨ψ(0)|ψ(t)⟩ |2.
+(20)
+The time evolution of fidelity is most clearly understood
+by considering the overlap of the initial state with all
+energy eigenstates. Let |φi⟩ be the i-th energy eigenstate
+with corresponding energy Ei and let ci be the inner
+product between the i-th energy eigenstate and the initial
+state ci = ⟨φi|ψ(0)⟩. The relation between fidelity and
+the expansion coefficients ci is highlighted by rewriting
+the fidelity according to
+F(t) =
+�
+i
+|ci|4 +
+�
+i̸=j
+|ci|2|cj|2ei(Ei−Ej)t
+(21)
+It is clear from this expression that the dynamics of fi-
+delity is sensitive to the distribution of |ci|2. We generally
+display this distribution along with the fidelity for clarity.
+We demonstrate the different dynamical behavior of
+the thermal and partial MBL phases by initializing a sys-
+tem of size L = 14 in a product state. First, we consider
+
+9
+0.0
+0.5
+1.0
+F
+(a)
+⟨F T⟩
+⟨F MBL⟩
+Fscar
+0.0
+0.5
+1.0
+F
+(b)
+0
+1
+2
+3
+4
+5
+t/Tscar
+0.0
+0.5
+1.0
+F
+(c)
+−25
+0
+25
+Ei
+0.0
+0.1
+|ci|2
+(d)
+−50
+0
+50
+Ei
+0.0
+0.5
+(e)
+−20
+0
+Ei
+0.0
+0.1
+(f)
+FIG. 6.
+(a) The average fidelity of a random product state in
+a thermal system at disorder strength W = 0.5. (b) The aver-
+age fidelity in a partially localized system at disorder strength
+W = 10. The system is initialized in a product state which
+fully localizes (solid line). For comparison, the system is ini-
+tialized in a random product state which only partially local-
+izes |ψ(0))⟩ = |D, n⟩ with ND = 5 (dashed line), 10 (dashed
+dotted line) and 35 (dotted line). (c) The system is initialized
+in the scar subspace at any disorder strength. The average
+fidelity is in all cases calculated over 103 disorder realizations.
+The bottom panel displays the distribution of expansion coef-
+ficients |ci|2 across energy in a single disorder realization. (d)
+For the thermal phase W = 0.5. (e) For partial MBL W = 10
+with initial state |ψ(0)⟩ = |D, n⟩ for ND = 5. (f) For the
+initial state being an equal weight linear combination of the
+scar states.
+a thermal system at disorder strength W = 0.5.
+The
+initial state is chosen as a random product state with
+all product states having the same probability of being
+drawn.
+We ensure the initial state resides outside the
+scar subspace by drawing a new product state if the first
+has non-zero overlap with a scar state. We consider 103
+disorder realizations and draw a random product state
+in each realization. In the i-th realization, the fidelity is
+computed as a function of time Fi(t) and Fig. 6(a) shows
+the average fidelity ⟨F(t)⟩ = 10−3 �103
+i=1 Fi(t) over all re-
+alizations. Figure 6(d) shows the expansion coefficients
+|ci|2 of a single disorder realization following the Gaus-
+sian distribution as expected [42, 43]. Since the initial
+state has large overlap with many different eigenstates,
+the second sum in Eq. (21) rapidly vanishes due to can-
+cellation between terms with different phase factors. As
+a consequence, the fidelity quickly decreases and satu-
+rates at Fi(t) ≈ �
+i |ci|4 ≈ 0 at long times Tscar ≪ t for
+all disorder realizations. These considerations agree with
+the observed time evolution of the average fidelity in Fig.
+6(a) which rapidly decreases to a value near zero.
+Next, we consider the same setup when the system is
+partially localized at large disorder W = 10. As discussed
+in Sec. III A, the spectrum separates into fully MBL
+eigenstates and partially localized eigenstates.
+Conse-
+quently, the dynamics depend greatly on the initial state.
+The solid blue line in Fig. 6(b) is the average fidelity
+over 103 disorder realizations when initialing the sys-
+tem in a random product state which fully localizes, i.e.
+|ψ(0)⟩ = |D, n⟩ with ND = 1. Fully MBL eigenstates
+have significant overlap with only one product state, and
+the average fidelity remains far from zero at all times as
+observed in Fig. 6(b).
+We note that a stronger disor-
+der strength is needed to achieve MBL in larger systems.
+Therefore, the average fidelity saturates significantly be-
+low unity in Fig. 6(b) even though all product states with
+ND = 1 in Fig. 1 are near identical to an energy eigen-
+state. The average fidelity saturates closer to unity at
+larger disorder strengths.
+When the initial state is chosen as a product state that
+only partially localizes, it has significant overlap with
+multiple eigenstates. Consequently, the average fidelity
+drops closer to zero as illustrated by the dashed and dot-
+ted curves in Fig. 6(b). For these curves, we choose the
+initial state randomly as |ψ(0)⟩ = |D, n⟩ with ND = 5, 10
+and 35. These initial states have significant support on
+up to ND eigenstates causing the average fidelity to de-
+crease with increasing ND.
+Figure 6(e) illustrates the
+distribution of |ci|2 for a single disorder realization for a
+random initial state |ψ(0)⟩ = |D, n⟩ with ND = 5. The
+distribution is more sparse than the thermal case.
+Finally, we consider the initial state being a linear com-
+bination of scar states
+|ψscar⟩ =
+1
+�
+L
+2 + 1
+L/2
+�
+n=0
+|Sn⟩ .
+(22)
+When the initial state is chosen within the scar subspace,
+the equal energy spacing causes the fidelity to display
+persistent periodic revivals. In particular, for the equal
+weight linear combination in Eq. (22), the fidelity is given
+by
+Fscar(t) =
+1
+L
+2 + 1
+�
+1 + 2
+L/2
+�
+n=1
+�
+1 −
+n
+L
+2 + 1
+�
+cos(n∆Et)
+�
+.
+(23)
+Revivals occur at times tℓ = Tscarℓ =
+2πℓ
+∆Escar where ℓ ∈ N
+and ∆Escar is the energy spacing between consecutive
+scar states.
+Figure 6(c) illustrates the fidelity of this
+initial state and Fig. 6(f) shows the distribution of the
+expansion coefficients.
+
+10
+In the thermal phase, states initialized respectively in-
+side and outside the scar subspace behave differently.
+The fidelity of states outside the scar subspace quickly
+drops to zero, while any linear combination of scar states
+display persistent revivals. In our analysis, we specifi-
+cally initialized the system as a product state, but the
+same conclusions hold for generic linear combinations of
+product states. In a partially localized background, the
+average fidelity distinguishes between states with sup-
+port inside and outside the scar subspace. The average
+fidelity of partially localized states saturates while scar
+states display revivals. Again, our analysis concerns the
+special case of initializing the system as a random prod-
+uct state. If instead the initial state is a generic linear
+combination of a large number of product states, the sec-
+ond term of Eq. (21) will generally vanish due to phase
+cancellation, and the average fidelity saturates near zero.
+While this is true for generic linear combinations, there
+exists particular states where the phase cancellation hap-
+pens exceptionally slowly. We discuss these special initial
+states in section VI and how to distinguish them from the
+scar states. Summing up, the average fidelity represents
+an effective tool for identifying scar states in both a ther-
+mal and localized background.
+Finally, we remark that the fidelity of individual dis-
+order realizations are enough to distinguish initial states
+with support inside and outside the scar subspace. This
+statement is simple in the thermal phase where initial
+states outside the scar subspace rapidly converges to zero.
+At large disorder, the fidelity of individual disorder re-
+alizations may oscillate rapidly contrary to the average
+fidelity. However, these oscillations are generally com-
+posed of frequencies different from the scar revivals. The
+amplitude of the oscillations are also typically different
+from the scar revivals. Thus, the scar states can be dis-
+tinguished from a partially localized background.
+V.
+DISORDER STABILIZATION OF SCAR
+REVIVALS
+We study the dynamics of initial states with support
+both inside and outside the scar subspace across all sym-
+metry sectors.
+In this case, we generally expect the
+scar revivals to diminish.
+The scar revivals are stabi-
+lized when the initial state only has support on product
+states with the same disorder indices as the scar states
+D0 = (−1, −1, . . . , −1). We demonstrate this behavior
+by initializing the system in a generic state only having
+support on product states with disorder indices D0
+|ψstable⟩ =
+1
+Nstable
+�
+|ψscar⟩ +
+�
+Ndw,n
+β(Ndw)
+n
+|Ndw, D0, n⟩
+�
+,
+(24)
+where Nstable is a normalization constant and β(Ndw)
+n
+are
+drawn
+randomly
+from
+the
+interval
+β(Ndw)
+n
+∈
+[0, 1.5/
+�
+N (Ndw)
+D0
+]. We reintroduce the index Ndw to de-
+scribe product states with the same disorder indices in
+different symmetry sectors. The time evolution of fidelity
+is investigated at weak and strong disorder in 103 real-
+izations. The coefficients β(Ndw)
+n
+are redrawn in each dis-
+order realization. Figure 7(a) displays the disorder aver-
+aged fidelity for a thermal system and a partially local-
+ized system. In both cases, the average fidelity displays
+persistent revivals with the revival amplitude decaying
+and eventually saturating at a value around 0.5.
+The fidelity amplitude quickly decays for a thermal
+system. The explanation can be found by studying the
+expansion coefficients |ci|2 as illustrated in Fig. 7(b). Be-
+cause the system is thermal, the initial state has support
+on many energy eigenstates. Consequently, terms with
+different phases quickly cancel causing the fidelity ampli-
+tude to saturate almost immediately.
+At large disorder, the fidelity amplitude decays at a
+much slower rate and only saturates alongside the ther-
+mal graph after many revivals t ∼ 7Tscar.
+We under-
+stand this behavior by recalling the spectral structure at
+large disorder. First, recall that the energy eigenstates
+{|ED0,m⟩ |m = 1, 2, . . . , ND0} are near degenerate and
+only have significant overlap with product states of the
+same disorder indices as described in Eq. (15). There-
+fore, the second term in Eq. (24) can be rewritten as a
+sum of near degenerate eigenstates,
+ND0
+�
+n=1
+β(Ndw)
+n
+|Ndw, D0, n⟩ ≈
+ND0
+�
+m=1
+γ(Ndw)
+m
+|ENdw,D0,m⟩ ,
+(25)
+with γ(Ndw)
+m
+= �
+n β(Ndw)
+n
+⟨ENdw,D0,m|Ndw, D0, n⟩. Fur-
+thermore, the scar states themselves are described by
+the disorder indices D0, so the eigenstates |ENdw,D0,m⟩
+are close in energy to a scar state.
+Consequently, the
+eigenstates outside the scar subspace having large over-
+lap with |ψstab⟩ are always close in energy to a scar state.
+We sketch this structure in Fig. 8 where the eigenstates
+|ENdw,D0,m⟩ have similar energy to the scar states for
+all Ndw. These considerations agree with the observed
+distribution of |ci|2 for a single disorder realization illus-
+trated in Fig. 7(c). The expansion coefficients are sharply
+peaked around the scar states and consequently the can-
+cellation of terms with different phases takes place at
+much larger times.
+In this way, the partially localized background stabi-
+lizes the scar revivals by rearranging the support outside
+the scar subspace. The stabilization takes place whenever
+the initial state is predominantly a linear combination of
+product states with the same disorder indices as the scar
+states D0. If product states with other disorder indices
+D′ ̸= D0 are included, the stabilization will be less pro-
+nounced.
+
+11
+10−4
+10−2
+100
+|ci|2
+(b)
+−50
+0
+50
+E
+10−4
+10−2
+100
+|ci|2
+(c)
+0
+2
+4
+6
+8
+t/Tscar
+0.00
+0.25
+0.50
+0.75
+1.00
+F
+(a)
+W = 10
+W = 0.5
+FIG. 7.
+A system of size L = 14 with parameters ∆ = 1, J = 5, λ = 1 is initialized according to Eq. (24) in the thermal phase
+at disorder strength W = 0.5 and the partial MBL phase at disorder strength W = 10. (a) The average fidelity over 103 disorder
+realizations when the system is thermal and partially MBL. The disorder protects the scar revivals and the fidelity amplitude
+decays much slower compared to the thermal case. The right panels illustrate the distribution of expansion coefficients |ci|2
+over energy Ei for a single disorder realization at disorder strength (b) W = 0.5 and (c) W = 10. The distribution of the
+expansion coefficients is wide in the thermal phase and consists of narrow peaks near the scar states in the localized phase.
+E
+Symmetry sectors
+∆Escar
+∆Escar
+∆Escar
+Ndw0
+Ndw1
+Ndw2
+Ndw3
+FIG. 8.
+At large disorder, the initial state Eq. (24) has
+significant overlap with a small number of energy eigenstates
+(black lines) as sketched in the figure. These eigenstates ap-
+pear in clusters around the energy of the scar states (green
+lines). A single cluster exists in every symmetry sector and
+the energy gap between two adjacent clusters equals the en-
+ergy gap between scar states ∆Escar.
+VI.
+DISORDER INDUCED APPROXIMATE
+SCARS
+Additional approximate scar states emerge as disorder
+is introduced. These approximate scars appear because
+some symmetry sectors contain energy eigenstates with
+the same disorder indices. For instance, the eigenstates
+|E2,D,1⟩ ≈ |↑↑↓↓↓↓⟩ and |E4,D,m⟩ ≈ αm1 |↑↑↓↑↓↓⟩ +
+αm2 |↑↑↓↓↑↓⟩ for m = 1, 2 have the same disorder in-
+dices D = (3, −1, −1, −1, −1, −1) but different number
+of domain walls Ndw. Recall from Sec. III A that the en-
+ergy of an eigenstate at large disorder is approximately
+given by,
+ENdw,D,m ≈ ∆MNdw,D + J
+�
+N (↑↑,↓↓)
+Ndw,D − N (↑↓,↓↑)
+Ndw,D
+�
++
+�
+i
+diDi,
+(26)
+If an eigenstate |ENdw,D,m⟩ is described by the values
+MNdw,D, N (↑↑,↓↓)
+Ndw,D and N (↑↓,↓↑)
+Ndw,D , then another eigenstate
+|ENdw+2,D,m⟩ with Ndw + 2 domain walls and identical
+disorder indices D is described by
+MNdw+2,D = MNdw,D + 2,
+(27a)
+N (↑↑,↓↓)
+Ndw+2,D = N (↑↑,↓↓)
+Ndw,D − 2,
+(27b)
+N (↑↓,↓↑)
+Ndw+2,D = N (↑↓,↓↑)
+Ndw,D + 2.
+(27c)
+Using Eq. (26) and (27), one can show the energy dif-
+ference between two eigenstates with the same disorder
+indices D and number of domain walls ND and ND + 2
+is approximately
+ENdw+2,D,m − ENdw,D,m ≈ ∆Escar,
+(28)
+where ∆Escar = 2(∆−2J) is the energy gap between the
+scar states. This calculation demonstrates that towers
+of approximate scar states appear in the spectrum as
+disorder is introduced.
+We demonstrate how the appearance of approximate
+scars generates non-trivial dynamics. The system is ini-
+tialized in a generic linear combination of product states
+with disorder indices D1 = (3, −1, −1, . . . , −1)
+|ψinduced
+D1
+⟩ =
+1
+Ninduced
+�
+Ndw,n
+ζ(Ndw)
+n
+|Ndw, D1, n⟩ .
+(29)
+The coefficients are chosen randomly from the interval
+ζ(Ndw)
+n
+∈ [0, 1] and Ninduced is a normalization constant.
+
+12
+0
+1
+2
+3
+4
+5
+t/Tscar
+0.0
+0.5
+1.0
+F
+(a)
+0
+1
+2
+3
+4
+5
+t/Tscar
+0.0
+0.5
+1.0
+(b)
+0
+1
+2
+3
+4
+5
+t/Tscar
+0.0
+0.5
+1.0
+(c)
+−50
+0
+50
+E
+0.000
+0.025
+|ci|2
+(d)
+−50
+0
+50
+E
+0.0
+0.1
+(e)
+−50
+0
+50
+E
+0.0
+0.2
+(f)
+FIG. 9.
+The average fidelity of the initial state Eq. (29) over 103 disorder realizations for system size L = 14 with parameters
+λ = ∆ = 1, J = 5 at disorder strength (a) W = 0.5, (b) W = 5 and (c) W = 10. The shaded areas show the interquartile range
+(middle 50%) of the disorder realizations. The corresponding distribution of expansion coefficients |ci|2 of a single disorder
+realization at disorder strength (d) W = 0.5, (e) W = 5 and (f) W = 10. At weak disorder, the initial state has significant
+overlap with many energy eigenstates and the average fidelity quickly decays to zero. As the disorder strength is increased,
+the initial state has significant overlap with a small number of energy eigenstates with equal energy spacing. Consequently, the
+average fidelity shows persistent revivals.
+We study this initial state because, at large disorder, it
+is a linear combination of an approximate scar tower.
+We consider 103 disorder realizations at different disor-
+der strengths and the fidelity is computed for each re-
+alization. Figure 9(a) displays the average fidelity of a
+thermal system at weak disorder W = 0.5. In this case,
+there is nothing special about the initial state in Eq. (29)
+and it quickly decays to zero similar to Fig. 6(a). The
+dynamical behavior changes remarkably as the disorder
+strength is increased as illustrated in Fig. 9(b)-(c). At
+stronger disorder, the initial state Eq. (29) has large over-
+lap with eigenstates that are approximately equidistant
+in energy.
+Consequently, the average fidelity oscillates
+with a period given by the energy gap Tscar =
+2π
+∆Escar .
+The revival amplitude increases with disorder strength.
+The shaded area in Fig. 9(a)-(c) displays the interquar-
+tile range of disorder realizations. Figures 9(d)-(f) shows
+the expansion of the initial state in energy eigenstates at
+(d) weak disorder W = 0.5, (e) strong disorder W = 5
+and (f) very strong disorder W = 10. As expected, the
+initial state is distributed over a wide range of eigenstates
+in the thermal phase similar to Fig. 6(d). As the disor-
+der strength increases, the initial state has higher and
+higher overlap with eigenstates in an approximate tower
+of equidistant states.
+Figure 9 demonstrates that it is possible to observe
+revivals from generic linear combinations of the states
+{|Ndw, D, n⟩ |Ndw = 0, 2, . . . ; n = 1, 2, . . .} at large dis-
+order. However, the effects may be enhanced by choosing
+the initial state more carefully. The initial state in Eq.
+(29) is, in some sense, the worst case scenario. When all
+product states with disorder indices D are included in
+the sum, the initial state generally has significant overlap
+with all relevant energy eigenstates {|ENdw,D,m⟩ |Ndw =
+0, 2, . . . ; m = 1, 2, . . .}. This causes a large spread in the
+distribution of |ci|2 resulting in a faster decay of the av-
+erage fidelity. If instead, we consider an initial state with
+exactly one product state from each symmetry sector, the
+spread of |ci|2 is smaller
+| ˜ψinduced
+D1
+⟩ =
+1
+�
+L
+2 − 1
+�
+|↑↑↓↓↓↓↓ . . . ↓⟩ + |↑↑↓↑↓↓↓ . . . ↓⟩
++ |↑↑↓↑↓↑↓ . . . ↓⟩ + . . . + |↑↑↓↑↓↑ . . . ↓↑↓↓⟩
+�
+.
+(30)
+Figure 10(a) shows the average fidelity of this initial state
+over 103 disorder realizations at strong disorder W =
+10 and Fig. 10(b) displays the distribution of |ci|2 for a
+single realization. As expected, the distribution of |ci|2
+is narrower and the revival amplitude larger compared to
+Fig. 9.
+The initial states Eq. (29) and (30) display revivals
+similar to the scar states. However, one may distinguish
+these initial states from the scar subspace by noting that
+the average fidelity in Fig. 9 and 10 decays to zero, while
+the amplitude in Fig. 6(c) and 7 remain strictly larger
+than zero. The different dynamical behavior is caused by
+Eq. (29) and (30) being composed of approximate scar
+towers while the original scars |Sn⟩ are exactly equally
+spaced in energy.
+
+13
+0
+2
+4
+t/Tscar
+0.0
+0.5
+1.0
+F
+(a)
+−50
+0
+50
+E
+0.0
+0.1
+|ci|2
+(b)
+FIG. 10.
+(a) Average fidelity of the initial state Eq. (30)
+over 103 disorder realizations with system size L = 14 and
+parameters λ = ∆ = 1, J = 5 and W = 10. The shaded
+area displays the interquartile range of the disorder realiza-
+tions. The average fidelity displays persistent revivals with
+larger amplitude compared to Fig. 7. (b) Expansion of the
+initial state across energy eigenstates. The coefficients |ci|2
+are sharply peaked around certain energies which are approx-
+imately equally spaced.
+VII.
+CONCLUSION
+Building on a known method to find parent Hamil-
+tonians, we proposed a way to determine Hamiltonians
+hosting a tower of QMBS. Starting from the model by
+Iadecola and Schecter, we used this method to identify
+all local 1- and 2-body Hamiltonians of the scar tower
+|Sn⟩. Among these Hamiltonians, we found operators fa-
+cilitating the implementation of local disorder while pre-
+serving the scar states. When introducing disorder, the
+mean level spacing statistics shifts from the GOE to the
+Poisson distribution and the entanglement entropy goes
+from volume-law to area-law scaling with system size. We
+conclude the system transitions from the thermal phase
+to being partially localized. A theory describing the par-
+tially localized eigenstates was developed and verified nu-
+merically.
+In total, we determined a system hosting a
+tower of scar states with the remaining spectrum being
+either thermal or partially localized depending on the
+disorder strength.
+We studied the properties of scar states embedded in a
+localized spectrum and compared with the corresponding
+features in a thermal spectrum. In contrast to thermal
+systems, the bipartite entanglement entropy does not en-
+able the identification of scar states in a localized back-
+ground. The average fidelity, on the other hand, effec-
+tively identifies the scar subspace.
+We investigated the effect of localization on initial
+states with support both inside and outside the scar sub-
+space. For a thermal system, the fidelity displays persis-
+tent revivals with rapidly decreasing amplitude. In con-
+trast, the revival amplitude decays slower for a partially
+localized system.
+Hence, partial localization stabilizes
+the persistent revivals of states initialized partly outside
+the scar subspace.
+Finally, we demonstrated how additional approximate
+scar states emerge as disorder is introduced. When ini-
+tializing the system as a superposition of these states, the
+average fidelity displays revivals with the same period as
+the true scar states. While this effect does not rely on
+fine-tuning the initial state, the revivals are amplified by
+choosing the initial state appropriately.
+ACKNOWLEDGMENTS
+This work has been supported by the Carlsberg Foun-
+dation under grant number CF20-0658.
+Appendix A: Proof that |Sn⟩ are eigenstates of all
+operators in Tab. I with equal energy spacing
+In section II C, we found L + 4 operators having the
+scar states as eigenstates equidistantly spaced in energy.
+Since this analysis was carried out for finite system sizes
+L = 8, 10, 12, 14, the validity of this statement is not
+guaranteed for larger system sizes. In this appendix, we
+rigorously prove the scar states |Sn⟩ are equally spaced
+eigenstates of all operators in Tab. I. Since the scar states
+are constructed iteratively by applying the operator Q†,
+we generally prove this statement using proof by induc-
+tion.
+First, we consider the operator ˆHz = �
+i ˆσz
+i .
+The
+lowest scar state |S0⟩ = |↓↓ . . . ↓⟩ is trivially an eigen-
+state of ˆHz.
+A straightforward calculation shows that
+[ ˆHz, ˆQ†] = 2 ˆQ† and by induction all other scar states are
+eigenstates because
+ˆHz |Sn+1⟩ ∝ ˆHz ˆQ† |Sn⟩
+=
+�
+Ez,n ˆQ† + 2 ˆQ†�
+|Sn⟩
+=
+�
+Ez,n + 2
+�
+|Sn+1⟩ ,
+(A1)
+where ˆHz |Sn⟩ = Ez,n |Sn⟩.
+The scar states are also
+equally spaced in energy En+1,z − En,z = 2.
+A simi-
+lar argument holds for ˆHodd
+zz
+since [ ˆHodd
+zz , ˆQ†] = −4 ˆQ†
+where the energy gap between scar states is −4.
+Next, we consider the operators ˆDi = ˆσz
+i + ˆσz
+i+1 +
+ˆσz
+i ˆσz
+i+1. Recall that ˆDi is related to the projection oper-
+ators through ˆDi = 4 ˆP ↑
+i ˆP ↑
+i+1 − ˆ1 where ˆP ↑
+i = (ˆ1 + ˆσz
+i )/2
+projects site i onto spin-up. First note that ˆDi |S0⟩ =
+(4 ˆP ↑
+i ˆP ↑
+i+1 − ˆ1) |↓↓ . . . ↓⟩ = − |↓↓ . . . ↓⟩. A simple calcu-
+lation shows that ˆDi commutes with ˆQ† by noting that
+
+14
+ˆP ↑
+i ˆP ↓
+i = 0
+[ ˆDi, ˆQ†] = 4
+L
+�
+j=1
+(−1)j�
+ˆP ↓
+j−1[ ˆP ↑
+i , ˆσ+
+j ] ˆP ↓
+j+1 ˆP ↑
+i+1
++ ˆP ↑
+i ˆP ↓
+j−1[ ˆP ↑
+i+1, ˆσ+
+j ] ˆP ↓
+j+1
+�
+= 4(−1)i�
+ˆP ↓
+i−1ˆσ+
+i ˆP ↓
+i+1 ˆP ↑
+i+1 − ˆP ↑
+i ˆP ↓
+i ˆσ+
+i+1 ˆP ↓
+i+2
+�
+= 0.
+(A2)
+Thus, for all scar states we have ˆDi |Sn⟩ = − |Sn⟩. Alter-
+natively, one may note that |Sn⟩ by construction does not
+contain adjacent sites being spin-up. Therefore, ˆP ↑
+i ˆP ↑
+i+1
+naturally annihilates the state.
+Next, we consider the operator ˆHalt
+xz . Before studying
+the action of ˆHalt
+xz on the scar states, we prove by in-
+duction that the commutator [ ˆHalt
+xz , ˆQ†] annihilates |Sn⟩.
+The commutator is given by
+[ ˆHalt
+xz , ˆQ†] =
+L
+�
+i=1
+�
+2
+� ˆP ↓
+i ˆσ+
+i+1ˆσ−
+i+2 − ˆσ+
+i ˆσ+
+i+1 ˆP ↓
+i+2
+�
++ i
+� ˆP ↓
+i ˆσ+
+i+1ˆσy
+i+2 + ˆσy
+i ˆσ+
+i+1 ˆP ↓
+i+2
++ ˆσz
+i ˆσy
+i+1ˆσ+
+i+2 ˆP ↓
+i+3 − ˆP ↓
+i ˆσ+
+i+1ˆσy
+i+2ˆσz
+i+3
+��
+,
+(A3)
+where ˆP ↓
+i = (ˆ1 − ˆσz
+i )/2 is the local projection onto spin-
+down. By direct calculation, one can show the lowest scar
+state is annihilated by this expression [ ˆHalt
+xz , ˆQ†] |S0⟩ = 0.
+A lengthy, yet straightforward, calculation also shows the
+nested commutator vanishes
+�
+[ ˆHalt
+xz , ˆQ†], ˆQ†�
+= 0.
+We
+now prove by induction that the commutator annihilates
+all scar states. Assume [ ˆHalt
+xz , ˆQ†] |Sn⟩ = 0 and consider,
+[ ˆHalt
+xz , ˆQ†] |Sn+1⟩ ∝ [ ˆHalt
+xz , ˆQ†] ˆQ† |Sn⟩
+=
+�
+ˆQ†[ ˆHalt
+xz , ˆQ†] +
+�
+[ ˆHalt
+xz , ˆQ†], ˆQ†��
+|Sn⟩
+= 0.
+(A4)
+Having shown this intermediate result, we prove by in-
+duction that the operator ˆHalt
+xz annihilates the scar states.
+First we show the operator ˆHalt
+xz annihilates |S0⟩
+ˆHalt
+xz |S0⟩ =
+L
+�
+i=1
+(−1)i(ˆσx
+i ˆσz
+i+1 + ˆσz
+i ˆσx
+i+1) |↓↓ . . . ↓⟩
+=
+L
+�
+i=1
+(−1)i+1(ˆσx
+i + ˆσx
+i+1) |↓↓ . . . ↓⟩
+= 0,
+(A5)
+where the second term cancels the first after changing
+summation index i + 1 → i. Next, we show by induction
+that the n-th scar state is annihilated by ˆHalt
+xy . Assume
+ˆHalt
+xz annihilates |Sn⟩ and consider
+ˆHalt
+xz |Sn+1⟩ ∝ ˆHalt
+xz ˆQ† |Sn⟩
+= ( ˆQ† ˆHalt
+xz + [ ˆHalt
+xz , ˆQ†]) |Sn⟩
+= 0.
+(A6)
+The first term vanishes by assumption and the second
+term is exactly what we considered in Eq. (A4). In total,
+we conclude ˆHalt
+xy has |Sn⟩ as eigenstates equidistantly
+separated in energy (with zero energy spacing).
+Finally we consider the operator ˆHalt
+yz . One can prove
+this operator annihilates the scar states using similar ar-
+guments to above. The commutator is given by
+[ ˆHalt
+yz , ˆQ†] =i
+L
+�
+i=1
+�
+2
+� ˆP ↓
+i ˆσ+
+i+1ˆσ−
+i+2 + ˆσ+
+i ˆσ+
+i+1 ˆP ↓
+i+2
+�
+− ˆσx
+i ˆσ+
+i+1 ˆP ↓
+i+2 − ˆP ↓
+i ˆσ+
+i+1ˆσx
+i+2
++ ˆP ↓
+i ˆσ+
+i+1ˆσx
+i+2ˆσz
+i+3 − ˆσz
+i ˆσx
+i+1ˆσ+
+i+2 ˆP ↓
+i+3
+�
+.
+(A7)
+Using induction, one can prove the commutator annihi-
+lates all scar states [ ˆHalt
+yz , ˆQ†] |Sn⟩ = 0 and the operator
+annihilates the lowest scar state ˆHalt
+yz |S0⟩ = 0. Retracing
+the steps in Eq. (A6), we find that ˆHalt
+yz annihilates all
+scar states.
+[1] J. M. Deutsch, Quantum statistical mechanics in a closed
+system, Phys. Rev. A 43, 2046 (1991).
+[2] M. Srednicki, Chaos and quantum thermalization, Phys.
+Rev. E 50, 888 (1994).
+[3] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization
+and its mechanism for generic isolated quantum systems,
+Nature 452, 854 (2008).
+[4] M. Rigol, Breakdown of thermalization in finite one-
+dimensional systems, Phys. Rev. Lett. 103, 100403
+(2009).
+[5] M. Rigol, Quantum quenches and thermalization in one-
+dimensional fermionic systems, Phys. Rev. A 80, 053607
+(2009).
+[6] L. F. Santos and M. Rigol, Onset of quantum chaos
+in one-dimensional bosonic and fermionic systems and
+its relation to thermalization, Phys. Rev. E 81, 036206
+(2010).
+[7] S. Sorg, L. Vidmar, L. Pollet, and F. Heidrich-Meisner,
+
+15
+Relaxation and thermalization in the one-dimensional
+Bose-Hubbard model: A case study for the interaction
+quantum quench from the atomic limit, Phys. Rev. A
+90, 033606 (2014).
+[8] C. Neuenhahn and F. Marquardt, Thermalization of
+interacting fermions and delocalization in Fock space,
+Phys. Rev. E 85, 060101 (2012).
+[9] R. Steinigeweg, A. Khodja, H. Niemeyer, C. Gogolin, and
+J. Gemmer, Pushing the limits of the eigenstate thermal-
+ization hypothesis towards mesoscopic quantum systems,
+Phys. Rev. Lett. 112, 130403 (2014).
+[10] K. R. Fratus and M. Srednicki, Eigenstate thermalization
+in systems with spontaneously broken symmetry, Phys.
+Rev. E 92, 040103 (2015).
+[11] R. Steinigeweg, J. Herbrych, and P. Prelovˇsek, Eigenstate
+thermalization within isolated spin-chain systems, Phys.
+Rev. E 87, 012118 (2013).
+[12] H. Kim, T. N. Ikeda, and D. A. Huse, Testing whether all
+eigenstates obey the eigenstate thermalization hypothe-
+sis, Phys. Rev. E 90, 052105 (2014).
+[13] R. Mondaini, K. R. Fratus, M. Srednicki, and M. Rigol,
+Eigenstate thermalization in the two-dimensional trans-
+verse field Ising model, Phys. Rev. E 93, 032104 (2016).
+[14] D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator
+transition in a weakly interacting many-electron system
+with localized single-particle states, Annals of Physics
+321, 1126 (2006).
+[15] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interact-
+ing electrons in disordered wires: Anderson localization
+and low-T transport, Phys. Rev. Lett. 95, 206603 (2005).
+[16] V. Oganesyan and D. A. Huse, Localization of interacting
+fermions at high temperature, Phys. Rev. B 75, 155111
+(2007).
+[17] A. Pal and D. A. Huse, Many-body localization phase
+transition, Phys. Rev. B 82, 174411 (2010).
+[18] M. Serbyn, Z. Papi´c, and D. A. Abanin, Local conserva-
+tion laws and the structure of the many-body localized
+states, Phys. Rev. Lett. 111, 127201 (2013).
+[19] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phe-
+nomenology of fully many-body-localized systems, Phys.
+Rev. B 90, 174202 (2014).
+[20] J. ˇSuntajs, J. Bonˇca, T. c. v. Prosen, and L. Vidmar,
+Ergodicity breaking transition in finite disordered spin
+chains, Phys. Rev. B 102, 064207 (2020).
+[21] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om-
+ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres,
+M. Greiner, V. Vuleti´c, and M. D. Lukin, Probing many-
+body dynamics on a 51-atom quantum simulator, Nature
+551, 579 (2017).
+[22] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn,
+and Z. Papi´c, Weak ergodicity breaking from quantum
+many-body scars, Nature Physics 14, 745 (2018).
+[23] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn,
+and Z. Papi´c, Quantum scarred eigenstates in a Rydberg
+atom chain: Entanglement, breakdown of thermalization,
+and stability to perturbations, Phys. Rev. B 98, 155134
+(2018).
+[24] C.-J. Lin and O. I. Motrunich, Exact quantum many-
+body scar states in the Rydberg-blockaded atom chain,
+Phys. Rev. Lett. 122, 173401 (2019).
+[25] T. Iadecola, M. Schecter, and S. Xu, Quantum many-
+body scars from magnon condensation, Phys. Rev. B
+100, 184312 (2019).
+[26] N. S. Srivatsa, R. Moessner, and A. E. B. Nielsen, Many-
+body delocalization via emergent symmetry, Phys. Rev.
+Lett. 125, 240401 (2020).
+[27] M. Iversen, N. S. Srivatsa, and A. E. B. Nielsen, Escap-
+ing many-body localization in an exact eigenstate, Phys.
+Rev. B 106, 214201 (2022).
+[28] N. S. Srivatsa, H. Yarloo, R. Moessner, and A. E. B.
+Nielsen, Mobility edges through inverted quantum many-
+body scarring (2022), arXiv:2208.01054.
+[29] T. Iadecola and M. Schecter, Quantum many-body
+scar states with emergent kinetic constraints and finite-
+entanglement revivals, Phys. Rev. B 101, 024306 (2020).
+[30] R. Vasseur, A. J. Friedman, S. A. Parameswaran, and
+A. C. Potter, Particle-hole symmetry, many-body local-
+ization, and topological edge modes, Phys. Rev. B 93,
+134207 (2016).
+[31] J. A. Kj¨all, J. H. Bardarson, and F. Pollmann, Many-
+body localization in a disordered quantum Ising chain,
+Phys. Rev. Lett. 113, 107204 (2014).
+[32] E. Chertkov and B. K. Clark, Computational inverse
+method for constructing spaces of quantum models from
+wave functions, Phys. Rev. X 8, 031029 (2018).
+[33] M. Greiter, V. Schnells, and R. Thomale, Method to iden-
+tify parent Hamiltonians for trial states, Phys. Rev. B 98,
+081113 (2018).
+[34] Q. Qu, J. Sun, and J. Wright, Finding a sparse vec-
+tor in a subspace: Linear sparsity using alternating di-
+rections, IEEE Transactions on Information Theory 62,
+5855 (2016).
+[35] X.-L. Qi and D. Ranard, Determining a local Hamilto-
+nian from a single eigenstate, Quantum 3, 159 (2019).
+[36] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,
+From quantum chaos and eigenstate thermalization to
+statistical mechanics and thermodynamics, Advances in
+Physics 65, 239 (2016).
+[37] T. Guhr, A. M¨uller–Groeling, and H. A. Weidenm¨uller,
+Random-matrix theories in quantum physics: Common
+concepts, Physics Reports 299, 189 (1998).
+[38] A. A. Abul-Magd and A. Y. Abul-Magd, Unfolding of the
+spectrum for chaotic and mixed systems, Physica A: Sta-
+tistical Mechanics and its Applications 396, 185 (2014).
+[39] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux,
+Distribution of the ratio of consecutive level spacings in
+random matrix ensembles, Phys. Rev. Lett. 110, 084101
+(2013).
+[40] D. N. Page, Average entropy of a subsystem, Phys. Rev.
+Lett. 71, 1291 (1993).
+[41] B. Bauer and C. Nayak, Area laws in a many-body lo-
+calized state and its implications for topological order,
+Journal of Statistical Mechanics:
+Theory and Experi-
+ment 2013, P09005 (2013).
+[42] L. F. Santos, F. Borgonovi, and F. M. Izrailev, Chaos and
+statistical relaxation in quantum systems of interacting
+particles, Phys. Rev. Lett. 108, 094102 (2012).
+[43] L. F. Santos, F. Borgonovi, and F. M. Izrailev, Onset of
+chaos and relaxation in isolated systems of interacting
+spins: Energy shell approach, Phys. Rev. E 85, 036209
+(2012).
+

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+arXiv:2301.00976v1  [hep-ph]  3 Jan 2023
+The Σ and Ξ electromagnetic form factors in the extended vector meson dominance model
+Bing Yan,1,2, ∗ Cheng Chen,2,3, † and Ju-Jun Xie2, 3, 4, ‡
+1College of Mathematics and Physics, Chengdu University of Technology, Chengdu 610059, China
+2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
+3School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
+4Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics,
+Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China
+We propose a phenomenological extended vector meson dominance model for the baryon electromagnetic
+structure, and it is found that the current experimental data on the Σ and Ξ electromagnetic form factors in the
+time-like region can be well described. Meanwhile, we can also reproduce the ratios of the total cross sections
+of reactions e+e− → Σ+ ¯Σ−, Σ0 ¯Σ0, and Σ− ¯Σ+, which are 9.7 ± 1.3 : 3.3 ± 0.7 : 1 at center-of-mass
+energies from 2.3864 to 3.02 GeV. We also analytically continue the expression of the form factors to space-
+like region and estimate the charge radii of the Σ and Ξ hyperons. The result for the Σ− is in agreement with
+the experimental date.
+I.
+INTRODUCTION
+The electromagnetic structure information of hadrons is
+characterized by the electromagnetic form factors (EMFFs),
+which are functions of the four-momentum transfer squared
+q2, with q the four-momentum carried by the exchanged vir-
+tual photon. Study of these EMFFs can lead to a better under-
+standing of fundamental structure of hadrons. On the experi-
+mental side, most commonly the baryon EMFFs in the space-
+like region (q2 < 0) were measured in the electron-baryon
+scattering [1–4]. While for these unstable hadrons, for ex-
+ample, these hyperons, their EMFFs in the space-like region
+are very difficult to be experimentally measured. However, in
+the time-like region (q2 > 0), their EMFFs can be measured
+through the electron-positron annihilation reactions by the
+BESIII and Belle collaborations [5–12]. Meanwhile, the ef-
+fective form factor Geff(q2) of hyperons can be extracted from
+the high-precision measured Born cross sections of the reac-
+tions e+e− → Y ¯Y (Y stands for hyperon; ¯Y is anti-hyperon).
+It was pointed out that these baryon EMFFs in the time-like
+region can be associated with the time evolution of the charge
+and magnetic distributions inside the baryon [13, 14].
+The hyperon effective form factors Geff(q2) are the func-
+tions that parametrize the γY ¯Y vertex generated by the strong
+interaction. While, the production vertex γY ¯Y is very poorly
+understood so far [15, 16].
+The vector meson dominance
+(VMD) model is a very successful tool for studying the nu-
+cleon electromagnetic form factors, in both the space-like and
+time-like regions [17–19]. Within a modified VMD model,
+the EMFFs of Λ hyperon were investigated in Refs. [20, 21].
+By considering the Y ¯Y final sate interactions, the EMFFs
+of hyperons in the time-like region have been studied in
+Ref. [22]. It is worth to mention that the enhancement of the
+∗Electronic address: yanbing@impcas.ac.cn
+†Electronic address: chencheng22@mails.ucas.ac.cn
+‡Electronic address: xiejujun@impcas.ac.cn
+effective form factor of the Λ hyperon seen in the e+e− →
+Λ¯Λ reaction, was reproduced within the two above different
+calculations in Refs. [20, 21] and Ref. [22], respectively. In
+the vector meson dominance model for studying the electro-
+magnetic form factors of baryons, there is a phenomenological
+intrinsic form factor g(q2). From these studies of the nucleon
+and hyperon EMFFs [17–29], it is found that a better choice
+of g(q2) is the dipole form
+g(q2) =
+1
+(1 − γq2)2 ,
+(1)
+with γ a free parameter. In the space-like region, the dipole
+form is consistent with the results obtained from perturbative
+quantum chromodynamics calculations [30, 31]. In the time-
+like region, it should be noticed that γ is a positive parameter,
+thus g(q2) will have a pole in the position γ = 1/q2, such
+pole could be restricted in the unphysical region, if γ satisfy
+γ > 1/(4m2
+Y ) for hyperon Y .
+For a long time, the simple dipole form paremetrization is
+very useful for the discussion of different baryons. For exam-
+ple, the dipole form of g(q2) can well describe the effective
+form factors of Λ [20, 21], Σ [32], and Ξ [27]. While for the
+nucleon, a good general review is given in Refs. [33–36], both
+from the theoretical and from the experimental points of view.
+However, these determined values of γ for different ground
+state octet baryons with spin 1/2 are much different, even for
+the triplet Σ+, Σ− and Σ0 [32]. The determined values of γ,
+from previous works, for nucleon, Λ, Σ, and Ξ0 baryons are
+collected in Table I. Nevertheless, the VMD model and the
+parametrization of g(q2) can give a reasonable description of
+the experimental data on the baryon EMFFs at the considered
+energy region.
+Various experimental and theoretical efforts have been con-
+tributed to the electromagnetic form factors. Very recently, the
+EMFFs of Σ+, Σ−, and Σ0 hyperons in the time-like region,
+have been measured with high-precision by the BESIII collab-
+oration through e+e− → Σ+ ¯Σ− [9], Σ− ¯Σ+ [9], and Σ0 ¯Σ0
+reactions [10] at center-of-mass energies from 2.3864 to 3.02
+GeV. The resulting ratios of total cross sections of these above
+
+2
+TABLE I: Values of γ (in GeV−2) for octet baryons used in previous
+works.
+Proton ([17–19]) Neutron ([24–27])
+Λ ([20])
+Λ ([21])
+γ
+1.408
+1.408
+0.336
+0.48 ± 0.08
+Σ+ ([32])
+Σ− ([32])
+Σ0 ([27])
+Ξ0 ([27])
+γ
+0.46 ± 0.01
+1.18 ± 0.13
+0.26 ± 0.01 0.21 ± 0.02
+three reactions are 9.7 ± 1.3 : 1 : 3.3 ± 0.7 [9, 37, 38], which
+disagree with various theoretical model predictions [39, 40].
+After the experimental measurements of e+e− → Σ+ ¯Σ− and
+Σ− ¯Σ+ [9], the effective form factors of Σ+ and Σ− were in-
+vestigated by using the VMD model [32], where the param-
+eter γ were taken with different values for Σ+ and Σ−. In
+Ref. [22], by considering the final state interactions of Y ¯Y ,
+the energy dependence of the three reactions e+e− → Σ+ ¯Σ−,
+Σ0 ¯Σ0, and Σ− ¯Σ+ at low energies can be roughly reproduced,
+and it was found that there is a strong interplay between
+Σ+ ¯Σ−, Σ0 ¯Σ0, and Σ− ¯Σ+ channel in the near-threshold re-
+gion, caused by the Σ¯Σ final state interactions. In the present
+work, we revisit the EMFFs at the time-like region of Σ and Ξ
+hyperons within an extended vector meson dominance model,
+where the affects of the isospin combinations from isovector
+ρ0 and isoscalar ω and φ mesons are taken into account. Fur-
+thermore, we assume that the values of model parameter γ
+are same for Σ and Ξ hyperons. In addition, a vector meson
+with mass around 2.7 GeV was considered for the sake of bet-
+ter fitting the EMFFs of the Ξ0 and Ξ− hyperons. We then
+progress to an analysis of the electromagnetic form factors in
+the space-like region and evaluate the electromagnetic radius
+of Σ hyperons. The theoretical result for the Σ− hyperon is in
+agreement with the experimental measurements. This article
+is organized as follows: in next section we will show the theo-
+retical formalism of the Σ and Ξ electromagnetic form factors
+in the VMD model. Numerical results about the effective form
+factors of Σ and Ξ and total cross sections of e+e− → Σ¯Σ and
+Ξ¯Ξ are shown in Sec. III, and a short summary is given in the
+final section.
+II.
+FORMALISM
+As already pointed out, as fixed-energy e+e− colliders,
+the EMFFs of hyperons in the time-like region was extracted
+from the data on the differential cross section of the process
+e+e− → Y ¯Y . For analysis the data, the BESIII Collaboration
+use the energy scan method [41–43], while the initial state
+radiation method was used by Belle Collaboration [12] and
+BABAR collaboration [44, 45]. Besides, the effective form
+factors Geff can be easily obtained from the data of the total
+cross sections. The module squared of effective form factor
+|Geff|2 is a linear combination of |GE|2 and |GM|2, and pro-
+portional to the total cross section of e+e− → Y ¯Y reaction.
+In this work, we study the EMFFs of Σ and Ξ baryons in the
+time-like region with the experimental measurements on the
+e+e− → Y ¯Y reactions. Based on parity conservation and
+Lorentz invariant, the electromagnetic current of the baryons
+with a spin of 1/2 characterize two independent scalar func-
+tions F1(q2) and F2(q2) depending on q2, which are the Dirac
+and Pauli form factors, respectively. While the corresponding
+electrical and magnetic form factors GE(q2) and GM(q2) are
+written as [38, 46, 47],
+GE(q2) = F1(q2) + τF2(q2),
+(2)
+GM(q2) = F1(q2) + F2(q2),
+(3)
+where M is the baryon mass and τ = q2/(4M 2).
+With
+GE(q2) and GM(q2), the magnitude of the effective form fac-
+tor |Geff(q2)| is defined as
+|Geff(q2)| =
+�
+2τ|GM(q2)|2 + |GE(q2)|2
+1 + 2τ
+.
+(4)
+In the time-like region, the effective form factors of hyper-
+ons are experimentally studied via the electron-positron anni-
+hilation processes. Under the one photon exchange approx-
+imation, the total cross section of the annihilation reaction
+e+e− → ¯Y Y can be expressed in terms of the effective form
+factor Geff as [44, 48, 49]
+σe+e−→ ¯Y Y = 4πα2βCY
+3s
+(1 + 1
+2τ ) | Geff(s) |2,
+(5)
+with α = e2/(4π) = 1/137.036 the fine-structure con-
+stant, and β =
+�
+1 − 4M 2
+Y /s is a phase-space factor. Here,
+s = q2 is the invariant mass square of the e+e− system. The
+coulomb enhancement factor CY accounts for the electromag-
+netic interaction of charged point-like fermion pairs in the fi-
+nal state [50], which is given by
+CY =
+�
+y
+1−e−y
+for Σ+, Σ−, and Ξ−,
+1
+for Σ0 and Ξ0,
+(6)
+with y = απ
+β
+2MY
+√s . Considering the CY factor, it is expected
+that the cross section of process e+e− → Y ¯Y is nonzero at
+the reaction threshold for charged hyperons pairs. As plotted
+in Fig. 1 for the case of Ξ− 1, the factor CY affects only at
+the energy region very close to the reaction threshold, and it
+decreases very quickly as the reaction energy growing and it
+follows that few MeV above the reaction threshold it is CY ∼
+1, then its effect can be neglected [50–53].
+A.
+The EMFFs of Σ hyperon
+In the VMD model, the virtual photon couples to Σ and ¯Σ
+through isovector ρ0 meson and isoscalar ω and φ mesons.
+Since both the ω and φ are far from the mass threshold of Σ¯Σ,
+the behavior of the contributions from them are similar, thus
+we combine their contributions. In this way, one can param-
+eterize Dirac and Pauli form factors for Σ+ and Σ− in the
+1 The numerical results for Σ+ and Σ− are similar.
+
+3
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+√
+s
+−
+2M
+Ξ
+−
+(MeV)
+0
+2
+4
+6
+8
+10
+12
+C
+Ξ
+−
+FIG. 1: The Coulomb factor for Ξ−.
+time-like region as follows [17, 19], 2
+F Σ+
+1
+= g(q2)(f Σ+
+1
++ βρ
+√
+2Bρ − βωφ
+√
+3 Bωφ),
+(8)
+F Σ+
+2
+= g(q2)(f Σ+
+2
+Bρ − αωφ
+√
+3 Bωφ),
+(9)
+F Σ−
+1
+= g(q2)(f Σ−
+1
+− βρ
+√
+2Bρ − βωφ
+√
+3 Bωφ),
+(10)
+F Σ−
+2
+= g(q2)(f Σ−
+2
+Bρ − αωφ
+√
+3 Bωφ),
+(11)
+F Σ0
+1
+= g(q2)(βωφ
+√
+3
+− βωφ
+√
+3
+Bωφ),
+(12)
+F Σ0
+2
+= g(q2)µΣ0Bωφ,
+(13)
+with
+Bρ =
+m2
+ρ
+m2ρ − q2 − imρΓρ
+,
+(14)
+Bωφ =
+m2
+ωφ
+m2
+ωφ − q2 − imωφΓωφ
+,
+(15)
+where the widths of ρ, ω and φ are taken into account. In this
+work, we take mρ = 0.775 MeV, Γρ = 149.1 MeV, Γωφ =
+2 We have followed:
+|Σ+ ¯Σ−⟩ =
+1
+√
+2
+|1, 0⟩ +
+1
+√
+3
+|0, 0⟩ +
+1
+√
+6
+|2, 0⟩ ,
+|Σ− ¯Σ+⟩ = − 1
+√
+2
+|1, 0⟩ +
+1
+√
+3
+|0, 0⟩ +
+1
+√
+6
+|2, 0⟩ ,
+|Σ0 ¯Σ0⟩ = − 1
+√
+3
+|0, 0⟩ +
+�
+2
+3 |2, 0⟩ ,
+(7)
+with the basis of |IΣ¯Σ, IZ
+Σ¯Σ⟩. In the one photon exchange approximation,
+there is no contributions from the isospin tensor terms.
+(Γω + Γφ)/2 = 6.4645 MeV, and mωφ = (mω + mφ)/2 =
+0.9005 GeV, which are quoted in the review of particle
+physics book [54].
+Besides, we take µΣ+ = 3.112ˆµΣ+,
+µΣ− = −1.479ˆµΣ−, µΣ0 = 2.044ˆµΣ0 in natural unit [54],
+i.e., ˆµ =
+e
+2MΣ . In addition, at q2 = 0, with the constraints
+GΣ+
+E
+= 1 and GΣ+
+M = µΣ+, GΣ−
+E
+= −1 and GΣ−
+M = µΣ−, the
+coefficients f Σ+
+1
+and f Σ+
+2
+, f Σ−
+1
+and f Σ−
+2
+can be calculated,
+f Σ+
+1
+= 1 − βρ
+√
+2 + βωφ
+√
+3 ,
+f Σ+
+2
+= 2.112 + αωφ
+√
+3 ,
+(16)
+f Σ−
+1
+= −1 + βρ
+√
+2 + βωφ
+√
+3 ,
+f Σ−
+2
+= −0.479 + αωφ
+√
+3 .(17)
+Finally, the model parameters γ, the coefficients βρ, βωφ, and
+αωφ will be determined by fitting them to the experimental
+data on the time-like effective form factors of Σ+, Σ0, and
+Σ−, which will be discussed in following.
+B.
+The EMFFs of Ξ hyperon
+For the case of e+e− → Ξ−¯Ξ+ and Ξ0¯Ξ0 reactions, since
+Ξ− and Ξ0 are isospin doublets, we express the Ξ−¯Ξ+ and
+Ξ0¯Ξ0 states in terms of isospin 0 and 1 components. The mix-
+tures of isoscalar and isovector for Ξ−¯Ξ+ and Ξ0¯Ξ0 of equal
+relative wight but different sign are imposed by the isospin
+symmetry as introduced by the underlying Clebsch-Gorden
+coefficients [54]. Then, the Dirac and Pauli form factors F1
+and F2 for Ξ− and Ξ0 can be easily obtained as before for the
+Σ hyperon,
+F Ξ−
+1
+= g(q2)(f Ξ−
+1
+− βρ
+√
+2
+Bρ − βV1
+√
+2
+BV1
+−βV2
+√
+2 BV2 + βωφ
+√
+2 Bωφ),
+(18)
+F Ξ−
+2
+= g(q2)(f Ξ−
+2
+Bρ − αV1
+√
+2 BV1
+−αV2
+√
+2 BV2 + αωφ
+√
+2 Bωφ),
+(19)
+F Ξ0
+1
+= g(q2)(f Ξ0
+1
++ βρ
+√
+2
+Bρ + βV1
+√
+2
+BV1
++βV2
+√
+2 BV2 + βωφ
+√
+2 Bωφ),
+(20)
+F Ξ0
+2
+= g(q2)(f Ξ0
+2 Bρ + αV1
+√
+2 BV1
++αV2
+√
+2
+BV2 + αωφ
+√
+2
+Bωφ),
+(21)
+with
+BV 1 =
+M 2
+V1
+M 2
+V1 − q2 − iMV1ΓV1
+,
+(22)
+BV 2 =
+M 2
+V2
+M 2
+V2 − q2 − iMV2ΓV2
+,
+(23)
+where we have considered contributions from two more ex-
+cited vector mesons, V1 and V2, in addition the contribu-
+tions from ground states ρ, ω and φ. Their mass and width
+
+4
+are MV1 (MV2) and ΓV1 (ΓV2), respectively. The mass MV2
+and width ΓV2 are taken as used in Ref. [7], which are:
+MV2 = 2.993 GeV and ΓV2 = 88 MeV.
+Besides, we
+take µΞ− = −0.915ˆµΞ−, and µΞ0 = −1.749ˆµΞ0 in natural
+unit [54]. Then the coefficients f Ξ−
+1
+, f Ξ−
+2
+, f Ξ0
+1 , and f Ξ0
+2
+can
+be calculated as
+f Ξ−
+1
+= −1 + βρ
+√
+2 + βV1
+√
+2 + βV2
+√
+2 − βωφ
+√
+2 ,
+(24)
+f Ξ−
+2
+= 0.085 + αV1
+√
+2 + αV2
+√
+2 − αωφ
+√
+2 ,
+(25)
+f Ξ0
+1
+= − βρ
+√
+2 − βV1
+√
+2 − βV2
+√
+2 − βωφ
+√
+2 ,
+(26)
+f Ξ0
+2
+= −1.749 − αV1
+√
+2 − αV2
+√
+2 − αωφ
+√
+2 .
+(27)
+The parameter γ will be fixed as the one determined from the
+case of Σ, while the other free parameters βωφ, βρ, βV1, βV2,
+αωφ, αV1, αV2, ΓV1, and MV1 are determined by fitting them
+to experimental data on the time-like effective form factors of
+Ξ− and Ξ0.
+III.
+NUMERICAL RESULTS
+Under the above formulations, we perform a four-parameter
+(γ, βρ, βωφ, αωφ)-χ2 fit to the experimental data on the ef-
+fective form factors Geff of Σ+, Σ0, and Σ− hyperons. There
+are 33 data points in total, which are extracted at the center-
+of-mass energies from 2.3864 to 3.0200 GeV. The fitted pa-
+rameters are: γ = 0.527 ± 0.024 GeV−2, βρ = 1.63 ± 0.07,
+βωφ = −0.08 ± 0.06, and αωφ = −3.18 ± 0.77. And the
+obtained χ2/dof is 1.69, where dof is the number of dimen-
+sion of the freedom. Note that the obtained χ2/dof is larger
+than 1, since we have fitted all the experimental data from
+BESIII [9, 10], Belle [12], and BABAR [45] Collaborations,
+by considering these contributions from only ground state of
+vector mesons. If we considered only these data of BESIII
+Collaboration [9, 10], the obtained χ2/dof is 1.17. In Fig. 2
+we show the theoretical results of the effective form factors of
+the Σ+, Σ0, and Σ−. The red, blue, and green curves stand
+for the results for Σ+, Σ0, and Σ−, respectively. The exper-
+imental data from BESIII [9, 10], Belle [12], and BABAR
+Collaboration [45] are also shown for comparing. One can
+see that, with same model parameters, we can describe these
+data on the effective form factors of Σ+, Σ0 and Σ− quite
+well, especially for the precise data measured by the BESIII
+Collaboration [9, 10]. The total cross sections of e+e− → Σ¯Σ
+are also calculated with these fitted parameters. The numeri-
+cal results are shown in Fig. 3, compared with the experimen-
+tal data. Since the effective form factors of Σ hyperons can
+be well reproduced with our model, the total cross sections
+of e+e− → Σ+ ¯Σ−, e+e− → Σ0 ¯Σ0 and e+e− → Σ− ¯Σ+
+reactions can be also well described.
+For the case of Ξ− and Ξ0 effective form factors, γ is taken
+as the result of fitting to Σ hyperon, i.e., γ = 0.527, we per-
+form nine-parameter (βωφ, βρ, βV1, βV2, αωφ, αV1, αV2, ΓV1,
+MV1)-χ2 fit to the experimental data on. There are totally 18
+2.3
+2.4
+2.5
+2.6
+2.7
+2.8
+2.9
+3.0
+3.1
+√
+s
+(GeV�
+10
+−3
+10
+−2
+10
+−1
+10
+0
+10
+1
+|Geff|
+Σ
++
+BESIII
+Σ
+0
+BESIII
+Σ
+−
+BESIII
+Σ
++
+Belle
+Σ
+0
+Belle
+Σ
+0
+BABAR
+FIG. 2: The obtained effective form factors of Σ+, Σ0, and Σ−,
+compared with the experimental data.
+2.3
+2.4
+2.5
+2.6
+2.7
+2.8
+2.9
+3.0
+3.1
+√
+s
+(GeV�
+10
+−1
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+σ(pb)
+Σ
++
+BESIII
+Σ
+0
+BESIII
+Σ
+−
+BESIII
+Σ
++
+Belle
+Σ
+0
+Belle
+Σ
+0
+BABAR
+FIG. 3: The total cross section of Σ+, Σ0 and Σ− hyperons com-
+pared with experimental data.
+data points, and these data correspond to the center-of-mass
+energies from 2.644 to 3.080 GeV. The fitted parameters are
+listed in Table II, with a reasonably small χ2/dof = 0.29.
+Since we have more free parameters and the experimental data
+points is limited, we did not get the uncertainties of these pa-
+rameters from the χ2 fit. In Fig. 4, we depict the effective form
+factor of the Ξ− and Ξ0 using the fitted parameters shown in
+Table II. The red curve stands for the results of Ξ0, while the
+green curve is the fitted results for Ξ−. Again, one can see that
+the experimental data on the effective form factors of Ξ− and
+Ξ0 can be well reproduced. It is worth to mention that the two
+resonances V1 and V2 are crucial to describe the experimental
+data, and without their contributions, we cannot get a good fit
+to the experimental data. In addition, the total cross section of
+e+e− → Ξ−¯Ξ+ and e+e− → Ξ0¯Ξ0 are also calculated with
+
+5
+TABLE II: Fitted model parameters for the effective form factors of
+Ξ− and Ξ0.
+Parameter
+Value
+Parameter
+Value
+βωφ
+−0.774
+αωφ
+9.346
+βρ
+0.616
+αV1
+−0.039
+βV1
+0.099
+αV2
+−0.113
+βV2
+0.115
+ΓV1 (MeV)
+71
+MV1 (GeV)
+2.742
+the fitted parameters shown in Table II, and the numerical re-
+sults are shown Fig. 5. The two peaks of V1 and V2 can be
+clear seen, and more precise data around 2744 and 2993 MeV
+are needed to further study their properties.
+We next pay
+2.6
+2.7
+2.8
+2.9
+3.0
+3.1
+√
+s
+(GeV�
+10
+−2
+10
+−1
+10
+0
+|Geff|
+Ξ
+−
+BESIII
+Ξ
+0
+BESIII
+FIG. 4: The obtained effective form factors of Ξ− and Ξ0 compared
+with the experimental data.
+attention to the EMFFs at the space-like region, which can
+be straightforwardly obtained with the these parameters de-
+termined from the experimental data in the time-like region.
+Since the EMFFs in the space-like region are real, thus we
+have to ignore the widths of the vector mesons. Then one can
+calculate the mean squared charge radius, which is defined by
+the relation [1, 40, 55]
+��
+r2
+ch
+�
+=
+
+
+
+
+
+
+
+−6
+GE(0)
+dGE(Q2)
+dQ2
+����
+Q2=0
+,
+for Σ+, Σ− and Ξ−,
+−6 dGE(Q2)
+dQ2
+����
+Q2=0
+,
+for Σ0 and Ξ0,
+(28)
+with Q2 = −q2. With the parameters fitted above, the cal-
+culated results of
+�
+r2
+ch
+�
+of Σ and Ξ hyperons are shown in
+Table III. Our result for Σ− is agreement with the experi-
+mental data within uncertainties:
+�
+r2
+ch
+�
+Σ− = 0.61 ± 0.12 ±
+0.09 [1],
+�
+r2
+ch
+�
+Σ− = 0.91 ± 0.32 ± 0.4 [2]. In Ref. [1] the
+Σ− charge radius was measured in the space-like Q2 range
+0.035 − 0.105 GeV2 by elastic scattering of a Σ− beam
+off atomic electrons. The measurement was performed with
+the SELEX (E781) spectrometer using the Fermilab hyperon
+2.6
+2.7
+2.8
+2.9
+3.0
+3.1
+√
+s
+(GeV�
+10
+−1
+10
+0
+10
+1
+10
+2
+10
+3
+σ(pb)
+Ξ
+−
+BESIII
+Ξ
+0
+BESIII
+FIG. 5: The total cross sections of e+e− → Ξ−¯Ξ+ and e+e− →
+Ξ0¯Ξ0 reactions compared with experimental data.
+beam at a mean energy of 610GeV. In Ref. [2] it was at-
+tracted from the elastic scattering of high energy Σ− off elec-
+trons from carbon and copper targets using the CERN hy-
+peron beam, where these events are identified using a maxi-
+mum likelihood technique exploring the kinematical relations
+of the scattered particles. Theoretical calculations with chi-
+ral perturbation theory (ChPT) [40, 56] and the nonlocal chi-
+ral effective theory (ChET) [57], and chiral constituent quark
+model (ChCQM) [58] are also listed for comparison. On can
+see that the orderings of the most charge radii calculated by
+other works are in agreement with our results. Moreover, our
+results are consistent with these calculations in Refs. [56–58]
+that
+�
+r2
+ch
+�
+Σ+ >
+�
+r2
+ch
+�
+Σ−. On the contrary, the results ob-
+tained with chiral perturbation theory predictions in Ref. [40]
+indicate that the charge radius of Σ− is larger than the one of
+Σ+. In addition, the charge radius of Ξ0 calculated here is
+small and negative, which is in agreement with the nonlocal
+chiral effective theory calculation in Ref. [57]. It is expected
+that these results can be tested by future experimental mea-
+surements.
+IV.
+SUMMARY
+In this work, we study the effective form factor of Σ and
+Ξ hyperons in time-like region within the vector meson dom-
+inance model, and we take a common model parameter γ. In
+addition, the effect of the isospin combination is taken into
+account. For the case of Σ hyperon, the contributions from ρ,
+ω and φ mesons are considered. Within same model parame-
+ters, we can simultaneously describe the current experimental
+data on the effective form factors of Σ+, Σ0 and Σ−. While
+for the case of Ξ+ and Ξ−, in addition to the contributions of
+the ground states ρ, ω and φ, it is found that one needs also
+contributions from two new vector states, and their masses and
+widths are: MV1 = 2.742 GeV, ΓV1 = 71 MeV, MV2 = 2.993
+
+6
+TABLE III: The obtained results for mean squared electromagnetic radii
+�
+r2
+ch
+�
+(fm2) for Σ and Ξ. The results from two ChPT calculations,
+ChET and, ChCQM as well as the experimental data are also listed.
+Baryon
+Ξ0
+Ξ−
+Σ+
+Σ0
+Σ−
+This work
+−0.07
+0.43
+0.78
+0.12
+0.65
+ChPT [40]
+0.13 ± 0.03
+0.49 ± 0.05
+0.60 ± 0.02
+−0.03 ± 0.01
+0.67 ± 0.03
+ChPT [56]
+0.36 ± 0.02
+0.61 ± 0.01
+0.99 ± 0.03
+0.10 ± 0.02
+0.780
+ChET [57]
+−0.015 ± 0.007 0.601 ± 0.127 0.719 ± 0.116 0.010 ± 0.004 0.700 ± 0.124
+ChCQM [58]
+0.091
+0.587
+0.825
+0.089
+0.643
+GeV, and ΓV2 = 88 MeV. It is expected that new precise ex-
+perimental data at BESIII [59] can be used to further study
+their properties. Finally, we would like to stress that thanks
+to the effects of the isospin combinations, the effective form
+factors of Σ+, Σ0 and Σ− can be simultaneously reproduced
+within the same model parameters by using the vector meson
+dominance model. Again, the theoretical results obtained here
+also indicate that the vecor meson dominance model is a valid
+tool for studying the baryonic electromagnetic form factors at
+the time-like region. More precise data on the e+e− → Y ¯Y
+reactions can be used to improve our knowledge of hyperon
+effective form factors.
+Acknowledgements
+We warmly thank Profs. Xiong-Fei Wang and Xiao-Rong
+Zhou for useful comments and discussions.
+This work is
+partly supported by the National Natural Science Founda-
+tion of China under Grant Nos. 12075288, 11735003, and
+11961141012. It is also supported by the Youth Innovation
+Promotion Association CAS.
+[1] I. M. Gough Eschrich et al. [SELEX], Phys. Lett. B
+522,
+233-239
+(2001)
+doi:10.1016/S0370-2693(01)01285-0
+[arXiv:hep-ex/0106053 [hep-ex]].
+[2] M. I. Adamovich et al. [WA89], Eur. Phys. J. C 8, 59-66 (1999)
+doi:10.1007/s100520050444
+[3] M. K. Jones et al. [Jefferson Lab Hall A], Phys. Rev.
+Lett. 84, 1398-1402 (2000) doi:10.1103/PhysRevLett.84.1398
+[arXiv:nucl-ex/9910005 [nucl-ex]].
+[4] O.
+Gayou
+et
+al.
+[Jefferson
+Lab
+Hall
+A],
+Phys.
+Rev.
+Lett. 88, 092301 (2002) doi:10.1103/PhysRevLett.88.092301
+[arXiv:nucl-ex/0111010 [nucl-ex]].
+[5] M. Ablikim et al. [BESIII], Phys. Rev. D 97, no.3, 032013
+(2018) doi:10.1103/PhysRevD.97.032013 [arXiv:1709.10236
+[hep-ex]].
+[6] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 124, no.3,
+032002
+(2020)
+doi:10.1103/PhysRevLett.124.032002
+[arXiv:1910.04921 [hep-ex]].
+[7] M. Ablikim et al. [BESIII], Phys. Rev. D 103, no.1, 012005
+(2021) doi:10.1103/PhysRevD.103.012005 [arXiv:2010.08320
+[hep-ex]].
+[8] M. Ablikim et al. [BESIII], Phys. Lett. B 820, 136557 (2021)
+doi:10.1016/j.physletb.2021.136557 [arXiv:2105.14657 [hep-
+ex]].
+[9] M. Ablikim et al. [BESIII], Phys. Lett. B 814, 136110 (2021)
+doi:10.1016/j.physletb.2021.136110 [arXiv:2009.01404 [hep-
+ex]].
+[10] M. Ablikim et al. [BESIII], Phys. Lett. B 831, 137187 (2022)
+doi:10.1016/j.physletb.2022.137187 [arXiv:2110.04510 [hep-
+ex]].
+[11] X. Wang and G. Huang, Symmetry 14, no.1, 65 (2022)
+doi:10.3390/sym14010065
+[12] [Belle], [arXiv:2210.16761 [hep-ex]].
+[13] M. A. Belushkin, H. W. Hammer and U. G. Meissner, Phys.
+Rev. C 75, 035202 (2007) doi:10.1103/PhysRevC.75.035202
+[arXiv:hep-ph/0608337 [hep-ph]].
+[14] E. A. Kuraev, E. Tomasi-Gustafsson and A. Dbeyssi, Phys. Lett.
+B 712, 240-244 (2012) doi:10.1016/j.physletb.2012.04.073
+[arXiv:1106.1670 [hep-ph]].
+[15] I. T. Lorenz, H. W. Hammer and U. G. Meißner, Phys. Rev.
+D 92, no.3, 034018 (2015) doi:10.1103/PhysRevD.92.034018
+[arXiv:1506.02282 [hep-ph]].
+[16] A. Mangoni, S. Pacetti and E. Tomasi-Gustafsson, Phys. Rev. D
+104, no.11, 116016 (2021) doi:10.1103/PhysRevD.104.116016
+[arXiv:2109.03759 [hep-ph]].
+[17] F. Iachello, A. D. Jackson and A. Lande, Phys. Lett. B 43, 191-
+196 (1973) doi:10.1016/0370-2693(73)90266-9
+[18] F. Iachello and Q. Wan, Phys. Rev. C 69, 055204 (2004)
+doi:10.1103/PhysRevC.69.055204
+[19] R. Bijker and F. Iachello, Phys. Rev. C 69, 068201 (2004)
+doi:10.1103/PhysRevC.69.068201
+[arXiv:nucl-th/0405028
+[nucl-th]].
+[20] Y. Yang, D. Y. Chen and Z. Lu, Phys. Rev. D 100, no.7, 073007
+(2019) doi:10.1103/PhysRevD.100.073007 [arXiv:1902.01242
+[hep-ph]].
+[21] Z. Y. Li, A. X. Dai and J. J. Xie, Chin. Phys. Lett.
+39, no.1, 011201 (2022) doi:10.1088/0256-307X/39/1/011201
+[arXiv:2107.10499 [hep-ph]].
+[22] J. Haidenbauer, U. G. Meißner and L. Y. Dai, Phys. Rev. D
+103, no.1, 014028 (2021) doi:10.1103/PhysRevD.103.014028
+[arXiv:2011.06857 [nucl-th]].
+[23] E. Tomasi-Gustafsson and M. P. Rekalo, Phys. Lett. B 504, 291-
+295 (2001) doi:10.1016/S0370-2693(01)00312-4
+[24] A. Bianconi and E. Tomasi-Gustafsson, Phys. Rev. Lett. 114,
+no.23, 232301 (2015) doi:10.1103/PhysRevLett.114.232301
+[arXiv:1503.02140 [nucl-th]].
+[25] A.
+Bianconi
+and
+E. Tomasi-Gustafsson,
+Phys.
+Rev.
+C
+93, no.3, 035201 (2016) doi:10.1103/PhysRevC.93.035201
+[arXiv:1510.06338 [nucl-th]].
+[26] M. Ablikim et al. [BESIII], Nature Phys. 17, no.11, 1200-1204
+(2021) doi:10.1038/s41567-021-01345-6
+[arXiv:2103.12486
+
+7
+[hep-ex]].
+[27] A. X. Dai, Z. Y. Li, L. Chang and J. J. Xie, Chin. Phys.
+C 46, no.7, 073104 (2022) doi:10.1088/1674-1137/ac5f9c
+[arXiv:2112.06264 [hep-ph]].
+[28] Q. H. Yang, L. Y. Dai, D. Guo, J. Haidenbauer, X. W. Kang and
+U. G. Meißner, [arXiv:2206.01494 [nucl-th]].
+[29] X. Cao, J. P. Dai and H. Lenske, Phys. Rev. D 105,
+no.7, L071503 (2022) doi:10.1103/PhysRevD.105.L071503
+[arXiv:2109.15132 [hep-ph]].
+[30] G. P. Lepage and S. J. Brodsky, Phys. Rev. Lett. 43, 545-
+549 (1979) [erratum: Phys. Rev. Lett. 43, 1625-1626 (1979)]
+doi:10.1103/PhysRevLett.43.545
+[31] G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980)
+doi:10.1103/PhysRevD.22.2157
+[32] Z. Y. Li and J. J. Xie, Commun. Theor. Phys. 73, no.5, 055201
+(2021)
+doi:10.1088/1572-9494/abea0f
+[arXiv:2012.02379
+[hep-ph]].
+[33] C.
+F.
+Perdrisat,
+V.
+Punjabi
+and
+M.
+Vander-
+haeghen,
+Prog.
+Part.
+Nucl.
+Phys.
+59,
+694-764
+(2007)
+doi:10.1016/j.ppnp.2007.05.001
+[arXiv:hep-ph/0612014
+[hep-ph]].
+[34] A. Denig and G. Salme, Prog. Part. Nucl. Phys. 68, 113-
+157 (2013) doi:10.1016/j.ppnp.2012.09.005 [arXiv:1210.4689
+[hep-ex]].
+[35] S.
+Pacetti,
+R.
+Baldini
+Ferroli
+and
+E.
+Tomasi-
+Gustafsson,
+Phys.
+Rept.
+550-551,
+1-103
+(2015)
+doi:10.1016/j.physrep.2014.09.005
+[36] V. Punjabi,
+C. F. Perdrisat, M. K. Jones,
+E. J. Brash
+and
+C.
+E.
+Carlson,
+Eur.
+Phys.
+J.
+A
+51,
+79
+(2015)
+doi:10.1140/epja/i2015-15079-x [arXiv:1503.01452 [nucl-ex]].
+[37] A.
+Mangoni,
+SciPost
+Phys.
+Proc.
+10,
+021
+(2022)
+doi:10.21468/SciPostPhysProc.10.021
+[arXiv:2110.12475
+[hep-ex]].
+[38] M. Irshad, D. Liu, X. Zhou and G. Huang, Symmetry 14, no.1,
+69 (2022) doi:10.3390/sym14010069
+[39] M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson and
+D. B. Lichtenberg, Rev. Mod. Phys. 65, 1199-1234 (1993)
+doi:10.1103/RevModPhys.65.1199
+[40] B. Kubis and U. G. Meissner, Eur. Phys. J. C 18, 747-756
+(2001)
+doi:10.1007/s100520100570
+[arXiv:hep-ph/0010283
+[hep-ph]].
+[41] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 124, no.4,
+042001
+(2020)
+doi:10.1103/PhysRevLett.124.042001
+[arXiv:1905.09001 [hep-ex]].
+[42] L. Xia, C. Rosner, Y. D. Wang, X. Zhou, F. E. Maas, R. B. Fer-
+roli, H. Hu and G. Huang, Symmetry 14, no.2, 231 (2022)
+doi:10.3390/sym14020231 [arXiv:2111.13009 [hep-ex]].
+[43] X. Zhou, L. Yan, R. B. Ferroli and G. Huang, Symmetry 14,
+no.1, 144 (2022) doi:10.3390/sym14010144
+[44] B. Aubert et al. [BaBar], Phys. Rev. D 73, 012005 (2006)
+doi:10.1103/PhysRevD.73.012005
+[arXiv:hep-ex/0512023
+[hep-ex]].
+[45] B. Aubert et al. [BaBar],
+Phys. Rev.
+D 76,
+092006
+(2007)
+doi:10.1103/PhysRevD.76.092006
+[arXiv:0709.1988
+[hep-ex]].
+[46] R.
+G.
+Sachs,
+Phys.
+Rev.
+126,
+2256-2260
+(1962)
+doi:10.1103/PhysRev.126.2256
+[47] J. R. Green, J. W. Negele, A. V. Pochinsky, S. N. Syrit-
+syn, M. Engelhardt and S. Krieg, Phys. Rev. D 90, 074507
+(2014)
+doi:10.1103/PhysRevD.90.074507
+[arXiv:1404.4029
+[hep-lat]].
+[48] S.
+Dobbs,
+A.
+Tomaradze,
+T.
+Xiao,
+K.
+K.
+Seth
+and
+G.
+Bonvicini,
+Phys.
+Lett.
+B
+739,
+90-94
+(2014)
+doi:10.1016/j.physletb.2014.10.025
+[arXiv:1410.8356
+[hep-
+ex]].
+[49] J. Haidenbauer, X. W. Kang and U. G. Meißner, Nucl. Phys.
+A 929, 102-118 (2014) doi:10.1016/j.nuclphysa.2014.06.007
+[arXiv:1405.1628 [nucl-th]].
+[50] A. B. Arbuzov and T. V. Kopylova, JHEP 04, 009 (2012)
+doi:10.1007/JHEP04(2012)009 [arXiv:1111.4308 [hep-ph]].
+[51] R. Baldini,
+S. Pacetti,
+A. Zallo and A. Zichichi,
+Eur.
+Phys. J. A 39, 315-321 (2009) doi:10.1140/epja/i2008-10716-1
+[arXiv:0711.1725 [hep-ph]].
+[52] R. Baldini Ferroli, S. Pacetti and A. Zallo, Eur. Phys. J. A 48,
+33 (2012) doi:10.1140/epja/i2012-12033-6 [arXiv:1008.0542
+[hep-ph]].
+[53] Q. F. Cao, H. R. Qi, Y. F. Wang and H. Q. Zheng, Phys. Rev. D
+100, no.5, 054040 (2019) doi:10.1103/PhysRevD.100.054040
+[arXiv:1906.00356 [hep-ph]].
+[54] R. L. Workman et al. [Particle Data Group], PTEP 2022,
+083C01 (2022) doi:10.1093/ptep/ptac097
+[55] H. Atac, M. Constantinou,
+Z. E. Meziani, M. Paolone
+and N. Sparveris, Nature Commun. 12, no.1, 1759 (2021)
+doi:10.1038/s41467-021-22028-z
+[arXiv:2103.10840
+[nucl-
+ex]].
+[56] A. N. Hiller Blin, Phys. Rev. D 96, no.9, 093008 (2017)
+doi:10.1103/PhysRevD.96.093008
+[arXiv:1707.02255
+[hep-
+ph]].
+[57] M. Yang and P. Wang, Phys. Rev. D 102, no.5, 056024
+(2020) doi:10.1103/PhysRevD.102.056024 [arXiv:2005.11971
+[hep-ph]].
+[58] G.
+Wagner,
+A.
+J.
+Buchmann
+and
+A.
+Faessler,
+Phys.
+Rev. C 58, 3666-3669 (1998) doi:10.1103/PhysRevC.58.3666
+[arXiv:nucl-th/9809015 [nucl-th]].
+[59] M. Ablikim et al. [BESIII], Chin. Phys. C 44,
+no.4,
+040001
+(2020)
+doi:10.1088/1674-1137/44/4/040001
+[arXiv:1912.05983 [hep-ex]].
+

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+EPJ manuscript No.
+(will be inserted by the editor)
+Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p
+Collisions
+R. Abou Yassine6,13, O. Arnold10,9, M. Becker11, P. Bergmann5, A. Blanco1, C. Blum8, M. B¨ohmer10, N. Carolino1,
+L. Chlad14,c, P. Chudoba14, I. Ciepał3, J. Dreyer7, W. Esmail5, L. Fabbietti10,9, P. Fonte1,a, J. Friese10, I. Fr¨ohlich8,
+T. Galatyuk6,5, J. A. Garz´on15, M. Grunwald17, M. Gumberidze5, S. Harabasz6,b, C. H¨ohne11,5, F. Hojeij13, R. Holzmann5,
+H. Huck8, M. Idzik2, B. K¨ampfer7,c, B. Kardan8, V. Kedych6, I. Koenig5, W. Koenig5, M. Kohls8, J. Kolas17, G. Korcyl4,
+G. Kornakov17, R. Kotte7, W. Krueger6, A. Kugler14, T. Kunz10, R. Lalik4, F. Linz6,5, L. Lopes1, M. Lorenz8, A. Malige4,
+J. Markert5, V. Metag11, J. Michel8, A. Molenda2, C. M¨untz8, M. Nabroth8, L. Naumann7, K. Nowakowski4, J. Orli´nski16,
+J.-H. Otto11, Y. Parpottas12, M. Parschau8, V. Pechenov5, O. Pechenova5, K. Piasecki16, J. Pietraszko5, A. Prozorov14,d,
+W. Przygoda4, B. Ramstein13, N. Rathod17, J. Ritman5, A. Rost6,5, A. Rustamov5, P. Salabura4, N. Schild6, E. Schwab5,
+F. Seck6, U. Singh4, S. Spies8, M. Stefaniak17,5, H. Str¨obele8, J. Stroth8,5, C. Sturm5, K. Sumara4, O. Svoboda14, M. Szala8,
+P. Tlusty14, M. Traxler5, H. Tsertos12, V. Wagner14, A.A. Weber11, C. Wendisch5, H.P. Zbroszczyk17, E. Zherebtsova5,e,
+M. Zielinski4, and P. Zumbruch5 (HADES collaboration)
+1 LIP-Laborat´orio de Instrumentac¸˜ao e F´ısica Experimental de Part´ıculas 3004-516 Coimbra, Portugal
+2 AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krak´ow, Poland
+3 Institute of Nuclear Physics, Polish Academy of Sciences, 31342 Krak´ow, Poland
+4 Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059 Krak´ow, Poland
+5 GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany
+6 Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany
+7 Institut f¨ur Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany
+8 Institut f¨ur Kernphysik, Goethe-Universit¨at, 60438 Frankfurt, Germany
+9 Excellence Cluster ’Origin and Structure of the Universe’, 85748 Garching, Germany
+10 Physik Department E62, Technische Universit¨at M¨unchen, 85748 Garching, Germany
+11 II.Physikalisches Institut, Justus Liebig Universit¨at Giessen, 35392 Giessen, Germany
+12 Department of Physics, University of Cyprus, 1678 Nicosia, Cyprus
+13 Laboratoire de Physique des 2 infinis Ir`ene Joliot-Curie, Universit´e Paris-Saclay, CNRS-IN2P3., F-91405 Orsay, France
+14 Nuclear Physics Institute, The Czech Academy of Sciences, 25068 Rez, Czech Republic
+15 LabCAF. F. F´ısica, Univ. de Santiago de Compostela, 15706 Santiago de Compostela, Spain
+16 Uniwersytet Warszawski - Instytut Fizyki Do´swiadczalnej, 02-093 Warszawa, Poland
+17 Warsaw University of Technology, 00-662 Warsaw, Poland
+a also at Coimbra Polytechnic - ISEC, Coimbra, Portugal
+b also at Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Darmstadt, 64390 Darmstadt, Germany
+c also at Technische Universit¨at Dresden, 01062 Dresden, Germany
+d also at Charles University, Faculty of Mathematics and Physics, 12116 Prague, Czech Republic
+e also at University of Wrocław, 50-204 Wrocław, Poland
+e-mail: hades-info@gsi.de
+Abstract The production of Σ0 hyperons in proton proton collisions at a beam kinetic energy of 3.5 GeV impinging
+on a liquid hydrogen target was investigated using data collected with the HADES setup. The total production cross
+section is found to be σ(pK+Σ0)[µb] = 17.7 ± 1.7(stat) ± 1.6(syst). Differential cross section distributions of the
+exclusive channel pp → pK+Σ0 were analyzed in the center-of-mass, Gottfried-Jackson and helicity reference frames
+for the first time at the excess energy of 556 MeV. The data support the interplay between pion and kaon exchange
+mechanisms and clearly demonstrate the contribution of interfering nucleon resonances decaying to K+Σ0. The Bonn-
+Gatchina partial wave analysis was employed to analyse the data. Due to the limited statistics, it was not possible
+to obtain an unambiguous determination of the relative contribution of intermediate nucleon resonances to the final
+state. However nucleon resonances with masses around 1.710 GeV/c2 (N∗(1710)) and 1.900 GeV/c2 (N∗(1900) or
+∆∗(1900)) are preferred by the fit.
+arXiv:2301.11766v1  [nucl-ex]  27 Jan 2023
+
+2
+R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+1 Introduction
+Strangeness production at intermediate energies in p+p and
+p+A collisions is of particular importance to the field of hadron
+physics. The production of baryons with strange quark content,
+i.e. hyperons, requires creating a new quark flavor, which can
+occur out of the vacuum from the quark sea in the colliding
+protons. The s-quark with mass
+O(ΛQCD) is distinguished
+from the light (u, d) quark flavors but much smaller than heavy
+(c, t, b) flavors. The resulting (approximate) SU(3) flavor
+symmetry in the u-d-s sector is therefore still a cornerstone of
+hadron physics. Since the entrance channel in p+p and p+A
+collisions carries no net strangeness, the emergence of an
+s-quark can unravel much of the flavor dynamics in hadronic
+reactions. The flavor-conserving strong interaction process
+requires associate strangeness production, e.g. realized by
+simultaneous creation of a single-strange hyperon, such as
+Λ or Σ0 and an associated kaon. Therefore, understanding
+the production mechanism of strange baryons near threshold
+deepens our knowledge of their internal structure and of the
+strong interaction in the non-perturbative regime. Strangeness
+production is also used as a probe to study hot and dense
+nuclear matter in heavy ion collisions both at medium-energy
+and in the late stages prior to freeze-out in high-energy
+collisions, e.g. at LHC [1].
+The production of the Λ hyperon in p+p and p+A reac-
+tions near threshold has been studied extensively by many ex-
+periments including HADES [2, 3, 4, 5, 6], yet there are only
+few experimental investigations on the Σ0 hyperon [2, 7]. De-
+spite there are considerable experimental results and numerous
+dedicated theoretical investigations, the strangeness production
+mechanism is not yet well understood. In the context of the bo-
+son exchange model [8, 9, 10, 11],
+it is assumed that the initial protons exchange a virtual me-
+son. The interaction between the meson and the initial protons
+results in the production of the final state particles, which can
+proceed directly or via an intermediate resonance.
+The exchange of a virtual meson can be put into one of
+two categories. The first category is strange meson exchange,
+where strangeness exchange occurs, and no resonances are
+involved. In this case, the reaction amplitude KN → KN
+is governed by t-channel diagrams. The second category is
+non-strange meson exchange, a pion exchange in its simplest
+form. At the same time the elementary reaction amplitude
+πN → KY is dominated by resonance excitations, which
+implies a strong and characteristic energy dependence, where
+Y stands for hyperons (Λ, Σ0, ...).
+Several experiments have studied the exclusive reaction
+pp → pK+Λ and proven that a pure phase space model de-
+scription of the data is not sufficient without taking the dynam-
+ics of the process into account [2, 6, 12, 13]. It was found that
+the Λ hyperon production is dominated by the excitation and
+subsequent decay of N∗ resonances to the K+Λ decay chan-
+nel. In particular N∗(1650) (JP= 1
+2
+−), N∗(1710) (JP= 1
+2
++)
+and N∗(1720) (JP= 3
+2
++) were found to contribute. This sup-
+ports a picture wherein the exchange of non-strange mesons
+is the leading process in the production mechanism. In addi-
+tion, a considerable Final State Interaction (FSI) was found
+to contribute [14, 15] leading to ΣN → ΛN conversion that
+is observed as a ΣN cusp effect in the Λ cross section [16].
+In the pp → pK+Σ0 reaction the proton–hyperon FSI seems
+to be negligible, especially at low energies near threshold and
+a pure phase space distribution describes the data reasonably
+well. The cross section ratio σ(pK+Λ) / σ(pK+Σ0) below ex-
+cess energies of ∼ 20 MeV is about 28 in agreement of the
+SU(6) prediction and reduces drastically to about 2.5 at excess
+energies above 300 MeV [17, 18]. This energy-dependence of
+the cross section ratio is strongly affected by FSI effects in the
+pp → pK+Λ reaction [19].
+Besides the energy dependence of the cross section, the
+differential cross sections at selected energies add much more
+stringent tests of the model descriptions. This study fills this
+gap and delivers such data which allow some clues about the
+involved exchange mesons and resonances, in particular by em-
+ploying a partial wave analysis.
+Furthermore,
+a
+theoretical
+study
+of
+the
+reaction
+pp → pK+Σ0
+based on a chiral dynamical study has
+been proposed in [20]. This approach uses the pion and kaon
+exchange mechanisms and chiral amplitudes in addition to all
+pairs FSI, where the contribution of nucleon resonances appear
+naturally using chiral unitary amplitudes.
+This paper is organized as follows. In Section 2, the experi-
+mental setup is briefly explained. Section 3 is devoted to the Σ0
+selection method, where the particle identification, the Λ hy-
+peron reconstruction and the kinematic refit methods were pre-
+sented. In Section 3.5 the method for efficiency correction and
+differential analysis is described. Sections 5 and 6 presents the
+calculated total production cross section and the partial wave
+analysis of the exclusive reaction pp → pK+Σ0. In Section 7
+a summary and a short outlook are given.
+2 The HADES experiment
+The data presented here were collected in April 2007 with the
+High Acceptance Di-Electron Spectrometer (HADES) located
+at the heavy ion synchrotron SIS18 at GSI Helmholtzzentrum
+f¨ur Schwerionenforschung in Darmstadt, Germany. HADES is
+characterized by six identical sectors covering almost the full
+azimuthal range and polar angles from θ = 18◦ to θ = 85◦. Each
+sector of the spectrometer contains a Ring-Imaging Cherenkov
+Detector (RICH) operating in a magnetic field-free region that
+allows lepton identification over a wide range of momenta. Two
+Multi-Wire Drift Chambers (MDCs) are placed in front of a
+toroidal magnetic field, and two outer MDCs are placed behind
+the magnetic field. The MDCs enable the momentum informa-
+tion and the specific energy loss dE/dx to be reconstructed for
+each particle track. Two scintillator hodoscopes, the Time Of
+Flight (TOF) and TOFino are also placed behind the magnet
+and provide a stop time (ts) signal. The TOF and TOFino sys-
+tem are used as input to the trigger systems to start the data
+readout. A detailed description of the HADES setup can be
+found in [21].
+In the present analysis, a proton beam with an intensity of
+107 particles/s and kinetic energy T = 3.5 GeV was incident on
+a liquid hydrogen target with an areal density of 0.35 g/cm2.
+The dimensions of the target were 15 mm in diameter and 50
+mm length located between -65 to -15 mm in the longitudi-
+nal direction. The data readout was started by a first level trig-
+
+R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+3
+ger requiring a charged particle multiplicity ≥ 3 (M3). In total,
+1.14 × 109 events were recorded under these conditions [3].
+During this experiment HADES included an additional For-
+ward Wall (FW) scintillator hodoscope that was placed 7 me-
+ters downstream the target in a magnetic field-free region and
+covered polar angles from θ = 0.33◦ to θ = 7.17◦ with full az-
+imuthal acceptance. The FW measured the hit position and ar-
+rival time of the particle track with a time resolution of about
+700 ps [22].
+3 Event selection method
+In this section, the exclusive reconstruction of the reaction
+pp → pK+Σ0 is presented. The Σ0 hyperon is reconstructed
+via its electromagnetic decay Σ0 → Λγ (BR ≈ 100%) and the
+daughter Λ hyperon is reconstructed with the decay mode
+Λ → pπ− (BR = 63.9%).
+The Σ0 reconstruction strategy includes the following steps:
+a) time of flight (tof) reconstruction, b) charged particle iden-
+tification (PID), c) the Λ hyperon reconstruction, and d) the Σ0
+hyperon reconstruction.
+3.1 Time of flight reconstruction
+The interaction of the high intensity proton beam with the
+START detector induced a background and prevented a stable
+operation of the RICH detector. Therefore, it was not possible
+to use the START detector information during this experiment.
+Consequently, the tof of particle tracks were not directly mea-
+sured since there was no common start time (t0) reference for
+tracks in the same event. The start time has to be reconstructed
+in order to obtain a proper time of flight measurement.
+The reconstruction algorithm is based on the assumption
+that at least one particle has been correctly identified. Since pi-
+ons are abundantly produced, it is assumed that any negatively
+charged particle track that is geometrically uncorrelated to a
+ring in the RICH detector is a π−. The common start time for
+each event is calculated by
+t0 = ts − d
+c ×
+�
+p2 + m2π
+p
+,
+where ts is the stop time of the π−, d is the distance to the TOF
+or TOFino hit, mπ is the pion mass, p is the momentum of the
+π− and c is the velocity of light. The tof of the other particles
+in the same event is the difference between the measured stop
+time ts and the common start time t0.
+3.2 Particle identification (PID)
+The reconstruction of the exclusive reaction pp → pK+pπ−γ
+only requires the identification of three particle species, pions
+(π−), kaons (K+) and protons (p), since the event is kinemati-
+cally complete even without measuring the photon (γ).
+As mentioned in the previous section, the π− is identified
+as any negatively charged track that is geometrically uncorre-
+lated to a ring in the RICH detector. Therefore, the problem
+reduces to identifying the positively charged tracks.
+In order to minimize systematic bias in the model output,
+an auto-encoder [23] implemented in PyTorch framework [24]
+is trained simultaneously with both simulated and real events
+[25]. The input features used to train the auto-encoder are the
+momentum components, the energy loss dE/dx in the MDC and
+TOF sub-systems, the reconstructed tof and the distance to the
+TOF/TOFino hit.
+A classification layer has been stacked on top of the
+bottleneck layer of the auto-encoder, which has three output
+nodes corresponding to the three classes (π+, K+ and p). Each
+node outputs a number between 0 and 1, where all output
+numbers sum to 1, so that each number can be interpreted as
+a probability of being a specific particle species. The network
+is trained by minimizing a cost function that is defined as the
+binary cross-entropy loss [26]. Because the network outputs
+three probabilities for each particle track, the node with the
+largest probability is chosen.
+The classification accuracy evaluated on a holdout data-set
+is 92% for pions, 76% for kaons and 98% for protons. It is
+much lower in the case of kaons since their production rate is
+suppressed with respect to the protons and pions.
+3.3 Λ hyperon reconstruction
+The next step after the PID is to reconstruct the intermediate Λ
+hyperon. In this analysis the Λ reconstruction method is two-
+fold. In the first case, which is referred to as the Spectrometer
+data-set, events with exactly 2 protons, 1 pion and 1 kaon are
+required to be within the main HADES detector acceptance.
+The other case, referred as the WALL data-set, events were ac-
+cepted if exactly 1 proton, 1 pion and 1 kaon are registered in
+HADES and in addition one hit in the FW. In the latter case, it is
+assumed that the hit registered in the FW is due to the daughter
+proton from the Λ decay (marked as pdecay).
+A common primary vertex in each event is then defined as
+the intersection point or the Point of Closest Approach (PCA)
+of the proton and kaon tracks. Since there is more than one pro-
+ton in each event in the Spectrometer data-set, the proton-kaon
+pair with the smaller Distance of Closest Approach (DCA) is
+used to define the primary vertex. To reduce the contribution
+from off-target events, a two dimensional selection is applied
+on the primary vertex position (x, y, z):
+a) -65 < z [mm] < -15 and
+b)
+�
+x2 + y2 < 5 [mm].
+The Spectrometer data-set
+Since the daughter Λ decays weakly (cτ = 7.89 cm), it can be
+identified by its displaced vertex. First, all possible combina-
+tions of the two p and π− candidates were made, leaving the
+decision about which is the decay proton (Λ → p π−) for later.
+For each combination the decay vertex (the displaced vertex) is
+defined as the PCA between the two tracks. The DCA between
+the p and π− tracks (marked as dpπ−) is expected to be small if
+the tracks stem from the same vertex. Therefore, an upper limit
+of dpπ− < 10 mm is imposed in order to reduce Combinatorial
+
+4
+R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+Figure 1: (a) The DCA distribution between the p and π− tracks. (b) The DCA distribution between the Λ track and the primary
+vertex. In both panels, data are shown by the black points, the blue histogram represents the true Λ and the red histogram
+represents the CB, where both the true Λ and the CB were estimated from the simulation. (c) Distribution of the DCA between
+the π− track and the primary vertex as a function of the DCA between the p track and the primary vertex. The arrows indicate
+the accepted regions.
+Background (CB), which originates from combining the wrong
+p and π− pairs. Considering momentum and energy conserva-
+tion, the p should be emitted in nearly the same direction as the
+Λ in the laboratory reference frame, while the π− will have a
+different direction. Thus, the DCA between the p track and the
+primary vertex (dp,pvtx) is required to be smaller than the DCA
+between the π− track and the primary vertex (dπ−,pvtx). Fi-
+nally, the DCA between the calculated Λ track and the primary
+vertex (dΛ,pvtx) is required to be < 6 mm. The distributions
+of the topological variables are shown in Figure 1, where the
+selection criteria are indicated by the vertical dashed lines. The
+proton used in the Λ reconstruction is tagged as the decay pro-
+ton (marked in the following as pdecay), while the other proton
+in the event is tagged as the scattered (primary) proton.
+To further purify the selected Λ sample, the event kinemat-
+ics were constrained to the Σ0 production range. The squared
+pΛ missing mass (MM2(ppdecayπ−)) is required to be > 0.2
+GeV/c2 in order to reject the multi-pion production channel
+as shown in Figure 2. In this figure, the experimental data
+are shown by the black points and the simulations (discussed
+in Section 3.4) by different colored histograms. Two peaks
+are visible, the first peak at 0.02 GeV/c2 corresponds to the
+multi-pion channel via the reaction pp → ppπ+π− (violet
+histogram), where a pπ− pair is incorrectly identified as a Λ
+candidate and the π+ is incorrectly identified as a K+. The
+other broader peak is the sum of pp → pK+Λ, pp → pK+Σ0
+and pp → pK+Λπ0 reactions shown by the red, blue and
+green histograms, respectively. The relative normalizations
+of the simulated channels have been chosen to best fit the
+experimental data as explained in Section 3.4.
+The pdecay π− invariant mass distribution is shown in
+Figure 3. A peak around the nominal Λ mass is visible on
+top of background. The signal has been parameterized by
+a Voigt distribution and the background is modeled by a
+fourth-order polynomial. Events are further processed if they
+are in the range of µ ± 3σ, where the calculated signal to
+background ratio in this range is S/B = 2.57 and the number
+of Λ candidates is NΛ = 6766.
+The WALL data-set
+In the WALL data-set the hit in the FW is assumed to be due
+to the decay proton. Since the FW is installed in a magnetic
+field-free region, the pdecay is reconstructed as a straight line
+trajectory from the primary vertex position to the hit in the FW.
+The track momentum is calculated from the tof and the dis-
+tance from the primary vertex and the FW detector hit, assum-
+ing the proton mass. In this case, the topological cuts are not
+as effective to suppress the background as in the Spectrometer
+data-set. Therefore, events fulfilling the following kinematical
+constraints were selected:
+(i) MM2(ppdecayπ−) > 0.2 GeV/c2 (Figure 4 a) and
+(ii) The squared missing mass of all charged particles is
+required to be in the following range:
+−0.02 < MM2(pK+pdecayπ−)[GeV2/c4] < 0.01
+be-
+cause
+only a photon is missing to completely measure the
+exclusive final state (Figure 4 b).
+
+(a)
+(b)
+(c)
+108
+108
+15
+×103
+L Data
+4.5
+HADES
+TrueA
+4
+p(3.5 GeV)+p → pK+z0
+106
+106
+CB
+3.5
+ww
+0.2
+3
+2.5
+104
+Events 
+2
+1.5
+102
+102
+1
+0.5
+0
+0
+50
+100
+0
+20
+40
+0
+5
+10
+15
+[mm]
+d(p, pvtx) [mm]R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+5
+Figure 2: The squared ppdecay π− missing mass distribu-
+tion after applying the topological selections. Black points are
+the Spectrometer data-set data. The violet histogram is the
+pp → ppπ+π− simulation. The pp → pK+Λ, pp → pK+Σ0
+and pp → pK+Λπ0 simulations are shown by the red, blue and
+green histograms, respectively. The vertical line and the arrow
+indicate the accepted region for the further analysis.
+The pdecayπ− invariant mass distribution for the WALL
+data-set is shown Figure 5 after applying the selections
+mentioned above. Once again, the peak has been fitted by
+a Voigt distribution and the background by a fourth-order
+polynomial. However, the mass resolution of the Λ peak of
+the Spectrometer data-set(Figure 3) is better than the signal
+of the WALL data-set, since in the latter case the proton was
+detected in the FW, which has a worse momentum resolution.
+Events are further processed if they are in the range of µ ± 3σ,
+where the calculated signal to background ratio in this range is
+S/B = 1.56 and the number of Λ candidates is NΛ = 2340.
+3.4 Σ0 hyperon reconstruction
+To further suppress the remaining background and to obtain a
+better mass resolution, a kinematic fit based on the Lagrange
+multiplier method is employed [27]. The fit χ2, expressed as
+χ2(η, λ) = (y − η)T V (y)(y − η) + 2λT f(η) ,
+is minimized by differentiating χ2 with respect to all measured
+variables. Here y is a vector containing the initial guesses for
+the measured quantities, which are the track parameters pro-
+vided by the tracking algorithm, η is an improved set of the
+track parameters and V is the covariance matrix comprising
+the estimated errors on the measured quantities. The constraint
+Figure 3: The pdecay π− invariant mass distribution. The verti-
+cal dashed lines indicate the selected mass range. The blue, red
+and green curves are for the signal, background and the total
+fit.
+equations are expressed as a function of η in f(η), where λi are
+a set of Lagrange multipliers.
+The spherical coordinates used in this analysis for the track
+parameterization are defined as follows
+y =
+�
+�
+1/p
+θ
+φ
+�
+� ,
+where 1/p is the inverse of the absolute momentum, θ and φ
+are the polar and azimthual angles of the track.
+Two constraints were applied to both data-sets. The first is
+the proton and pion from the Λ decay are constrained to the Λ
+mass (MΛ = 1.1157 GeV/c2). The second constraint is that
+the missing mass of all the charged final state particles is con-
+strained to the photon mass (Mγ = 0 GeV/c2).
+The probability that a χ2 of the theoretical distribution is
+greater than or equal to the χ2 value found from the fit is known
+as the p-value (P(χ2)). The p-value distributions of the Spec-
+trometer and the WALL data-sets are shown in Figure 6. Be-
+cause both Λ and Σ0 have MM(pK+Λ) = 0, they have simi-
+lar distributions, which makes these two reactions difficult to
+distinguish. On the other hand, the reaction pp → pK+Λπ0
+should ideally have zero p-value. However, due to the limited
+detector resolution it has p-values greater than zero, which is
+more pronounced in the WALL data-set. The signal events show
+an almost flat distribution between 0 and 1, while events that do
+not satisfy the constraint equations have a prominent yield of
+p-values close to 0. Therefore, events with p-values > 0.01 are
+selected, where the cut was optimized based on a significance
+analysis.
+
+X103
+1.2
+ Data
+HADES
+pK+0
+p(3.5 GeV)+p -→ pK+z0
+Pp元+元
+pK+^
+pK+^元°
+0.8
+0.01
+0.6
+Events /
+0.4
+0.2
+-0.5
+0
+0.5
+MM?
+(pp
+元)[GeV2/c4]
+decayX103
+1.
+Mean
+1.114
+Sigma
+0.002
+HADES
+1.2
+S/B
+2.57
+p(3.5 GeV)+p → pK+≥0
+Na
+6766
+2
+/ 0.001 GeV/c
+0.8
+0.6
+Events /
+0.4
+0.2
+1.09
+1.1
+1.11
+1.12
+1.13
+1.14
+1.15
+M.
+[GeV/c"]
+decay6
+R. Abou Yassine et al.: Investigation of the ��0 Production Mechanism in p(3.5 GeV)+p Collisions
+Figure 4: (a) The squared ppdecay π− missing mass distribution of WALL data-set. (b) The squared ppdecay π− K+ missing mass
+distributions. The pp → ppπ+π−, pp → pK+Λ, pp → pK+Σ0 and pp → pK+Λπ0 simulations are shown by the violet, red,
+blue and green histograms, respectively. The arrows indicate the accepted regions.
+Simulation scaling to the experimental data
+By inspecting the pK+ missing mass distribution of the com-
+bined data-set shown in Figure 7, two peaks corresponding to
+the Λ and the Σ0, as well as other minor contributions in the
+high mass region, are plainly evident. In order to quantify the
+different contributions an incoherent cocktail has been simu-
+lated using the Pluto event generator [28]. All the simulated re-
+actions have been processed using the same full scale analysis
+employed for the experimental data, thus taking into account
+the efficiency of the trigger condition, the tracking algorithm
+and the analysis procedure. The particle decays, the acceptance
+and the particle interactions with the materials of HADES and
+the FW have been considered by using GEANT3 [29].
+To determine the contributions of the different channels, a
+fit of the simulations to the measured missing mass spectrum
+(MM(pK+)) has been carried out by minimizing the quantity
+χ2 =
+nbins
+�
+i
+(ndata − �
+ch(f ch × nch
+simulation))2
+σ2
+data + σ2
+simulation
+,
+where the summation runs over the number of bins of the miss-
+ing mass spectrum, ndata is the number of data events in each
+bin, nch
+simulation is the number of simulated events in each bin
+for each channel and fch is a scaling factor for each channel.
+The uncertainty for the data and the simulations in each bin is
+σdata and σsimulation, respectively.
+As can be seen from Figure 7, the experimental data is
+primarily described by contributions of pp → pK+Λ,
+pp → pK+Σ0 and pp → pK+Λπ0 indicated by the red, blue
+and the green histogram, respectively. The other simulated
+channels have minor contributions. In total 2613 Σ0 candidates
+were collected within the pK+ missing mass range of 1.170-
+1.220 GeV/c2, 58% of them are within the main HADES
+acceptance and 42% within the FW acceptance. The signal
+purity in the mass window calculated from the simulation is
+found to be 81%, where the main background contributions are
+the reactions pp → pK+Λ (14%) and pp → pK+Λπ0 (5%).
+3.5 Efficiency and acceptance correction
+The reconstructed experimental distributions are corrected for
+the limited detector acceptance and efficiency by using a sim-
+ulated phase space distribution that were assigned a weight de-
+termined by the best partial wave solution (discussed in Sec-
+tion 6), then the events were filtered through the full scale simu-
+lation and analysis. The efficiency correction is done in one di-
+mension whereas the other three degrees of freedom on which
+the efficiency depends are integrated.
+The 1D correction matrix (M) is calculated given the ini-
+tial 4π distribution (T) for each observable and after filtering
+
+(a)
+(b)
+X103
+X103
+ Data
+45
+HADES
+pK+0
+5
+40
+p(3.5 GeV)+p → pK+≥0
+pp元+元
+35
+pK+A
+Events / 0.002 GeV2/c4
+ pK+A元0
+4
+30
+25
+3
+20
+2
+15
+10F
+5
+0
+-1
+-0.5
+0
+0.5
+-0.1
+-0.05
+0
+0.05
+0.1
+MM2 (pp
+元) [GeV?/c4]
+MM? (pK*p.,) [GeV2/c*]R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+7
+Figure 5: The pwallπ− invariant mass distribution. The vertical
+dashed lines indicate the selected mass range. The blue, red and
+green curves are for the signal, background and the total fit.
+through the full scale simulation and analysis (R). To put it an-
+other way, a distinct correction matrix M = R/T is constructed
+for each angular distribution shown in Figure 8. The inverse
+of the correction matrix is then calculated using the Singular
+Value Decomposition (SVD) technique [30] implemented in
+RooUnfold framework [31].
+3.6 Absolute normalization and systematic
+uncertainties
+The production cross section of Σ0 can be calculated by nor-
+malizing the corrected Σ0 yield to the p+p elastic scattering
+yield measured in the same experimental run [32]. This nor-
+malization results in a systematic uncertainty of 7%. In addi-
+tion, there might be other sources of systematic uncertainty.
+The systematic error associated to the exclusive event selection
+has been estimated by varying the selection ranges and recal-
+culating the cross section.
+To test the influence of different selection cuts on the calcu-
+lated cross section (see section 5), the whole analysis chain has
+been repeated many times under different cut combinations.
+Each cut is varied in two steps in either direction. the cross
+section for each combination is then calculated by integrating
+the yield of the cosθ∗
+Σ0 angular distribution. Following this pro-
+cedure the obtained systematic error, defined as the 1σ interval
+of the cross sections distribution, is found to be ≈ 2%.
+Another source of the systematic errors is the PID, which
+is evaluated by activating the dropout layers of the neural net-
+work during the inference time as this is equivalent to doing
+a Bayesian approximation [33]. The estimated size of the PID
+systematic is ≈ 5%.
+Table 1: Coefficients of Legendre polynomials determined by
+fitting the angular distributions presented in Figure 8.
+Angle
+A0 [µb]
+A1 [µb]
+A2 [µb]
+cosθ∗
+Σ0
+8.55 ± 0.31
+0.00
+2.75 ± 0.73
+cosθ∗
+p
+10.01 ± 0.50
+0.00
+4.33 ± 1.27
+cosθ∗
+K+
+9.83 ± 0.43
+0.00
+-0.13 ± 1.02
+cosθFRpΣ0
+pb,t,p
+10.40 ± 0.80
+-0.64 ± 1.73
+2.79 ± 1.85
+cosθFRK+Σ0
+pb,t,K+
+8.55 ± 0.71
+-1.61 ± 1.54
+0.66 ± 1.63
+cosθFRK+p
+pb,t,K+
+10.30 ± 1.00
+1.91 ± 1.18
+0.50 ± 2.69
+cosθFRK+Σ0
+p,Σ0
+8.70 ± 0.30
+3.17 ± 0.59
+-0.73 ± 0.75
+cosθFRpΣ0
+p,K+
+8.75 ± 0.29
+-3.52 ± 0.50
+0.37 ± 0.67
+cosθFRK+p
+K+,Σ0
+8.81 ± 0.31
+4.84 ± 0.56
+-0.98 ± 0.75
+4 Angular Distributions
+This section presents the differential cross section of the reac-
+tion pp → pK+Σ0, namely the angular distributions of final
+state particles in the center-of-mass (CMS) frame, as well as in
+both the Gottfried-Jackson and helicity frames of all two-body
+subsystems. All distributions are acceptance and efficiency cor-
+rected and then fit with Legendre polynomials dσ/dcosθ =
+�
+l Al · Pl, with l = 0, 1, 2. The coefficients A1 and A2 are
+used to judge the asymmetries and anisotropies of the observed
+distributions. The best description of the distribution (indicated
+by the blue histogram in Figure 8) was found when the sim-
+ulations have been weighted simultaneously with the angular
+distribution of the Σ0 hyperon in the CMS frame and the pro-
+ton Gottfried-Jackson angular distribution measured in the pΣ0
+rest frame obtained from the data.
+Center of mass frame
+The angular distributions of the three final state particles in the
+CMS are shown in the top row of Figure 8. The Legendre poly-
+nomial coefficients obtained from the fits of the angular distri-
+butions are listed in Table 1. Since the initial p+p is a symmetric
+system, the A1 Legendre parameters of all CMS distributions
+were set to zero. The angular distribution of the Σ0 hyperon
+(Figure 8 (a)) and proton (Figure 8 (b)) shows an anisotropy,
+where it is more pronounced for the proton as quantified by the
+A2 parameter listed in Table 1. From the observed anisotropies
+and the fit parameters one deduces that a non-zero orbital an-
+gular momentum (L) is observed in both the p − K+Σ0 and
+Σ0 − pK+ sub-systems. This is in contrast to the kaons, where
+the angular distribution is compatible with isotropy. For pure
+pion exchange, the final state proton is the leading particle,
+since the exchange pion has a small mass, implying a small
+4-momentum transfer so that the proton is preferably emitted
+in the direction of the initial protons, which could explain the
+anisotropy in the proton angular distribution. In this picture, the
+Σ0 CMS angular distribution reflects the proton one, while the
+kaon has a broader distribution.
+The angular distributions in the overall CMS are not suited
+to directly draw conclusions on resonant production, which
+proceeds as a two step process pp → pR, R → K+ Σ0, where R
+
+Mean
+1.114
+600
+Sigma
+0.005
+HADES
+S/B
+1.56
+p(3.5 GeV)+p -→ pK+Z0
+NA
+2340
+500
+2
+ieV/
+5400
+G
+.001
+0300
+Events
+2200
+100
+1.09
+1.1
+1.11
+1.12
+1.13
+1.14
+1.15
+1.08
+M.
+[GeV/c']
+元
+decay8
+R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+Figure 6: (a) The p-value distributions for the HADES data-set and for (b) the WALL data-set. The insets display the region
+of small p-values, where the dashed line and the arrow indicates the accepted region. The pp → pK+Λ, pp → pK+Σ0 and
+pp → pK+Λπ0 simulations are shown by the red, blue and green histograms, respectively.
+stands for every kind of nucleon resonance, that can be either
+an isospin 1/2 N∗ state or an isospin 3/2 ∆∗ state. Therefore
+in the following the Gottfreid-Jackson and helicity frames are
+presented as a more natural choice for the Lorentzian reference
+frames in order to study the reaction properties due to resonant
+production.
+Gottfried-Jackson frames
+The Gottfried-Jackson (G-J) frame first introduced in [34] is
+the rest frame of two out of the three produced particles. In the
+G-J frame, the G-J angle is defined as the angle between one
+of the rest frame particles (e.g. the Σ0) and the initial proton
+θRF K+Σ0
+pb,t,Σ0
+, where the label RF stands for reference frame, the
+superscript indicates which rest frame is used and the subscript
+stands for the two particles, between which the angle is mea-
+sured. It should be noted that the two initial protons are indis-
+tinguishable. Therefore, the angular distribution is calculated
+by using the angle to both protons (pb,t).
+In the case of kaon (pion) exchange, the K+p (K+Σ0) rest
+frame is equivalent to the rest frame of the exchanged meson
+and the initial proton. In this way, the initial 2 → 3 reaction is
+reduced to a pure 2 → 2 reaction. If there is a resonant produc-
+tion, the internal angular momentum of the resonance is then
+reflected in this observable. It has to be noted that the distri-
+butions in the G-J frames do not have to be symmetric. The
+reason is the asymmetric reaction system, where either a kaon
+or a pion collides with a proton. The angular distributions in
+the G-J frames are shown in the middle row of Figure 8.
+An anisotropy is observed in the pΣ0 G-J frame (Figure 8
+(d)), which could be due to a relative angular momentum in
+the pΣ0 system. This effect is related to the above mentioned
+anisotropies of the p and Σ0 CMS angular distributions since
+they are kinematically related. The angular distribution in the in
+K+Σ0 G-J frame (Figure 8 (e)) tends to be asymmetric, which
+could be caused by the excitation of nucleon resonances decay-
+ing into the K+Σ0 channel [2]. Many of N∗ or ∆∗ resonances
+could contribute to the reaction. All these resonances have large
+widths and may also contribute through their broad tails to the
+reaction. The angular distribution of a true two-body resonance
+reaction is asymmetric only if resonances with both parities are
+simultaneously excited through interfering amplitudes. Hence,
+this distribution in the K+Σ0 G-J frame indicates that more
+than one nucleon resonance with opposite parity participates in
+the production process [2]. As explained earlier, the K+p rest
+frame is equivalent to the rest frame of the exchanged kaon.
+Therefore, the deviation from isotropy in the cosθFRK+p
+pb,t,K+ an-
+gular distribution could be explained by kaon exchange com-
+ponent [25]. For a pure pion exchange, the Treiman-Yang (T-
+Y) angle measured in the K+Σ0 rest frame is expected to be
+an isotropic distribution [35]. Therefore, if a kaon exchange
+contributes to the production mechanism it should reflect it-
+
+(a)
+(b)
+108
+103
+ Data
+107
+Spectrometer
+107
+Wall
+HADES
+TTTT
+data-set
+pp → pK+z0
+data-set
+103
+pp →pK+△
+106
+106
+102
+Pp →pK+A元°
+105
+Events
+10
+104
+10
+10
+E103
+103
+0
+0.005
+0.01
+0.015
+0.02
+0
+0.005
+0.01
+0.015
+0.02
+102
+102
+10
+10
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+P(x2)
+P(×2)R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+9
+Figure 7: The pK+ missing mass distribution. The colored
+histograms represent the simulated channels, where Y∗ refers
+to an excited hyperon (Σ(1385), Λ(1405) or Λ(1520)). The
+two peaks are due to the exclusive reactions pp → pK+Λ and
+pp → pK+Σ0 as shown by the red and the blue histograms,
+respectively. The vertical dashed lines mark the mass window
+used to select candidate events of the pp → pK+Σ0 final state.
+self in this distribution. The Σ0 hyperon T-Y angle measured in
+the K+Σ0 rest frame, shown in Figure 9, shows a clear devia-
+tion from isotropy, which could be an indication of a significant
+kaon exchange contribution to the reaction mechanism.
+Helicity frames
+The helicity angle is defined in a similar way as the G-J angle,
+but instead of calculating the angle of the respective particle
+to the initial proton, the helicity angle is calculated between
+one of the rest frame particles and the third produced parti-
+cle. The helicity angular distribution thus interrelates the kine-
+matics of the three final state particles and it is thus a linear
+transformation projection of the Dalitz plot. A uniformly popu-
+lated Dalitz plot results in isotropic helicity angle distributions.
+On the other hand, if dynamical effects distort the Dalitz plot,
+then the helicity angular distribution will be anisotropic. The
+helicity angular distributions are shown in the bottom row of
+Figure 8. All the distributions are significantly non-isotropic,
+which indicates that the reaction is dominated by intermediate
+resonances. Therefore, an inclusion of intermediate resonances
+is necessary in order to quantitatively describe experimental an-
+gular distributions.
+Comparison to lower energy
+A comparison of the normalized Legendre coefficients between
+this measurement and data collected at a lower value of excess
+energy ϵ = 162 MeV [2] is listed Table 2. The two sets of coeffi-
+cients show striking differences for few coefficients indicating
+that the Σ0 production mechanism changes between these val-
+ues of excess energy. The CMS distributions are more forward-
+backward peaked for the proton and the Σ0 hyperon and less
+peaked for the kaon, pointing to a larger relative contribution
+of pion with respect to kaon exchange at larger energies. In ad-
+dition, the helicity angle distributions have a significant asym-
+metry at the highest energy, in contrast with the lower energy
+results.
+5 Total Cross Section
+The total production cross section as function of the excess
+energy ϵ is used as a tool to compare the experimental
+data to the different theoretical approaches. The result on
+the pp → pK+Σ0 production cross section, obtained by
+integrating the cosθ∗
+Σ0 angular distribution, is
+σ(pK+Σ0)[µb] = 17.7 ± 1.7(stat) ± 1.6(syst) .
+The cross section value is included in Figure 10, which
+shows a compilation of the pp → pK+Σ0 cross sections as
+a function of the excess energy. The present data point corre-
+sponds to ϵ = 556 MeV, which is depicted by the green square
+and existed in a region where no other measurements have been
+performed. This behaviour can not be described by phase space
+within experimental uncertainty as clearly seen by the solid
+curve σpK+Σ0 = Kϵ2, where the quadratic excess-energy de-
+pendence is attributed to a pure (i.e. trivial) three-body phase
+space and K is the fit free parameter.
+An alternative parametrization proposed by F¨aldt and
+Wilkin in [43] that takes the proton-hyperon FSI interaction
+into account
+σ = C ·
+ϵ2
+(1 +
+�
+1 + ϵ/α)2 ,
+where the parameters C = 7.82 × 102µb GeV −2 and α =
+4.57 × 10−2GeV are related to the FSI strength. Interestingly,
+the deviations to the pure phase space behavior start showing
+up at ϵ > 200 MeV. The displayed data in that region could also
+be approximated by σ ≈ 10 µb.
+A more appropriate paramerization proposed by Tsushima
+in [44] shown by the dotted line is based on a resonance model,
+where the hyperon is produced via an intermediate nucleon res-
+onance N∗ or ∆∗. This paramerization describes all data points
+near threshold up to 1.4 GeV fairly well.
+Using
+the
+pp → pK+Λ
+cross
+section
+measured
+by
+the HADES collaboration [22], the cross section ratio
+σ(pK+Λ)/σ(pK+Σ0) is determined to be 1.90 ± 0.41.
+Based on the coupled channel calculation, where the interfer-
+ence of the pion and kaon exchange is taken in account, the
+cross section ratio can be reproduced by selecting the relative
+
+X103
+Data
+1.2
+pK+0
+HADES
+pK+^
+ p(3.5 GeV)+p → pK+0
+pK+E元0
+pK++元
+pK+(Y* → A)
+0.8
+pK+(Y* → A元°)
+pK+(Y* → +元)
+50.6
+Simulation Sum
+Events /
+0.4
+0.2
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+MM (pK*) [GeV/c?]10
+R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+Figure 8: The corrected angular distributions in the CMS (top row), Gottfried-Jackson (middle row) and helicity frames (bottom
+row). The experimental data are shown by the black points, where the error bars are the square root of the quadratic sum of
+the statistical and systematic uncertainties. The blue histogram represent the weighted pp → pK+Σ0 phase space simulation
+described in the text and the dotted pink histogram indicates the best partial wave analysis solution (discussed in Section 6).
+sign for these two mechanism [17]. Figure 11 shows the cross
+section ratio as a function of the excess energy together with
+a compilation of other measurements [42]. The solid curve
+is the ratio of the paramerization of both channels, where
+the paramerization proposed by F¨aldt and Wilkin [43] based
+on phase space and FSI is used for pp → pK+Λ and the
+Tsushima paramerization [44] based on a resonance model is
+used for the pp → pK+Σ0 channel.
+The observed cross section ratio in the present p+p data is
+similar to the corresponding value measured in p+Nb data [7],
+despite the large difference in the individual cross sections, thus
+corroborating the importance of FSI for these reactions.
+6 Partial Wave Analysis
+From the results presented above, it was concluded that
+the experimental data on angular distributions can not be
+described by pure phase space production, but there must be a
+resonant component as anticipated in [2]. Therefore, a Partial
+Wave Analysis (PWA) using the Bonn-Gatchina Partial Wave
+Analysis (Bo-Ga PWA) framework [45] has been applied
+with the goal to quantify the relative contributions of different
+partial waves.
+The Bo-Ga PWA framework takes a list of possible transi-
+tion waves as an input that may contribute to the final state. The
+non-resonant production proceeds as follows: the proton (JP=
+
+20
+C
+a
+HADES
+15
+p(3.5 GeV)+p -→ pK+≥0
+10F
+5
+-0.5
+0.5
+1 -1
+-0.5
+0.5
+1 -1
+-0.5
+0
+0
+0.5
+0
+cos Q,.
+cos Ok+
+20
+(d)
+(f)
+(e)
+15
+[ub]
+0
+5
+-0.5
+0.5
+0.5
+1 -1
+0.5
+0
+-1
+-0.5
+0.5
+RFpLC
+COS ORF K*20
+cos 0'
+COS
+Pb.t. K*
+20
+(g)
+(h)
+(i)
+15
+10
+5
+-0.5
+-0.5
+0.5
+1-1
+-0.5
+0.5
+1-1
+0.5
+1
+0
+RF Ktp
+COS ORF K*20
+p, kt
+p, oR. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+11
+Table 2: Comparison of the normalized Legendre coefficients between the present measurement and the data collected by COSY-
+TOF experiment at ϵ = 162 MeV [2].
+ϵ = 162 MeV
+ϵ = 556 MeV
+A1/A0
+A2/A0
+A1/A0
+A2/A0
+cosθCMS
+Σ0
+0.0 ± 0.0
+0.03 ± 0.24
+0.0 ± 0.0
+0.32 ± 0.09
+cosθCMS
+p
+0.0 ± 0.0
+0.25 ± 0.29
+0.0 ± 0.0
+0.43 ± 0.13
+cosθCMS
+K+
+0.0 ± 0.0
+0.48 ± 0.22
+0.0 ± 0.0
+-0.01 ± 0.1
+cosθFRpΣ0
+pb,t,p
+0.0 ± 0.0
+0.11 ± 0.15
+-0.06 ± 0.17
+0.27 ± 0.18
+cosθFRK+Σ0
+pb,t,K+
+-0.04 ± 0.04
+0.14 ± 0.18
+-0.19 ± 0.18
+0.08 ± 0.19
+cosθFRK+p
+pb,t,K+
+-0.07 ± 0.07
+0.57 ± 0.18
+0.19 ± 0.12
+0.05 ± 0.26
+cosθFRK+Σ0
+p,Σ0
+0.27 ± 0.27
+-0.15 ± 0.15
+0.36 ± 0.07
+-0.08 ± 0.09
+cosθFRpΣ0
+p,K+
+-0.22 ± 0.22
+0.0 ± 0.15
+-0.4 ± 0.06
+0.04 ± 0.08
+cosθFRK+p
+K+,Σ0
+-0.11 ± 0.11
+0.11 ± 0.18
+0.55 ± 0.07
+-0.11 ± 0.09
+Figure 9: The Σ0 Treiman-Yang angular distribution measured
+in the K+Σ0 reference frame. The blue histogram represents
+the weighted pp → pK+Σ0 phase space simulation and the
+dotted histogram indicates the best partial wave analysis so-
+lution (discussed in Section 6).
+1
+2
++) and the hyperon (in this case Σ0 with JP= 1
+2
++) are com-
+bined into a two particle sub-system and then the kaon (JP=
+0−) is combined with this sub-system to produce the three-
+body final state. In case of the resonant production, the proton
+is combined with one of the resonances listed in Table 3 N∗-p,
+or ∆∗-p to produce the final state pp → pK+Σ0. Resonance
+masses and widths were fixed to the PDG values [46] in order
+to reduce the number of the free fit parameters.
+The strength (α1) and the phase (α2) of each transition
+wave are determined by fitting the partial wave amplitudes
+to the experimental data on an event-by-event basis in an
+Figure 10: Compilation of cross sections of the reaction
+pp → pK+Σ0 from different experiments: COSY-11 [36, 37,
+38, 39, 40, 41], COSY-TOF [2] and data points from Landolt-
+B¨ornstein (LB) [42]. The production cross section of Σ0 deter-
+mined here is shown by the green square. The solid curve rep-
+resents a pure phase space fit, the dotted curve is a parametriza-
+tion based on the resonance model and the dashed curve is
+phase space and FSI as described in the text.
+unbinned fit. The fit is based on a log-likelihood minimization
+and the fitting procedure is repeated for many iterations until
+there is no further improvement of the log-likelihood value.
+By comparing the log-likelihood value of many fits the best fit
+can be determined through the largest negative value. As an
+output, the BG-PWA returns the fitted values of the parameters
+α1 and α2 and a list of simulated events that have been used
+as an input but with each event being assigned a weight factor,
+which gives the contribution of this event to the total yield.
+Since the signal region contains background events (mainly
+pp → pK+Λ and pp → pK+Λπ0), and because the Bo-Ga
+
+103
+HaDEs
+p(3.5 GeV)+p → pK+z0
+102
+10
+[ub]
+COSY-11
+COSY-TOF
+ LB
+HADES
+10-
+ Phase Space
+- Phase Space + FSi
+ Resonance Model
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+E[GeV].25
+do/Φ [degree]
+0.2
+0.15
+0.
+0.05
+50
+100
+150
+RF K+0
+[degree]HAPESp(3.5 GeV)+p -→ pK+Z0do/db [ub/degree12
+R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+Figure 11: Experimental cross section ratio of the present data
+point together with a compilation of the world data: COSY-
+11 [36, 37, 38, 39, 40, 41], COSY-TOF [2] and data points
+from Landolt-B¨ornstein (LB) [42]. The present data square is
+shown by the green square. The solid curve is the ratio of the
+paramerization of both channels [43, 44].
+Table 3: A list of N∗ and ∆∗ resonances that might contribute
+to the pp → pK+Σ0 reaction. The mass, width and spin-parity
+quantum numbers were taken from [46].
+Resonance
+Mass
+[GeV/c2 ]
+Width
+[GeV/c2 ]
+JP
+N∗(1710)
+1.710
+0.140
+1
+2
++
+N∗(1875)
+1.875
+0.200
+3
+2
+−
+N∗(1880)
+1.880
+0.300
+1
+2
++
+N∗(1895)
+1.895
+0.120
+1
+2
+−
+N∗(1900)
+1.920
+0.200
+3
+2
++
+∆∗(1900)
+1.860
+0.250
+1
+2
+−
+∆∗(1910)
+1.900
+0.300
+1
+2
++
+∆∗(1920)
+1.920
+0.300
+3
+2
++
+PWA method works on an event-by-event basis, it is important
+to identify whether a particular event belongs to the signal or
+the background. The pp → pK+Λ contribution is three times
+larger than pp → pK+Λπ0 inside the signal region. Therefore,
+the pp → pK+Λ channel is considered the main contributing
+background and its kinematics is modeled by performing a
+PWA on the pp → pK+Λ-like events. The solutions published
+in [22] have been tested and solution No. 8/1 was found
+to provide the best description of the experimental data by
+including the p+p initial waves 2S+1LJ = 1S0, 3P0, 3P1 and
+1D2.
+The solution No. 8/1 is then applied to the Λ 4π-phase
+space simulations and these events are filtered through the
+full simulation and analysis chain. After reconstructing the Λ
+events that have been assigned a PWA weight, the missing mass
+MM(pK+) spectrum was investigated and the Λ contribution
+in the signal region 1.170 < MM(pK+)[GeV/c2] < 1.220
+was determined to be 292 events. Those events are then added
+to the signal list with a negative weight.
+After subtracting the Λ contribution, the PWA technique
+is applied to the pp → pK+Σ0 events. A systematic variation
+of the input partial waves was performed and, in addition, the
+number of non-resonant and resonant final partial waves was
+varied and the quality of the PWA solution was determined by
+the negative log-likelihood value of the fit.
+The best PWA solution shown by the dashed histograms in
+Figures 8 and 9 was obtained by including p+p initial waves
+2S+1LJ = 2S0, 3P0, 3P1, 3P2, 1D2 and 3F2. In addition,
+nucleon resonances N∗(1710), N∗(1900) and ∆∗(1900) were
+found to contribute as well as non-resonant partial waves.
+However, due to the limited statistics and the large number
+of free fit parameters, an unambiguous determination of
+the contributions of each resonance is not possible since
+these contributions vary significantly for different solutions.
+Nevertheless, resonances with masses around 1.710 GeV/c2
+(N∗(1710)) and 1.900 GeV/c2 (N∗(1900) or ∆∗(1900)) are
+certainly preferred by the fit.
+7 Conclusion and Outlook
+The exclusive reconstruction of the reaction pp → pK+Σ0 at
+a beam kinetic energy of 3.5 GeV has been presented and the
+pp → pK+Σ0 total production cross section was determined
+with an accuracy better than 10 % in a region where no data
+existed. The dynamics of the reaction was investigated by
+studying the angular distributions in the CMS, G-J and helicity
+frame. The corrected CMS distributions of the hyperon and
+the proton show anisotropies, which it is more pronounced
+in the case of the proton. This is the expected behavior if
+the pion exchange mechanism dominates the particle pro-
+duction process in a simple one-boson exchange formalism.
+In addition, an investigation of the Σ0 T-Y angle measured
+in the K+Σ0 reference frame, deviates from isotropy, which
+hints to a non-negligible contribution of the of kaon exchange
+mechanism.
+The helicity angular distributions are not isotropic,
+which indicates that a pure phase space description with-
+out momentum-dependent matrix element(s) is by far not
+appropriate. The influence of different nucleon resonances
+has been tested by means of a PWA using the Bo-Ga PWA
+framework. The best solution was obtained by including the
+initial p+p configuration 1S0, 3P0, 3P1, 3P2, 1D2 and 3F2.
+Due to the limited statistics, it was not possible to obtain the
+exact strength of the individual nucleon resonances. However,
+nucleon resonances N∗(1710), N∗(1900) and ∆∗(1900) are
+preferred by the fit.
+Recently, the HADES setup has been upgraded by an elec-
+tromagnetic calorimeter (ECAL) and a Forward Detector (FD)
+based on PANDA experiment straw tubes [47]. The new data
+that was collected in February 2022 offers the opportunity to
+perform the same measurement with an upgraded setup at a
+higher proton beam energy of 4.5 GeV. This upgrade will allow
+the identification of the daughter photon in Σ0 → Λγ via the
+ECAL. In addition, it will improve the mass resolution of the
+
+102
+HADES
+(pK+A)/ (pK+0
+10
+0
+COSY-11
+COSY-TOF
++ LB
+HADES
+10-1
+E[GeV]R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+13
+Λ hyperon in the FD acceptance and consequently improve the
+quality of the kinematic refit. Furthermore, the collected data
+will provide sufficient statistics to extract quantitative contri-
+butions of the different nucleon resonances and a measurement
+of their K+Σ0 branching ratios, which will certainly improve
+the current measurement.
+8 Acknowledgment
+The HADES collaboration gratefully acknowledges the support by
+SIP JUC Cracow, Cracow (Poland), 2017/26/M/ST2/00600; WUT
+Warsaw (Poland) No: 2020/38/E/ST2/00019 (NCN), IDUB-POB-
+FWEiTE-3; TU Darmstadt, Darmstadt (Germany), VH-NG-823,
+DFG GRK 2128, DFG CRC-TR 211, BMBF:05P18RDFC1, HFHF,
+ELEMENTS 500/10.006, GSI F&E, EMMI at GSI Darmstadt;
+Goethe-University,
+Frankfurt
+(Germany),
+BMBF:05P12RFGHJ,
+GSI F&E, HIC for FAIR (LOEWE), EMMI at GSI Darmstadt;
+JLU Giessen, Giessen (Germany),BMBF:05P12RGGHM; IJCLab
+Orsay, Orsay (France), CNRS/IN2P3; NPI CAS, Rez, Rez (Czech
+Republic), MSMT LTT17003, MSMT LM2018112, MSMT OP VVV
+CZ.02.1.01/0.0/0.0/18 046/0016066;
+European
+Union’s
+Horizon
+2020, no. 824093 (STRONG2020).
+This project has received funding from the programme ”Netzwerke
+2021”, an initiative of the Ministry of Culture and Science of the State
+of Northrhine Westphalia. The sole responsibility for the content of
+this publication lies with the authors.
+The following colleagues from Russian institutes did contribute
+to the results presented in this publication but are not listed as
+authors following the decision of the HADES Collaboration Board
+on March 23, 2022: G. Agakishiev, A. Belyaev, O. Fateev, A.
+Ierusalimov, V. Ladygin, T. Vasiliev, M. Golubeva, F. Guber, A.
+Ivashkin, T. Karavicheva, A. Kurepin, A. Reshetin, A. Sadovsky and
+A.V.Sarantsev.
+References
+[1]
+G. E. Brown et al., Phys. Rev. C 43 (1991), 1881–1892.
+DOI: 10.1103/PhysRevC.43.1881.
+[2]
+M. Abdel-Bary et al., Eur. Phys. J. A 46 (2010), 27–44.
+DOI: 10.1140/epja/i2010-11023-0.
+[3]
+J. Adamczewski-Musch et al., Phys. Rev. C 95 (2017),
+015207. DOI: 10.1103/PhysRevC.95.015207.
+[4]
+G. Agakishiev et al., Eur. Phys. J. A 50 (2014), 81. DOI:
+10.1140/epja/i2014-14081-2.
+[5]
+J. T. Balewski et al., Phys. Lett. B 388 (1996), 859–865.
+DOI: 10.1016/S0370-2693(96)01360-3.
+[6]
+R. M¨unzer et al., Phys. Lett. B 785 (2018), 574–580.
+DOI: 10.1016/j.physletb.2018.08.068.
+[7]
+J. Adamczewski-Musch et al., Phys. Lett. B 781 (2018),
+735–740. DOI: 10.1016/j.physletb.2018.02.
+043.
+[8]
+G. F. Chew and F. E. Low, Phys. Rev. 113 (1959), 1640–
+1648. DOI: 10.1103/PhysRev.113.1640.
+[9]
+J. Sakurai, Nuovo Cim 20 (1961), 1212–1216. DOI:
+https://doi.org/10.1007/BF02732532.
+[10]
+R. Machleidt, K. Holinde, and C. Elster, Phys. Rept. 149
+(1987), 1–89. DOI: 10.1016/S0370- 1573(87)
+80002-9.
+[11]
+R. Machleidt, Adv. Nucl. Phys. 19 (1989), 189–376.
+[12]
+S. Abd El-Samad et al., Phys. Lett. B 688 (2010), 142–
+149. DOI: 10.1016/j.physletb.2010.03.076.
+[13]
+A. Sibirtsev et al., Eur. Phys. J. A 27 (2006), 269–285.
+DOI: 10.1140/epja/i2005-10268-x.
+[14]
+A. Budzanowski et al., Phys. Lett. B 687 (2010), 31–35.
+DOI: 10.1016/j.physletb.2010.02.082.
+[15]
+M. R¨oder et al., Eur. Phys. J. A 49 (2013), 157. DOI:
+10.1140/epja/i2013-13157-9.
+[16]
+S. Abd El-Samad et al., Eur. Phys. J. A 49 (2013), 41.
+DOI: 10.1140/epja/i2013-13041-8.
+[17]
+P. Kowina et al., Eur. Phys. J. A 22 (2004), 293–299.
+DOI: 10.1140/epja/i2003-10236-6.
+[18]
+T. Rozek et al., Phys. Lett. B 643 (2006), 251–256. DOI:
+10.1016/j.physletb.2006.07.066.
+[19]
+A. Sibirtsev et al., Eur. Phys. J. A 29 (2006), 363–367.
+[20]
+J.-J. Xie, H.-X. Chen, and E. Oset, Phys. Rev. C 84
+(2011), 034004. DOI: 10 . 1103 / PhysRevC . 84 .
+034004.
+[21]
+G. Agakishiev et al., Eur. Phys. J. A 41 (2009), 243–277.
+DOI: 10.1140/epja/i2009-10807-5.
+[22]
+G. Agakishiev et al., Phys. Lett. B 742 (2015), 242–248.
+DOI: 10.1016/j.physletb.2015.01.032.
+[23]
+C.-Y. Liou et al., Neurocomputing 139 (2014), 84–96.
+[24]
+A. Paszke et al., Adv. Neural Inf. Process Syst. 32
+(2019), 8024–8035.
+[25]
+W. Esmail, EPJ Web Conf. 271 (2022), 08013. DOI: 10.
+1051/epjconf/202227108013.
+[26]
+I. J. Good, J. R. Stat. Soc.: Series B 14 (1952). DOI:
+https://doi.org/10.1111/j.2517-6161.
+1952.tb00104.x.
+[27]
+S. Taylor et al., Phys. Rev. C 71 (2005-05). DOI: 10.
+1103/PhysRevC.71.054609.
+[28]
+I. Fr¨ohlich et al., PoS ACAT (2007), 076. DOI: 10 .
+22323/1.050.0076.
+[29]
+R. Brun et al., CERN-W5013 (1994). DOI: 10.17181/
+CERN.MUHF.DMJ1.
+[30]
+A. Hocker and V. Kartvelishvili, Nucl. Instrum. Meth.
+A 372 (1996), 469–481. DOI: 10 . 1016 / 0168 -
+9002(95)01478-0.
+[31]
+T. Adye, PHYSTAT 2011 (2011), 313–318. DOI: 10.
+5170/CERN-2011-006.313.
+[32]
+G. Agakishiev et al., Eur. Phys. J. A 48 (2012), 64. DOI:
+10.1140/epja/i2012-12064-y.
+[33]
+Y. Gal and Z. Ghahramani, ICML2016 48 (2016), 1050–
+1059. DOI: https://dl.acm.org/doi/10.
+5555/3045390.3045502.
+[34]
+K. Gottfried and J. D. Jackson, Nuovo Cim. 33 (1964),
+309–330. DOI: 10.1007/BF02750195.
+[35]
+E. Ferrari and S. Serio, Phys. Rev. 167 (1968), 1298–
+1308. DOI: 10.1103/PhysRev.167.1298.
+[36]
+D. Grzonka and K. Kilian, Nucl. Phy. A 626 (1997), 41–
+54. DOI: https://doi.org/10.1016/S0375-
+9474(97)00519-8.
+[37]
+J. Balewski et al., Nucl. Phy. A 626 (1997), 85–92.
+DOI: https : / / doi . org / 10 . 1016 / S0375 -
+9474(97)00524-1.
+[38]
+J. Balewski et al., Phys. Lett. B 420 (1998), 211–216.
+DOI: https : / / doi . org / 10 . 1016 / S0370 -
+2693(97)01527-X.
+
+14
+R. Abou Yassine et al.: Investigation of the Σ0 Production Mechanism in p(3.5 GeV)+p Collisions
+[39]
+S. Sewerin et al., Phys. Rev. Lett. 83 (1999), 682–685.
+DOI: 10.1103/PhysRevLett.83.682.
+[40]
+S. Abd El-Samad et al., Phys. Lett. B 632 (2006),
+27–34. DOI: https://doi.org/10.1016/j.
+physletb.2005.09.086.
+[41]
+Y. Valdau and C. Wilkin, Phys. Lett. B 696 (2011), 23–
+25. DOI: 10.1016/j.physletb.2010.11.072.
+[42]
+H. Schopper. Landolt-B¨ornstein. New series. Group 1
+Nuclear and particle physics. Berlin: Springer, 1988.
+DOI: 10.1007/b35211.
+[43]
+G. Faldt and C. Wilkin, Z. Phys. A 357 (1997), 241–243.
+DOI: 10.1007/s002180050239.
+[44]
+K. Tsushima et al., Phys. Rev. C 59 (1999), 369–387.
+DOI: 10.1103/PhysRevC.59.369.
+[45]
+A. V. Sarantsev et al., Eur. Phys. J. A 25 (2005), 441–
+453. DOI: 10.1140/epja/i2005-10121-4.
+[46]
+M. Tanabashi et al., Phys. Rev. D 98 (2018), 030001.
+DOI: 10.1103/PhysRevD.98.030001.
+[47]
+J. Adamczewski-Musch et al., Eur. Phys. J. A 57 (2021),
+138. DOI: 10.1140/epja/s10050-021-00388-
+w.
+

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+Designing Covalent Organic Framework-based Light-driven Microswimmers 
+towards Intraocular Theranostic Applications 
+ 
+Varun Sridhar1,+, Erdost Yildiz1,+, Andrés Rodríguez-Camargo,2,3, Xianglong Lyu1, Liang Yao2, Paul 
+Wrede1, Amirreza Aghakhani1, Mukrime Birgul Akolpoglu1, Filip Podjaski2,4,5,*, Bettina V. 
+Lotsch2,3,5,6,*, Metin Sitti1,7,8,*         
+ 
+1 Physical Intelligence Department, Max Planck Institute for Intelligent Systems, 70569 Stuttgart, 
+Germany 
+2 Nanochemistry Department, Max Planck Institute for Solid State Research, 70569 Stuttgart, 
+Germany 
+3 Department of Chemistry, University of Stuttgart, 70569 Stuttgart, Germany 
+4 Department of Chemistry, Imperial College London, W12 0BZ London, United Kingdom 
+5 Cluster of Excellence E-conversion, Lichtenbergstrasse 4, 85748 Garching, Germany 
+6 Department of Chemistry, University of Munich (LMU), Munich, Germany 
+7 Institute for Biomedical Engineering, ETH Zurich, 8092 Zurich, Switzerland 
+8 School of Medicine and College of Engineering, Koç University, 34450 Istanbul, Turkey 
+ 
++ These authors contributed equally to this article. 
+* Correspondence to: sitti@is.mpg.de, f.podjaski@imperial.ac.uk, b.lotsch@fkf.mpg.de  
+ 
+ 
+
+Abstract  
+Even micromachines with tailored functionalities enable targeted therapeutic applications in 
+biological environments, their controlled motion in biological media and drug delivery functions 
+usually require sophisticated designs and complex propulsion apparatuses for practical 
+applications. Covalent organic frameworks (COFs), new chemically versatile and nanoporous 
+materials, offer microscale multi-purpose solutions, which are not explored in light-driven 
+micromachines. We describe and compare two different types of COFs, uniformly spherical TABP-
+PDA-COF sub-micron particles and texturally highly nanoporous, irregular, micron-sized TpAzo-
+COF particles as light-driven microrobots. They can be used as highly efficient visible-light-driven 
+drug carriers in aqueous ionic and cellular media, even in intraocular fluids. Their absorption 
+ranging down to red light enables phototaxis even in deeper biological media and the organic 
+nature of COFs enables their biocompatibility. The inherently porous structure with ~2.5 nm 
+structural pores, and large surface areas allow for targeted and efficient drug loading even for 
+insoluble drugs and peptides, which can be released on demand. Also, indocyanine green (ICG) 
+dye loading in the pores enables photoacoustic imaging or optical coherence tomography and 
+hyperthermia in operando conditions. The real-time visualization of the drug-loaded COF 
+microswimmers enables new insights into the function of porous organic micromachines, which 
+will be useful to solve various drug delivery problems. 
+ 
+Keywords: Covalent organic framework, light-driven, microswimmer, targeted drug delivery, 
+optical coherence tomography  
+ 
+
+Introduction 
+Microrobots are tiny machines that are tailored to be controlled externally to perform individual 
+tasks. In order to achieve external control as well as multi-purpose functionality, micro/nanobots 
+typically require a sophisticated and specifically adapted design enabling targeted control and 
+applications. Their primary area of use is the large field of biomedicine.1, 2, 3, 4, 5 The microrobots 
+should not elicit any immune response and should be compatible with the cells to enable 
+biomedical applications.6, 7 Also, the propulsion method should be as noninvasive as possible, 
+and non-toxic, excluding the use of dedicated toxic fuels.8, 9 For an efficient microrobot function, 
+motion control is the first requirement, which can become more and more challenging if the 
+liquid they are propelled in contains species that hinder propulsion or external control. Wireless 
+motion control requires external energy input and is typically realized by magnetic or acoustic 
+actuation,10 but can also be realized by ultraviolet (UV) or blue light, even in biological conditions, 
+as evidenced very recently.11 However, UV light, which is typically used for light-driven 
+microswimmers12, 13 is incompatible with biological tissues. Also, visible light control, usually 
+reported with high-intensity blue light,12 limits applications to transparent conditions since tissue 
+penetration requires more red light or near-infrared light, which is a significant challenge to the 
+field. 
+The critical tasks of mobile microrobots are cargo uptake and delivery, often linked to 
+biopharmaceutical classes and properties of drugs, after actively navigating to a target diseased 
+tissue region.4, 7, 14, 15 Drug uptake and its controlled release were typically realized efficiently by 
+encapsulation structures being a separate part of the microrobots; these were then opened in 
+the desired conditions or where a release could be triggered otherwise. More recently, inherently 
+porous structures, such as metal-organic frameworks16, 17, 18 and porous carbon nitrides were 
+used for such applications since their sizeable inner pore volume, reminiscent of a sponge, 
+enables high and, even environmentally stable drug loading.11 However, porous particle 
+structures with many textural pores of different sizes as part of their inner surface area leave 
+challenges for controlled loading and release from their volume.   
+As a last critical step to clinical applications, cell viability, the absence of foreign body reactions, 
+and tissue biocompatibility are necessary conditions for microrobots to be used in biological 
+
+contexts, which is not always easy to ensure, with all the desired functions being fulfilled at a 
+time.4, 7, 14 For this purpose, typically biocompatible metal coatings, such as gold, titanium, or 
+polymers are employed, but also organic-based materials are up-and-coming and were used 
+recently without coatings, such as carbon nitrides.11, 17, 19 Compared to inorganic structures, 
+organic materials not only offer potential biodegradability but also the high flexibility of chemical 
+design of organic materials, especially in terms of surface functionalities and porosity, might 
+enable more efficient and targeted biomedical applications, such as drug delivery or 
+hyperthermia.20 Even the most sophisticated designs with biocompatible metallic structures fail 
+during actuation inside heterogenic biological fluids and specific-targeted drug delivery and 
+imaging in live tissues.21, 22, 23 Because of that, new materials and actuation methods should be 
+investigated for the basic tasks for the clinical obstacles, such as intraocular motion and drug 
+delivery.     
+In this work, we introduce covalent organic frameworks (COFs) as a tailorable active component 
+to the field of micro- and nanomachines, or more precisely, light-driven microswimmers. These 
+highly porous and crystalline materials can fulfill all the requirements listed above since their 
+molecular structure, and also their morphology, can be designed and tuned bottom-up while 
+enabling targeted properties.20, 24, 25 Simultaneously, they can use visible light for photocatalytic 
+reactions with their environment, which can also be used for active particle propulsion.19, 26, 27 
+Depending on the propulsion mechanism and particle structure, the propulsion can be self-
+diffusiophoretic or self-electrophoretic, while allowing light-induced directional motion control 
+via phototaxis.28 The tailorable properties of the COF building blocks enable tuning of not only 
+their absorption wavelength but also pore size and volume, surface polarity, and chemical 
+affinity, which enable the loading of large and small molecule cargo based on the specific 
+application requirements. Since such organic structures are non-magnetic per se, unless so-called 
+Janus or hybrid (encapsulation) structures were to be employed, the use of light is a highly 
+promising and convenient method not only to propel them but also to trigger functions within 
+them and to image their behavior.29 Thanks to their promising and ion-tolerant visible light-
+driven propulsion properties, we especially focused on their use in ophthalmological 
+applications. In addition to their light-driven propulsion, their configurable particle sizes enable 
+
+them to pass through the fibrillar mesh of vitreous humor (~500 nm pore size).30 For this purpose, 
+we selected therapeutic agents and imaging modalities accordingly. 
+Here, these possibilities are investigated and exemplified. We study and compare two very 
+different modified COFs, namely TAPB-PDA-COF made from the condensation of 1,3,5-tris(4-
+aminophenyl)benzene 
+and 
+terephthaldehyde,31 
+and 
+TpAzo-COF 
+made 
+from 
+1,3,5-
+triformylphloroglucinol and 4,4′-azodianiline,32 as light-driven microswimmer examples, in order 
+to explore the microrobotic possibilities for this class of materials and to establish versatile 
+applications. We describe their light-controlled propulsion in biological media, their 
+biocompatibility, as well as their uptake and release of drugs that can be physisorbed to the pores 
+of the material.25, 33 Since these two COFs have distinct structures and morphologies, we derive 
+design guidelines for their propulsion and cargo-related functions. For theranostic delivery 
+functions of microrobots, we used doxorubicin, insulin, and indocyanine green (ICG), which 
+covers the breath of small molecules and peptides used as therapeutic and imaging agents. We 
+further image the motion of the COFs in real-time and potential in-vivo conditions using optical 
+coherence tomography as well as photoacoustic imaging of COF particle swarms loaded with a 
+near-infrared active dye (ICG). The water-soluble or insoluble drugs and the contrast agents can 
+be loaded to track their release, allowing for first insights into the action of porous drug carriers 
+in real-time clinical imaging modalities. In this way, we designed and investigated in detail the 
+first photoactive intraocular drug carriers for various theranostic applications.   
+ 
+Results and Discussion 
+COF synthesis and characterization 
+The COF structures to be used as microswimmers were selected based on their structural and 
+optical properties and synthesized akin to a procedure reported earlier.31, 32 In brief, the  TAPB-
+PDA-COF nanospheres were obtained by using TAPB (1,3,5-tris(4-aminophenyl)benzene) and 
+PDA (terephthaldehyde) as building blocks, with acetonitrile and Sc(Otf)3 as solvent and catalyst, 
+respectively (see Materials & Methods section for more details) to yield a two-dimensional (2D), 
+imide linked organic network (Fig. 1a). These 2D sheets are stacked on each together forming an 
+
+ordered 3D structure with hexagonal pore channels of a diameter of approx. 3.4 nm, as reported 
+earlier and confirmed for the here modified synthesis by nitrogen sorption, powder XRD, and FT-
+IR analysis (Fig. 1b and Fig. S1).34 The obtained COF submicron particles (henceforth called 
+nanoparticles for better discrimination) have an almost perfectly spherical shape (Fig. 1c), which 
+is beneficial for propulsion in fluids.35 The synthesis yields a very homogeneous product with a 
+Brunauer, Emmett, and Teller (BET) surface area as high as 685 m²/g (Fig. S1c) and a narrow size 
+distribution of approx. 452 ± 74 nm (Fig. S2a,b). TEM analysis reveals that these nanoparticles 
+consist of agglomerated individual crystallites with sizes between 50 and 100 nm (Fig. S1c). 
+The TpAzo-COF presented here for comparison is synthesized by solvothermal condensation 
+between 1,3,5-triformylphloroglucinol (Tp) and 4,4´-azodianiline (Azo), forming a tautomeric 
+ketoenamine COF (Fig. 1d). A highly crystalline product with a 2D molecular structure is obtained 
+(see Fig. S3 for structural analysis), with slightly smaller structural pores of 2.6 nm and a similar 
+BET surface area of 635 m²/g (Fig. 1e). In contrast to the first COF, the particle morphology is 
+much less defined, leaving open large textural voids reminiscent of a sponge (Fig. 1f and Fig. S4). 
+The overall particle sizes are broadly distributed (7 ± 18 µm), hence being much larger. The 
+primary crystallites forming the COF particle are only 20 nm (Fig. S4d), and the TpAzo-COF 
+particles appear to be agglomerates of those. 
+Light-induced swimming in aqueous media 
+To enable light-triggered propulsion, we first investigate the light absorption properties. UV-Vis 
+spectroscopy and Kubelka-Munck analysis show that TAPB-PDA-COF and TpAzo-COF have an 
+optical band gap of 464 nm (Fig. 2a) and 616 nm (Fig. 2f), respectively with a small absorption 
+tail commonly arising from defect states. Hence, visible light propulsion can be extended up to a 
+wavelength of ~470 nm (blue light) for the small TABP-PDA particles, and to green or even red 
+parts of the spectrum for the large TpAzo-COF. Their light-induced propulsion was studied under 
+a microscope in a microfluidic chamber under ambient conditions to test the phototaxis 
+capabilities while diluting them to 100 µg/ml. First, we focus on propulsion in distilled water. 
+While in the dark, COF particles show only local Brownian motion with a mean displacement 
+speed of 4.5 µm/s for the TAPB-PDA and 3.7 µm/s for the TpAzo-COF, respectively (Fig. 2b,g, 
+dashed line). When light from the photodiode is focused on the microswimmers through the 
+
+microscope, their propulsion speed is significantly enhanced and becomes ballistic, as seen in 
+Video S1. The particles move towards the center of the light, then upwards. This way, the light-
+driven collective assembly or trapping of the microswimmers is made possible. UV excitation at 
+385 nm propels the TABP-PDA-COF with 13.2 ± 2.4 µm/s, while 470 nm blue light gives an even 
+increased speed of 16.4 ± 3.1 µm/s (~36 bodylengths/s (BLPS)). At 510 nm illumination, no light-
+enhanced swimming was observed, consistent with the absorption spectrum (Fig. 2b). The Tp-
+Azo-COF is propelled with 4.9 ± 1.2, 12.1 ± 2.1 (~2 BLPS), 8.2 ± 1.7, and 4.2 ± 1.2 µm/s at 385, 
+470, 560 nm, and 630 nm, respectively (Fig. 2g). UV and red light hence do not increase 
+propulsion significantly below Brownian motion and the absolute propulsion speed is lower, but 
+it can be triggered even by yellow light (560 nm).  
+Ion tolerance for light-driven microswimmers and phototaxis behavior 
+Ionic conditions represent a major challenge for light-driven microswimmers.36, 37, 38 The 
+presence of pores, both structural and textural (=morphological), was suggested previously to 
+enable the propulsion of microswimmers in ionic environments, which includes most of the 
+biological fluids and cell culture media.11 To confirm this and to widen the insights from different 
+and better controlled structural features present in our model COFs, we first tested them in 
+increasing concentrations of salt (NaCl), see Fig. 2c,h. 
+When propelled at 470 nm, the TABP-PDA-COF does not decrease the speed compared to 
+distilled water up to concentrations as large as 1000 mM. Therefore, the ionic concentration in 
+the media at which the microswimmers’ speed is halved (EI50) cannot be attributed.39 Also, we 
+observe a slightly increasing propulsion speed between 0.5 and 10 mM, with a maximum value 
+of 25.2 ± 3.7 µm/s  (54% increase vs. distilled water, 0 mM) at 1 mM NaCl (Fig. 2c). An explanation 
+for this non-linear behavior remains to be found. Ionic interactions can be influencing the 
+Helmholtz and Debye layers, as well as the materials' inner space charge layer. As such, light-
+induced charge carrier stability or recombination will also be affected. On the other hand, the 
+chlorine evolution reaction due to dissolved NaCl may another photocatalytic pathway possibly 
+increasing the reaction rate, and thereby the propulsion speed.11, 40, 41, 42 These factors currently 
+cannot be studied or disentangled on such size and complex reaction interface. However, their 
+
+propulsion speed even surpasses our previously reported PHI microswimmers, the only reported 
+system with comparable ionic tolerance.11 
+Similarly, for the TpAzo-COFs, an increased propulsion speed compared to pure water is observed 
+in all ionic conditions (1-1000 mM) at 560 nm illumination, peaking at 1 mM (14.7 ± 2.7 µm/s, 
+79% increase vs. distilled water) and followed by a 28% relative decay to 10.5 µm/s at 1000 mM. 
+When increasing the wavelength, no active propulsion is observed for TABP-PDA-COF (Fig. 2b), 
+but the Tp-AZO-COF exhibits slightly enhanced propulsion even at 630 nm (4.2 µm/s) (Fig. 2g).  
+Next, three standard biological media are studied, namely, Dulbecco’s phosphate-buffered saline 
+(dPBS), minimum essential medium (MEM), and MEM plus fetal bovine serum (FBS) (Fig. 2d,i), 
+which slightly differ in their components: dPBS contains NaCl, KCl, Na2HPO4, and KH2PO4 at ca. 10 
+g/L (~150 mM) in total; MEM contains the same components as dPBS and additional two amino 
+acids, vitamins, and glucose, some of which can be redox-active agents that help extract not only 
+electrons but especially holes from the microswimmers under illumination to power them.11, 43 
+FBS, slightly more viscous, adds nutrients for cell growth and imitates the conditions found within 
+the body.11 At 470 nm illumination, the mean speeds of the TAPB-PDA-COF microswimmers in 
+dPBS, MEM, and MEM + FBS are 20.4 ± 4.5 µm/s, 20.9 ± 2.8µm/s and 17.6 ± 3.3 µm/s, 
+respectively. These speeds are again higher than in distilled water (16.4 ± 3.1 µs/s). The slight 
+decrease upon FBS addition can be attributed to the increasing viscosity or other surface 
+interactions with the proteins present in the FBS.  
+Very similar behavior is observed with the TpAzo-COF at 560 nm, where the swimming speeds 
+are equivalent to the maximum value in 1mM NaCl, or even slightly higher (13.8 ± 3.4 µm/s, 16.2± 
+3.4 µm/s, 14.7 ± 3.6 µm/s, and 11.8 ± 2.2 µm/s in dPBS, MEM (with and without glucose), and 
+MEM + FBS). A difference however is observed when glucose, a well-oxidizable fuel,11, 43 is absent 
+– the speed is reduced. Its vital role as fuel for propulsion is clearly visible when illuminating 
+TpAzo at 630 nm in MEM that contains glucose, where efficient propulsion, independent of FBS, 
+is observed (10 ± 2.7 µm/s and 7.4 ± 1.8 µm/s respectively). This purely red light-induced 
+photocatalytic motion in the presence of high ion concentrations and without using potent and 
+toxic fuels is unprecedented.29, 44 However, the still efficient propulsion at 560 nm without 
+
+glucose in MEM confirms that the other ingredients (including dissolved oxygen11, 19) may also 
+assist motion induced by photocatalysis, or at least do not hamper it. These experiments not only 
+show the superiority in performance over current inorganic microswimmers in high-salinity 
+media but also highlight how crucial facile redox species are that can act as fuel for propulsion, 
+akin to photocatalysis in general, and especially if sub-band gap trap states might be partially 
+involved (630 nm illumination).43, 45, 46 Such a substantial shift toward the red part of the spectrum 
+that can penetrate deeper tissues makes organic and small band gap microswimmers (especially 
+with trap states in the gap) attractive for micromachines not just in-vitro, but even for in-vivo 
+conditions. 
+Light-driven directional propulsion control 
+Phototaxis is the property by which microswimmers swim towards or away from the direction of 
+incident light (i.e., positive or negative phototaxis), which often depends on their surface 
+charge.47, 48 It enables direction control, opposite to random ballistic displacement usually 
+observed with Janus particles.11, 49 When the COF microswimmers were illuminated by a directed 
+light source from the side with a 45° angle, both TABP-PDA-COF and TpAzo-COF microswimmers 
+exhibit positive phototaxis, and swim toward the light that can propel them (Fig. 2 e, j, and video 
+S2). TABP-PDA-COF and TpAzo-COF particles move with mean speeds of 13.3 ± 1.8 µm/s and 7.6 
+± 0.8 µm/s, at 470 nm and 630 nm illumination in water and MEM, respectively. This apparent 
+increase in the particle speed compared to vertical illumination could be attributed to the larger 
+parallel component of the light direction to the propulsion direction when the samples were 
+illuminated from the side. When the samples are illuminated from the bottom,  only the side-
+wise motion component is measured as a common standard, artificially decreasing the actual 
+velocity.50, 51 Similar findings have been found on carbon nitride microswimmers, which were 
+discussed in more detail in our previous study.11   The required symmetry breaking is created by 
+the side-wise illumination and, thereby, an artificially created Janus structure results from the 
+self-shadowing of the microswimmers.13, 47  
+Biocompatibility of COFs 
+In order to be used in potential biomedical applications and to ascertain biocompatibility, 
+microswimmers should have no significant cytotoxicity. Hence, we tested the cytotoxicity of the 
+
+microswimmers with human umbilical vein endothelial cells (HUVEC) in dMEM with FBS. 
+Different concentrations of TAPB-PDA-COF and TpAzo-COF microswimmers (3.1-25 µg/ml) were 
+incubated with HUVECs in the dark, and their viability was investigated with calcein-based 
+live/dead fluorescence staining of the cells after 24 hours. The cells with TABP-PDA COF were 
+completely viable, and they did not show any significant decrease in viability even at high 
+concentrations, both with illumination and without illumination at 470 nm with maximum light 
+intensity, 10 mW/cm2, for 30 minutes), as seen in Fig. 3 a, which is visible also in live cell 
+fluorescent images in Fig. 3 b. TpAzo-COF (Fig. 3 c,d) shows lower cell viability in comparison with 
+TAPB-PDA-COF, with 93% and 75%  HUVEC cell viability in 25 µg/ml concentration (in dark and 
+with 630 nm illumination (10 mW/cm2), respectively). Also, at concentrations of 3.1 µg/ml, the 
+viability is decreased to 88% in comparison to the TABP-PDA COF. However, this fairly good 
+viability indicates that also the TpAZo COF can be used at lower concentrations for drug delivery 
+applications. Generally, illumination seems not to affect the viability at low concentrations (3.1 
+and 6.25 µg/ml), and only slightly at 12.5 and 25 µg/ml for both COFs. These results also suggest 
+that light-induced propulsion induces only minimal cytotoxicity in the range of light-driven 
+propulsion periods. Compared to carbon nitride microswimmers, which have a larger band gap 
+(2.5 eV, 450 nm) and a very low-lying valance band, and therefore enable more redox reactions 
+with organic matter, including cells in principle, the use of 470 nm or 630 nm light with our TABP-
+PDA COFs and TpAzo COFs shows potential for reduced cell death [with 97% and 88% cell viability 
+after 30 minutes of light in 3.1 µg/ml concentrations of TAPB-PDA-COFs and TpAzo-COFs, 
+respectively] and makes especially the TABP-PDA COFs more applicable to practical applications 
+such as drug delivery.11   A previous study with primary cells from mouse splenocytes further 
+confirmed no detectable level of IL-12 (a pro-inflammatory cytokine) in the untreated samples in 
+concentrations used above in the dark.52   
+Drug loading, drug delivery, and hyperthermia 
+To explore the COF microswimmer’s applicability to biological environments, we also studied 
+their potential as drug carriers with different pharmacological agents. The differently 
+pronounced textural and structural porosity of the TABP-PDA and TpAzo-COFs (see Fig. 1 and Fig. 
+S1-S4), which enables ionic tolerance (Fig. 2c,h), is not only beneficial for motion but also as space 
+
+to take up, transport and deliver therapeutic drugs. We studied and compared how the structural 
+features enable interactions with such cargo in the following experiments. For this reason, we 
+chose an imaging agent, indocyanine green (ICG), and two different pharmacological agents with 
+different Biopharmaceutics Classification System (BCS) classes: doxorubicin (DOX) (Class III) and 
+insulin (Class I).53 Also both pharmacological agents are currently used to treat common ocular 
+disorders.54 
+First, we tested the loading of DOX, a chemotherapeutic agent against various cancer types, 
+including retinoblastoma.55 200 µg of DOX was added to a suspension of 100 µg of COF 
+microswimmers dispersed in 1 mL MEM, resulting in 138 µg  DOX encapsulated (loading efficiency 
+of 138%) on the TABP-PDA-COF microswimmers after 24 hours, and 75% for TpAzo-COF. Due to 
+the small molecular size of DOX (~1.1 nm approximate molecular diameter), the molecule should 
+fit into the structural pores of both COF structures (3.4 nm and 2.5 nm), while adsorbing also on 
+the inner textural surface. The overall negative surface charge on both COF microswimmers 
+attracts the positively charged DOX molecules in physiological pH values and gives rise to stable 
+loading. Since the overall surface areas are similar within 10%, it appears that differences in 
+polarity or hydrogen bonding, possibly mediated by the carbonyl groups of TpAzo-COF, enable 
+electrostatic repulsions with the DOX molecules and interfere with DOX uptake in TpAzo-COF 
+structures, which is also correlated with the zeta potential measurements. While the positive 
+zeta potential of the TABP-PDA-COF (ζTABP-PDA-COF = 12.13 ± 1.28 mV) reduces agglomeration and 
+enables sufficient drug loading values, the negative zeta potential of the TpAzo-COF (ζTpAzo-COF = -
+19.67 ± 0.68 mV) leads to agglomerations and reduces drug loading due to electrostatic 
+repulsions.56 In addition, a lower crystallinity and thereby, possibly decreased accessible pore 
+volume of TpAzo-COF are expected to lead to reduced DOX uptake. Overall, the DOX uptake of 
+both COF materials is among the highest reported, relative to other artificial structures using 
+physical encapsulation.11, 57  
+The DOX release can be achieved by changing the pH to slightly more acidic conditions, i.e., from 
+pH = 7.2 to 5.5 (Fig. 3 e, g), which is achieved by adding HCl to PBS. The TABP-PDA microswimmers 
+release 95 µg of DOX within 60 minutes, which is significantly boosted compared to the weak, 
+passive release also observed (12 µg). The passive release is commonly observed when drugs 
+
+such as DOX are not entirely trapped or encapsulated within porous structures but physisorbed 
+to the surface. Encapsulation within the TpAzo-COF, with a more open texture, appears more 
+stable, as evidenced by the lower passive release at pH 7.2 (5 µg in 60 minutes). In line, a 
+reduction of pH to 5 only releases 7% in 60 min, whereas a pH 3.5 yields 25% and is more 
+reasonable as a release trigger. The acid-triggered DOX release in the TABP-PDA-COF and TpAzo-
+COF microswimmers can be seen in fluorescence imaging in Figure 3f,h, respectively. The 
+enhanced drug delivery of microswimmers at lower pH has the potential to enable the targeted 
+therapy in tumor or infection environments, which typically have acidic pHs.58, 59  
+We also studied the loading and release of peptide (insulin), a frequently used drug in diabetic 
+retinopathy and convenient for light-controlled drug release applications.60, 61 Its larger molecular 
+size of ~3 nm makes larger pore sizes on the COFs desirable to allow for an efficiently 
+encapsulated loading. Indeed, insulin loading was observed on both COFS, 60% for TABP-PDA-
+COF (3.5 nm pore size] and 40% for TpAzo-COF (2.5 nm pore size) (Fig. 3i,k), which suggests that 
+physisorption of the drugs occurs on the outer surface of the textural pores and that the 
+structural pores can assist stable uptake. 
+Similar to DOX release from the COF structures, changing pH enables insulin release from both 
+COFs. While the TABP-PDA-COF shows a continuously increasing cumulative release of 
+approximately 35 µg/ml within 60 min at pH 5 already, which may be desirable for slower dosing, 
+the TpAzo-COF releases its cargo rather instantly (within 10 min), and at lower amounts (~10 
+µg/ml in more acidic pH 3.3 again). With both drugs, no visible light-triggered release was 
+observed, opposite to the carbon nitride systems reported earlier with DOX. However, as seen 
+herein, the absence of such a property can be very beneficial since it enables the decoupling of 
+motion control and drug release, which would otherwise have to co-occur.11  
+As a third theranostic agent to load onto COFs, we used ICG dye, commonly used in diagnosing 
+retinal diseases.62 Firstly, we investigated ICG loading and near-infrared laser-induced 
+hyperthermia capabilities; then, we focused on medical imaging of ICG-loaded COF 
+microswimmers with photoacoustic imaging and optical coherence tomography. As is the case 
+with drug delivery, TABP-PDA-COF has a pore size larger than the size of the ICG (~2.9 nm 
+molecular diameter on its longest axis); hence, the drug is presumably loaded better into the 
+
+structural pores of the TABP-PDA COF (3.5 nm), while in the case of TpAzo-COF, it appears to 
+dominantly bond to the bigger, textural pores (Fig. 4a). After ICG was loaded onto the both, 
+TpAzo-COF and TAPB-PDA-COF microswimmers at two different loading levels (50% and 100%, 
+w/w), they were irradiated with a near-infrared (NIR) laser at 808 nm.63 ICG-loaded TABP-PDA-
+COFs achieved quick heating to 66 oC and 69 oC after only 3 minutes of 808 nm NIR irradiation for 
+50% and 100% loading, respectively. Compared to TABP-PDA-COFs, ICG-loaded TpAzo-COFs 
+heated up to 42 oC and 45 oC for 50% and 100% loading under the same NIR illumination 
+conditions (Fig 4 b, c). Heat generation and accumulation are always affected by heat transport 
+to the environment. Assuming similar absorption and hence heat generation at the same 
+loadings, these findings indicate that indeed, ICG transfers the heat slightly better by binding to 
+TABP-PDA COF, and that the TpAzo COF dissipates accumulated heat faster to the environment 
+due to its more open shape, and thereby reaches lower temperatures over extended times. In 
+both cases, this NIR-controlled hyperthermia behavior of both COFs could be helpful for novel 
+intraocular photodynamic therapy application, which is already in the clinical trial phase for ICG 
+dye.64 Compared to other novel intraocular photothermal therapy agents in the recent literature, 
+especially  TABP-PDA-COFs with pores enabling ICG uptake into the material’s structural pores 
+and intense heating from 25 oC to 69 oC in 3 minutes, shows significant potential for the 
+photodynamic combined therapy applications that are used to degrade cells by heat 
+generation.65, 66 
+Photoacoustic imaging and optical coherence tomography  
+Imaging microswimmers as they move in different fluids is one of the most critical enablers for 
+their potential in vivo applications.67 For this purpose, we selected to study two clinical imaging 
+methods: optical coherence tomography (OCT) and photoacoustic (PA) imaging. OCT is the gold 
+standard high-resolution clinical imaging method to observe intraocular structures and is 
+accessible in most ophthalmology clinics worldwide.68 PA is an emerging imaging technique that 
+combines the resolution of optical imaging with the depth of penetration of ultrasound imaging. 
+In recent years, PA has been used in ophthalmology as it shows significant advantages in imaging 
+deep ocular structures, such as lymphatic drainage and choroidal vasculature.69, 70 While in the 
+PA imaging method ICG was used as a contrast agent to enhance the visualization of COF 
+
+microswimmers in the complex environment of intraocular fluids, COFs were imaged in 
+intraocular structures and ocular fluids without any contrast agent during OCT imaging. Different 
+concentrations of ICG are loaded onto the COF microswimmers and imaged under photoacoustic 
+imaging (Fig. 4d,e). While TAPB-PDA-COFs achieve up to 500 mean pixel intensity (MPI) at 815 
+nm, which is the highest peak in the emission spectrum of ICG, TpAzo-COFs achieve 250 MPI 
+under the same imaging conditions. These signal intensity increases correlate with the 
+concentration of the ICG in the COF loading suspension and also the drug uptake ability of both 
+COFs, which correlate with other drug loading experiments. Imaging uptake and delivery of 
+therapeutic agents on microswimmers will be helpful in the targeted in vivo drug delivery 
+experiments.71   
+As a next step, the light-driven propulsion of the COF microswimmers in intraocular fluids was 
+observed using PA imaging. In both vitreous and aqueous fluids, COFs were illuminated in the 
+same fashion as in the light-induced swimming experiments in various media and then observed 
+with photoacoustic imaging for 30 minutes (Fig. 5a-c). Except for TpAzo-COF in vitreous humor 
+under 630 nm light illumination, an increased ICG emission signal was observed in the focus areas 
+for all experimental groups. These results indicate that the light-driven collective motion of both 
+COF microswimmer types could be trackable under PA imaging.  
+For clinical applicability, we observed and measured the light-driven swimming of COF 
+microswimmers in intraocular fluids under real-time OCT. While the mean speeds of the smaller 
+and spherical TAPB-PDA-COFs were 12.1 ± 1.7 µm/s in aqueous humor and 7.6 ± 0.8 µm/s (~16.8 
+BLPS) in the vitreous humor, mean speeds of TpAzo-COFs were slightly increased to 14.2 ± 1.5 
+µm/s in aqueous humor and 8.8 ± 1.0 µm/s (~1.25 BLPS) in the vitreous humor under 470 nm 
+light illumination (Fig. 5d and Video S3). Compared to the previous intraocular microrobotic 
+studies employing magnetic actuation of helical microswimmers, the speed of the 
+microswimmers in terms of BLPS was significantly higher, ~16.8 BLPS in the current study vs. ~5.3 
+BLPS for the fastest magnetic intraocular microswimmers previously.23 Light-driven accumulation 
+behavior of both COF microswimmer types in the focus of the light was trackable under real-time 
+OCT imaging without any contrast agent loading (Fig. 5e and Video S4). Additionally, their light-
+driven propulsion in 470 nm wavelength light was also trackable even inside an ex vivo porcine 
+
+eye with anterior segment OCT imaging (Video S5). The COF-based microswimmers are the first 
+intraocular microswimmers that can swim and be trackable inside the eye without any contrast 
+agent or surface modification. TpAzo-COFs were actuated faster, opposite to the previous 
+experiments, which highlights that a perfectly spherical shape of TAPB-PDA-COFs alone is not of 
+dominating benefit for mesh-like heterogeneous structures. Although the reasons for this 
+inverted swimming speed remain to be clarified and likely depend on photocatalytic reaction 
+rates in the respective environment, it is possibly also linked to the increased viscosity and 
+fibrillary mesh structures in the aqueous and vitreous humor that overall decrease the propulsion 
+speed of both COF microswimmer types compared with previous aqueous conditions.30 These 
+results show that both COF microswimmer types are suitable microrobotic drug delivery agents 
+under both PA and OCT imaging, while enabling actual biomedical applications inside body fluids, 
+especially for intraocular structures.  With the help of their promising drug delivery and NIR-
+based hyperthermia abilities, they could solve the active retinal drug delivery problems in various 
+ocular disorders.54 They could be easily loaded with DOX for chemotherapy without adverse 
+effects on retinoblastoma patients or with insulin to treat increased ocular pressure.72, 73 COF-
+based microswimmers can easily be controllable with visible light, instead of other passive 
+nanomedicine agents in ophthalmology clinics and they do not require complex and unalterable 
+magnetic coil setups with narrow working spaces.21, 23   
+ 
+Conclusion and Outlook 
+In this manuscript, we have studied two structurally and texturally distinct COF microswimmer 
+types with tunable nanopore sizes towards their potential intraocular medical applications as 
+multifunctional microswimmers. This comparison of COFs from two different families with 
+distinct morphologies and drug loading capabilities yielded promising results in terms of 
+biocompatibility, imaging, drug delivery, and visible light-induced propulsion in ionic and 
+biological media, surpassing the applicability of current magnetically actuated microswimmer-
+based systems – without a need of further structural modification or sophisticated structural 
+engineering. Simultaneously, the COF microswimmers can be propelled by visible and even red 
+
+light in ionic and biological conditions (Fig. 2). Although some medium-dependent propulsion 
+trends at low salt concentrations remain to be clarified, their porous structure, coupled with 
+photocatalytic activity, seems key to efficient photocatalytic motion without dedicated toxic fuels 
+or harm to the tissue. A compact spherical shape, as achieved by the size-modified synthesis of 
+the TABP-PDA COFs, appears beneficial for fast propulsion, enabling bubble-free motion at 36 
+BLPS while opening up possibilities for mobility in the intraocular region. On the other hand, large 
+and texturally more porous structures, as observed for the TpAZo-COF, enable similar absolute 
+propulsion speeds in ionic conditions, albeit at a much-reduced speed relative to their size (~2 
+BLPS). The explanation for this behavior remains to be found and rationalized by numerical 
+models, especially since simple fluid dynamics and the applicability of Reynolds numbers, which 
+do not include inner flow, are not suited for these systems.74 Both microswimmers allow for 
+precise motion control as single particles by their phototactic properties, enabling complex 
+curvilinear navigation around obstacles in principle and collective motion for particle (re-
+)assembly (Fig. 2).11, 19, 75  We show that large structural and textural pores enable the loading of 
+different drugs and dyes (e.g., insulin, ICG, and DOX) but that the pore size itself only plays a 
+partial role in (stable) uptake, since textural surface area also contributes to drug binding, as 
+clearly visible by the different uptake properties of the small drug DOX (Fig. 3), whereas larger 
+molecules, such as insulin or ICG, can stay more stably bound, even in lower loading amounts 
+(Fig. 4). Since the drug binding and release is also affected by chemical interactions between the 
+COF backbone and the drug, independent of pore size and surface area, future material design 
+should focus on optimizing these interaction factors to broaden our insights. 
+The versatility of COFs, not only on the morphological but especially on a molecular level, is 
+anticipated to enable tailored approaches to tune the adsorption and desorption properties of 
+drugs, akin to their use on gas sorption.76 Modifications of these interactions, especially by 
+external stimuli, such as pH changes, light, viscosity changes, and oxygen content in the vicinity, 
+can enable the desired interaction strength with the cargo and its release kinetics.11 This 
+possibility is anticipated to enable tailored, targeted, and especially semi-autonomous therapy 
+not only for in vitro but also for in vivo applications.77, 78 
+
+We further demonstrated medical imaging of the ICG-loaded COFs, enabled by photoacoustic 
+imaging and optical coherence tomography. In principle, both of them enable the visualization of 
+swarms and motion of large individual particles, providing more detailed insights into local 
+propulsion and release properties inside the eye or soft tissues where visible light cannot 
+penetrate easily. Since the ICG loading can be kept very low in the porous COFs while maintaining 
+a high signal intensity (Fig. 4). Optical coherence tomography inside eye tissue also enables real-
+time imaging studies of drug-loaded microswimmers and evaluation in intraocular fluids and 
+structures, laying the grounds for a more detailed understanding of release properties and burst 
+kinetics for various theranostic agents. By decoupling COF microswimmers’ motion control and 
+release mechanism, a broad range of independent functionalities is made possible on these 
+porous organic structures in parallel. We anticipate that especially simultaneous imaging, drug 
+release, and NIR light-assisted photothermal therapy capabilities will offer additional theranostic 
+abilities beyond what current state-of-art noninvasive photodynamic therapy techniques could 
+achieve.79 In the near future, they could be functionalized in ophthalmology clinics for 
+multimodal therapy and imaging of retinal diseases, such as retinoblastoma, diabetic 
+retinopathy, or glaucoma. 
+ 
+Materials & Methods 
+Synthesis and preparation of covalent organic frameworks 
+Synthesis of TAPB-PDA-COF was carried out according to a previous report with minor changes.34 
+In a typical colloidal reaction, 1,3,5- tris(4-aminophenyl)benzene (TAPB) (0.030 mmol, 10.4 mg) 
+and terephthaldehyde (PDA) (0.044 mmol, 5.96 mg) were dissolved in 14 mL acetonitrile. After 
+10 minutes of sonication, a solution of Sc(OTf)3 (0.014 mmol, 7.00 mg) in 7 mL acetonitrile was 
+added dropwise at room temperature under slight stirring. After 24 hours of reaction, the solvent 
+was exchanged for distilled water by centrifugation for five times (795 g for 10 minutes each). 
+For solids characterization, the particles were precipitated by adding 0.5 mL of 1 M NaCl solution, 
+washed with methanol, and dried by supercritical CO2 on a Leica EM CPD300 instrument. TpAzo-
+COF was synthesized according to a previous report.80 
+
+Brunauer–Emmett–Teller (BET) measurements and analysis  
+Nitrogen sorption measurements were performed on a Quantachrome Instruments Autosorb iQ 
+MP at 77 K. Before the gas adsorption studies, the samples were degassed for 12 h at 120 °C 
+under a vacuum. Multipoint BET surface area calculations and pressure ranges were chosen 
+according to the linear region on the BET plot in the range between 0.05 and 0.35 P/P0. Pore size 
+distribution was determined from Nitrogen adsorption isotherms using the NLDFT cylindrical 
+pores in the carbon model for nitrogen at 77 K. 
+PXRD measurements and analysis 
+Powder X-ray diffraction experiments were performed on a Stoe Stadi P diffractometer (Cu-Kα1, 
+Ge(111) in Debye-Scherrer geometry. The samples were measured in sealed glass capillaries (OD 
+= 1.0 mm) and spun for improved particle statistics. 
+Transmission electron microscopy (TEM) and scanning electron microscopy (SEM) 
+Transmission electron microscopy was performed with a Philips CM30 ST (300kV, LaB6 cathode). 
+The samples were prepared dry onto a copper lacey carbon grid (Plano). Images were recorded 
+with a TVIPS TemCam-F216 CMOS camera. The program EM-Menu 4.0 Extended was used for 
+analysis. 
+SEM images were obtained on a Zeiss Merlin or a VEGA TS 5130MM (TESCAN) with an InLens 
+detector using electron energy of 1.5 kV. The samples were cast on indium-doped tin oxide (ITO) 
+substrates, and a 3 nm-thick iridium film was sputtered on them to reduce charging. 
+UV-VIS measurements and analysis 
+For diffuse reflectance UV–visible absorption, spectra were collected on a Cary 5000 
+spectrometer (referenced to barium sulfate). Absorption spectra were calculated from the 
+reflectance data using the Kubelka-Munk and assuming a direct band gap.81 
+Zeta potential measurements 
+The Z potential was determined using a Malvern nano Zs zetasizer. Dispersions of 0.5 mg/mL COF 
+in 10 mM aqueous NaCl were sonicated 15 min before zeta potential experiments. Surface charge 
+values represent the mean of 3 experiments and their standard deviation is indicated.    
+
+Light-driven propulsion experiments 
+The spectral irradiance of the illumination in the microscope was measured at the place of the 
+sample chamber with a calibrated Ocean Optics OCEAN-FX-XR1-ES spectrophotometer after 
+attenuation by a neutral density filter. The results have been normalized to the filter attenuation 
+and the spot size of the light beam in the microscope. It was measured to be 2.0 ± 0.5 mm in 
+diameter, resulting in a relative experimental error of 50% after the error propagation 
+calculation. In the case of visible light propulsion, a broad-spectrum low-intensity white LED is 
+illuminated from the top, and lights with various wavelengths (385 nm, 470 nm, 510 nm, 560 nm, 
+and 630 nm) are illuminated through the microscope objective. The intensity of the microscope 
+light (1 mW/cm2 for the control experiments in the dark and 2 mW/cm2 for imaging during UV 
+light-based propulsion) was increased to 10 mW/cm2 for visible light propulsion. For 
+photocatalytic and PEC experiments, a calibrated Thorlabs S425C/PM100D optical power meter 
+directly measured the light intensity.19 All light intensities are used in the light propulsion 
+experiments under the ocular safety limit (54 mW/cm2) for ophthalmic devices.82  
+Biocompatibility experiments 
+Human umbilical vein endothelial cells (CRL-1730 [HUVEC], ATCC, Manassas, VA) were grown in 
+dMEM supplemented with 10% (v/v) FBS and 1% (v/v) penicillin/streptomycin (Gibco, Grand 
+Island, NY, USA) at 37°C in a 5% CO2, 95% air-humidified atmosphere. Cells were reseeded after 
+growing to confluence into μ-Slide eight-well plates (Ibidi GmbH, Gräfelfing, Germany) at a cell 
+density of 25 x 103 cells/well and incubated for two days. HUVEC cells were incubated with  TAPB-
+PDA or TpAzo COF microswimmers at varying concentrations (3.1 to 25 μg/ml) for cytotoxicity 
+testing. Then, the cell viability was measured using a LIVE/DEAD assay (Thermo Fisher Scientific, 
+Waltham, MA) incorporating calcein-AM (green) and ethidium homodimer-1 (red) dyes. After 24 
+hours of incubation with the COF microswimmers, live-dead cell numbers were calculated from 
+fluorescence microscopy images. Furthermore, cytotoxicity of microswimmers during light 
+actuation (470 nm for TAPB-PDA and 630 nm for TpAzo, 10 mW/cm2 and 4 mW/cm2, respectively) 
+was tested by live/dead staining of HUVEC cells right after and 24 hours after actuation of COF 
+microswimmers for 30 minutes.11 
+
+Drug & ICG loading and release tests 
+The loading efficiency was measured by centrifuging the DOX (44583, Sigma-Aldrich, St. Louis, 
+USA) or insulin (I3661, Sigma-Aldrich, St. Louis, USA) loaded microswimmers and comparing the 
+optical density (OD) of the supernatant with the precalibrated OD of DOX or insulin (200 μg/ml) 
+at 480 nm. Both COF microswimmers (100 μg/ml) were dispersed with DOX or insulin (200 
+μg/ml), and this solution was stirred in the dark for 24 hours to allow the drugs to be adsorbed. 
+After 24 hours, the suspension was centrifuged, and the supernatant was used for measuring the 
+drug loading. The drug-loaded COF solution was washed three times with water and stored in 
+dPBS at +4°C for further delivery experiments. For the pH release, the pH of the resulting HCl-
+diluted PBS solution was checked using a pH meter to confirm the stability of the pH during the 
+release experiments.11 
+NIR-based remote heating of ICG-loaded COF particles 
+TpAzo-COF and TAPB-PDA-COF loaded with 50% and 100% ICG were loaded in microtubes and 
+irradiated with a NIR laser (808 nm, 0.6 W/cm2). Thermal images were obtained, and temperature 
+information was recorded with a thermal infrared camera (ETS320, FLIR Systems). 
+Photoacoustic imaging measurements and analysis 
+The photoacoustic (PA) signal characterizations were performed inside a Multispectral 
+Optoacoustic Tomography device (MSOT 512-element transducer, iThera Medical) system with 
+three scanning steps of 0.2 mm at different wavelengths. The samples with different 
+concentrations were prepared inside a transparent stripe and embedded in an agar phantom (1.5 
+g/100 mL agar-DI water). The same preparation was done for the control sample. The agar 
+phantom was placed at the center of the transducer arrays. The measurements were then taken 
+for a range of wavelengths (660 – 980 nm), and each image was repeated three times for each 
+laser pulse and then averaged. A circular region of interest (ROI) was chosen for calculating the 
+PA signal at each wavelength. Finally, the diagrams were plotted against the control sample for 
+all concentrations. 
+For PA imaging of light-induced motion of nanoparticles, a handheld 3D photoacoustic probe 
+(256-element transducer, iThera Medical) was used for real-time tracking. The laser wavelength 
+was set at 800 nm, and the image sequences were taken at 10 frames per second. Then, a 
+
+volumetric image of 20 × 20 × 20 mm³ was constructed from three orthogonal imaging planes. 
+The real-time change in the signal intensity at the light actuation spot indicated the movement 
+of the nanoparticles.  
+Optical coherence tomography (OCT)  
+The fresh porcine eyes were purchased from Ulmer Fleisch food factory, Ulm, Germany. Within 
+six hours after the euthanasia of the animals, a set of enucleated eyes stabilized to the holder, 
+and COFs were injected with a 30G syringe in the anterior chambers of the porcine eyes before 
+OCT imaging.  Besides that, aqueous humor was removed from another set of fresh porcine eyes 
+with the help of 30G trocar and cannula. For vitreous collection, a classical vitrectomy procedure 
+is followed.83  The intraocular fluids with COFs were injected into a cylindrical tubing and 
+observed via OCT (TEL320C1 – Spectral Domain OCT System, Thorlabs). The motion inside the leg 
+was recorded with an image speed at a medium sensitivity (76 kHz). The refractive index was set 
+to 1.00, and the Hann filter was used for the apodization window. The A-scan averaging was set 
+to 1, and the B-scan averaging to 1 with a pixel size of 6.5 μm. 
+ 
+Author contributions: F.P., V.S., B.V.L., and M.S. conceived and designed the project. F.P., V.S., 
+and E.Y. wrote the manuscript, with input and corrections from all authors. A.R. and L.Y 
+synthesized and characterized the materials. V.S. and X.L. performed the light propulsion 
+experiments and analyzed the data. E.Y. performed and analyzed in vitro biocompatibility tests. 
+X.L. and M.B.A. performed and analyzed drug loading experiments. B. A. performed and analyzed 
+NIR hyperthermia experiments. A.A., E.Y., and P.W. performed and analyzed the photoacoustic 
+imaging. E.Y. isolated porcine intraocular fluids and performed optical coherence tomography. 
+M.S., F.P., and B.V.L. supervised the research. All authors contributed to the discussion of the 
+data and overall results.  
+Data availability: All data are available from the corresponding author upon reasonable request. 
+Acknowledgments: The authors acknowledge Viola Duppel for SEM and TEM image acquisition. 
+We thank Julia Kröger for the fruitful discussions. Support by the Max Planck Society, the Bavarian 
+Research Network SolTech (B.V.L.), and the Deutsche Forschungsgemeinschaft (DFG) via the 
+
+cluster of excellence “e-conversion” (project number EXC2089/1–390776260) is gratefully 
+acknowledged. F.P. has received and acknowledges UKRI funding under the grant reference 
+EP/X027449/1.  E.Y. has received funding from the European Union’s Horizon 2020 research and 
+innovation program under the Marie Skłodowska-Curie grant agreement [PHOTODOCTOR].  
+ 
+References 
+1. 
+M.Sitti. Mobile microrobotics. MIT Press, Cambridge, MA 2017. 
+ 
+2. 
+Erkoc P, Yasa IC, Ceylan H, Yasa O, Alapan Y, Sitti M. Mobile Microrobots for Active 
+Therapeutic Delivery. Advanced Therapeutics 2019, 2(1): 1800064. 
+ 
+3. 
+Dupont PE, Nelson BJ, Goldfarb M, Hannaford B, Menciassi A, O'Malley MK, et al. A 
+decade retrospective of medical robotics research from 2010 to 2020. Sci Robot 2021, 
+6(60): eabi8017. 
+ 
+4. 
+Wang B, Kostarelos K, Nelson BJ, Zhang L. Trends in Micro-/Nanorobotics: Materials 
+Development, Actuation, Localization, and System Integration for Biomedical 
+Applications. Adv Mater 2021, 33(4): e2002047. 
+ 
+5. 
+Sitti M, Ceylan H, Hu W, Giltinan J, Turan M, Yim S, et al. Biomedical Applications of 
+Untethered Mobile Milli/Microrobots. Proc IEEE Inst Electr Electron Eng 2015, 103(2): 
+205-224. 
+ 
+6. 
+Mujtaba J, Liu J, Dey KK, Li T, Chakraborty R, Xu K, et al. Micro-Bio-Chemo-Mechanical-
+Systems: Micromotors, Microfluidics, and Nanozymes for Biomedical Applications. Adv 
+Mater 2021, 33(22): e2007465. 
+ 
+
+7. 
+Soto F, Wang J, Ahmed R, Demirci U. Medical Micro/Nanorobots in Precision Medicine. 
+Adv Sci (Weinh) 2020, 7(21): 2002203. 
+ 
+8. 
+Wu Z, Chen Y, Mukasa D, Pak OS, Gao W. Medical micro/nanorobots in complex media. 
+Chem Soc Rev 2020, 49(22): 8088-8112. 
+ 
+9. 
+Sitti M, Wiersma DS. Pros and Cons: Magnetic versus Optical Microrobots. Adv Mater 
+2020, 32(20): e1906766. 
+ 
+10. 
+Srivastava SK, Clergeaud G, Andresen TL, Boisen A. Micromotors for drug delivery in 
+vivo: The road ahead. Adv Drug Deliv Rev 2019, 138: 41-55. 
+ 
+11. 
+Sridhar V, Podjaski F, Alapan Y, Kroger J, Grunenberg L, Kishore V, et al. Light-driven 
+carbon nitride microswimmers with propulsion in biological and ionic media and 
+responsive on-demand drug delivery. Sci Robot 2022, 7(62): eabm1421. 
+ 
+12. 
+Kong L, Mayorga-Martinez CC, Guan J, Pumera M. Photocatalytic Micromotors Activated 
+by UV to Visible Light for Environmental Remediation, Micropumps, Reversible 
+Assembly, Transportation, and Biomimicry. Small 2020, 16(27): e1903179. 
+ 
+13. 
+Wang J, Xiong Z, Tang J. The Encoding of Light‐Driven Micro/Nanorobots: from Single to 
+Swarming Systems. Advanced Intelligent Systems 2021, 3(4): 2000170. 
+ 
+14. 
+Schmidt CK, Medina-Sanchez M, Edmondson RJ, Schmidt OG. Engineering microrobots 
+for targeted cancer therapies from a medical perspective. Nature communications 2020, 
+11(1): 5618. 
+
+ 
+15. 
+Vargason AM, Anselmo AC, Mitragotri S. The evolution of commercial drug delivery 
+technologies. Nature Biomedical Engineering 2021, 5(9): 951-967. 
+ 
+16. 
+Lin G, Richardson JJ, Ahmed H, Besford QA, Christofferson AJ, Beyer S, et al. 
+Programmable Phototaxis of Metal-Phenolic Particle Microswimmers. Adv Mater 2021, 
+33(13): e2006177. 
+ 
+17. 
+Vikrant K, Kim K-H. Metal–organic framework micromotors: perspectives for 
+environmental applications. Catalysis Science & Technology 2021, 11(20): 6592-6600. 
+ 
+18. 
+Li J, Yu X, Xu M, Liu W, Sandraz E, Lan H, et al. Metal-Organic Frameworks as 
+Micromotors with Tunable Engines and Brakes. J Am Chem Soc 2017, 139(2): 611-614. 
+ 
+19. 
+Sridhar V, Podjaski F, Kroger J, Jimenez-Solano A, Park BW, Lotsch BV, et al. Carbon 
+nitride-based light-driven microswimmers with intrinsic photocharging ability. 
+Proceedings of the National Academy of Sciences of the United States of America 2020, 
+117(40): 24748-24756. 
+ 
+20. 
+Haase F, Lotsch BV. Solving the COF trilemma: towards crystalline, stable and functional 
+covalent organic frameworks. Chem Soc Rev 2020, 49(23): 8469-8500. 
+ 
+21. 
+Ullrich F, Bergeles C, Pokki J, Ergeneman O, Erni S, Chatzipirpiridis G, et al. Mobility 
+Experiments With Microrobots for Minimally Invasive Intraocular Surgery. Investigative 
+ophthalmology & visual science 2013, 54(4): 2853-2863. 
+ 
+
+22. 
+Kim M-S, Lee H-T, Ahn S-H. Laser Controlled 65 Micrometer Long Microrobot Made of 
+Ni-Ti Shape Memory Alloy. Advanced Materials Technologies 2019, 4(12): 1900583. 
+ 
+23. 
+Wu Z, Troll J, Jeong H-H, Wei Q, Stang M, Ziemssen F, et al. A swarm of slippery 
+micropropellers penetrates the vitreous body of the eye. Science Advances 2018, 4(11): 
+eaat4388. 
+ 
+24. 
+Vyas VS, Lotsch BV. Materials chemistry: Organic polymers form fuel from water. Nature 
+2015, 521(7550): 41-42. 
+ 
+25. 
+Zhao W, Xia L, Liu X. Covalent organic frameworks (COFs): perspectives of 
+industrialization. CrystEngComm 2018, 20(12): 1613-1634. 
+ 
+26. 
+Wang H, Wang H, Wang Z, Tang L, Zeng G, Xu P, et al. Covalent organic framework 
+photocatalysts: structures and applications. Chem Soc Rev 2020, 49(12): 4135-4165. 
+ 
+27. 
+McCracken JM, Donovan BR, White TJ. Materials as Machines. Adv Mater 2020, 32(20): 
+e1906564. 
+ 
+28. 
+Zhou D, Zhuang R, Chang X, Li L. Enhanced Light-Harvesting Efficiency and Adaptation: A 
+Review on Visible-Light-Driven Micro/Nanomotors. Research 2020, 2020: 6821595. 
+ 
+29. 
+Zhou D, Zhuang R, Chang X, Li L. Enhanced Light-Harvesting Efficiency and Adaptation: A 
+Review on Visible-Light-Driven Micro/Nanomotors. Research (Wash D C) 2020, 2020: 
+6821595. 
+ 
+
+30. 
+Lee B, Litt M, Buchsbaum G. Rheology of the vitreous body. Part I: Viscoelasticity of 
+human vitreous. Biorheology 1992, 29(5-6): 521-533. 
+ 
+31. 
+Li RL, Flanders NC, Evans AM, Ji W, Castano I, Chen LX, et al. Controlled growth of imine-
+linked two-dimensional covalent organic framework nanoparticles. Chem Sci 2019, 
+10(13): 3796-3801. 
+ 
+32. 
+Cote AP, El-Kaderi HM, Furukawa H, Hunt JR, Yaghi OM. Reticular synthesis of 
+microporous and mesoporous 2D covalent organic frameworks. J Am Chem Soc 2007, 
+129(43): 12914-12915. 
+ 
+33. 
+Lohse MS, Bein T. Covalent Organic Frameworks: Structures, Synthesis, and 
+Applications. Advanced Functional Materials 2018, 28(33): 1705553. 
+ 
+34. 
+Li Rebecca L, Flanders NC, Evans AM, Ji W, Castano I, Chen LX, et al. Controlled growth 
+of imine-linked two-dimensional covalent organic framework nanoparticles. Chemical 
+Science 2019, 10(13): 3796-3801. 
+ 
+35. 
+Wu Z, Zhang Y, Ai N, Chen H, Ge W, Xu Q. Magnetic Mobile Microrobots for Upstream 
+and Downstream Navigation in Biofluids with Variable Flow Rate. Advanced Intelligent 
+Systems 2022, 4(7): 2100266. 
+ 
+36. 
+Wei M, Zhou C, Tang J, Wang W. Catalytic Micromotors Moving Near Polyelectrolyte-
+Modified Substrates: The Roles of Surface Charges, Morphology, and Released Ions. ACS 
+Appl Mater Interfaces 2018, 10(3): 2249-2252. 
+ 
+
+37. 
+Zhan X, Wang J, Xiong Z, Zhang X, Zhou Y, Zheng J, et al. Enhanced ion tolerance of 
+electrokinetic locomotion in polyelectrolyte-coated microswimmer. Nature 
+communications 2019, 10(1): 3921. 
+ 
+38. 
+Wang J, Xiong Z, Zheng J, Zhan X, Tang J. Light-Driven Micro/Nanomotor for Promising 
+Biomedical Tools: Principle, Challenge, and Prospect. Acc Chem Res 2018, 51(9): 1957-
+1965. 
+ 
+39. 
+Liang Z, Shen R, Ng YH, Fu Y, Ma T, Zhang P, et al. Covalent organic frameworks: 
+Fundamentals, mechanisms, modification, and applications in photocatalysis. Chem 
+Catalysis 2022, 2(9): 2157-2228. 
+ 
+40. 
+Xiao K, Chen L, Chen R, Heil T, Lemus SDC, Fan F, et al. Artificial light-driven ion pump for 
+photoelectric energy conversion. Nature communications 2019, 10(1): 74. 
+ 
+41. 
+Xiao K, Giusto P, Wen L, Jiang L, Antonietti M. Nanofluidic Ion Transport and Energy 
+Conversion through Ultrathin Free-Standing Polymeric Carbon Nitride Membranes. 
+Angew Chem Int Ed Engl 2018, 57(32): 10123-10126. 
+ 
+42. 
+Xu F, Wei M, Zhang X, Wang Y. Ion Rejection in Covalent Organic Frameworks: Revealing 
+the Overlooked Effect of In-Pore Transport. ACS Appl Mater Interfaces 2019, 11(48): 
+45246-45255. 
+ 
+43. 
+Gouder A, Jiménez-Solano A, Vargas-Barbosa NM, Podjaski F, Lotsch BV. 
+Photomemristive sensing via charge storage in 2D carbon nitrides. Materials Horizons 
+2022, 9(7): 1866-1877. 
+ 
+
+44. 
+Wang J, Xiong Z, Zhan X, Dai B, Zheng J, Liu J, et al. A Silicon Nanowire as a Spectrally 
+Tunable Light-Driven Nanomotor. Adv Mater 2017, 29(30). 
+ 
+45. 
+Kröger J, Podjaski F, Savaşçı G, Moudrakovski I, Jimenez-Solano A, Terban MW, et al. 
+Conductivity mechanism in ionic 2D carbon nitrides: from hydrated ion motion to 
+enhanced photocatalysis. ChemRxiv 2021. 
+ 
+46. 
+Kröger J, Jiménez-Solano A, Savasci G, Lau VWh, Duppel V, Moudrakovski I, et al. 
+Morphology Control in 2D Carbon Nitrides: Impact of Particle Size on Optoelectronic 
+Properties and Photocatalysis. Advanced Functional Materials 2021, 31(28): 2102468. 
+ 
+47. 
+Chen C, Mou F, Xu L, Wang S, Guan J, Feng Z, et al. Light-Steered Isotropic 
+Semiconductor Micromotors. Adv Mater 2017, 29(3): 1603374. 
+ 
+48. 
+Dai B, Wang J, Xiong Z, Zhan X, Dai W, Li CC, et al. Programmable artificial phototactic 
+microswimmer. Nat Nanotechnol 2016, 11(12): 1087-1092. 
+ 
+49. 
+Uspal WE. Theory of light-activated catalytic Janus particles. The Journal of Chemical 
+Physics 2019, 150(11): 114903. 
+ 
+50. 
+Uspal WE. Theory of light-activated catalytic Janus particles. J Chem Phys 2019, 150(11): 
+114903. 
+ 
+51. 
+You M, Chen C, Xu L, Mou F, Guan J. Intelligent Micro/nanomotors with Taxis. Acc Chem 
+Res 2018, 51(12): 3006-3014. 
+ 
+
+52. 
+Zhou Y, Liu S, Hu C, Cai L, Pang M. A covalent organic framework as a nanocarrier for 
+synergistic phototherapy and immunotherapy. Journal of Materials Chemistry B 2020, 
+8(25): 5451-5459. 
+ 
+53. 
+Mehta M. Biopharmaceutics Classification System (BCS): Development, Implementation, 
+and Growth. Wiley, 2017. 
+ 
+54. 
+Kompella UB, Amrite AC, Pacha Ravi R, Durazo SA. Nanomedicines for back of the eye 
+drug delivery, gene delivery, and imaging. Progress in Retinal and Eye Research 2013, 
+36: 172-198. 
+ 
+55. 
+Dimaras H, Kimani K, Dimba EAO, Gronsdahl P, White A, Chan HSL, et al. 
+Retinoblastoma. The Lancet 2012, 379(9824): 1436-1446. 
+ 
+56. 
+Lestari WA, Wahyuningsih S, Gomez-Ruiz S, Wibowo FR. Drug loading ability and release 
+study of various size small mesoporous silica nanoparticle as drug carrier. Journal of 
+Physics: Conference Series 2022, 2190(1): 012032. 
+ 
+57. 
+Ibrahim M, Abuwatfa WH, Awad NS, Sabouni R, Husseini GA. Encapsulation, Release, 
+and Cytotoxicity of Doxorubicin Loaded in Liposomes, Micelles, and Metal-Organic 
+Frameworks: A Review. Pharmaceutics 2022, 14(2). 
+ 
+58. 
+Lin Q, Pilewski JM, Di YP. Acidic Microenvironment Determines Antibiotic Susceptibility 
+and Biofilm Formation of Pseudomonas aeruginosa. Frontiers in Microbiology 2021, 12. 
+ 
+59. 
+Justus C, Dong L, Yang L. Acidic tumor microenvironment and pH-sensing G protein-
+coupled receptors. Frontiers in Physiology 2013, 4. 
+
+ 
+60. 
+Broichhagen J, Schönberger M, Cork SC, Frank JA, Marchetti P, Bugliani M, et al. Optical 
+control of insulin release using a photoswitchable sulfonylurea. Nature communications 
+2014, 5(1): 5116. 
+ 
+61. 
+Reiter CEN, Gardner TW. Functions of insulin and insulin receptor signaling in retina: 
+possible implications for diabetic retinopathy. Progress in Retinal and Eye Research 
+2003, 22(4): 545-562. 
+ 
+62. 
+Yannuzzi LA. Indocyanine Green Angiography: A Perspective on Use in the Clinical 
+Setting. American journal of ophthalmology 2011, 151(5): 745-751.e741. 
+ 
+63. 
+Gowsalya K, Yasothamani V, Vivek R. Emerging indocyanine green-integrated 
+nanocarriers for multimodal cancer therapy: a review. Nanoscale Adv 2021, 3(12): 3332-
+3352. 
+ 
+64. 
+Yannuzzi LA, Slakter JS, Gross NE, Spaide RF, Costa DL, Huang SJ, et al. Indocyanine green 
+angiography-guided photodynamic therapy for treatment of chronic central serous 
+chorioretinopathy: a pilot study. Retina 2003, 23(3): 288-298. 
+ 
+65. 
+Li L, Zeng Z, Chen Z, Gao R, Pan L, Deng J, et al. Microenvironment-Triggered Degradable 
+Hydrogel for Imaging Diagnosis and Combined Treatment of Intraocular Choroidal 
+Melanoma. ACS nano 2020, 14(11): 15403-15416. 
+ 
+66. 
+Liu D, Wu Q, Chen W, Chen K, Lin H, Liu F, et al. Nanoporous Gold Ring-Integrated 
+Photothermal Intraocular Lens for Active Prevention of Posterior Capsular Opacification. 
+Small 2022, 18(34): 2201098. 
+
+ 
+67. 
+Aziz A, Pane S, Iacovacci V, Koukourakis N, Czarske J, Menciassi A, et al. Medical Imaging 
+of Microrobots: Toward In Vivo Applications. ACS nano 2020, 14(9): 10865-10893. 
+ 
+68. 
+Drexler W, Morgner U, Ghanta RK, Kärtner FX, Schuman JS, Fujimoto JG. Ultrahigh-
+resolution ophthalmic optical coherence tomography. Nature Medicine 2001, 7(4): 502-
+507. 
+ 
+69. 
+Jeon S, Song HB, Kim J, Lee BJ, Managuli R, Kim JH, et al. In Vivo Photoacoustic Imaging 
+of Anterior Ocular Vasculature: A Random Sample Consensus Approach. Scientific 
+reports 2017, 7(1): 4318. 
+ 
+70. 
+Yücel YH, Cardinell K, Khattak S, Zhou X, Lapinski M, Cheng F, et al. Active Lymphatic 
+Drainage From the Eye Measured by Noninvasive Photoacoustic Imaging of Near-
+Infrared Nanoparticles. Investigative ophthalmology & visual science 2018, 59(7): 2699-
+2707. 
+ 
+71. 
+Aziz A, Holthof J, Meyer S, Schmidt OG, Medina-Sánchez M. Dual Ultrasound and 
+Photoacoustic Tracking of Magnetically Driven Micromotors: From In Vitro to In Vivo. 
+Advanced Healthcare Materials 2021, 10(22): 2101077. 
+ 
+72. 
+Fujiwara K, Yasuda M, Ninomiya T, Hata J, Hashimoto S, Yoshitomi T, et al. Insulin 
+Resistance Is a Risk Factor for Increased Intraocular Pressure: The Hisayama Study. 
+Investigative ophthalmology & visual science 2015, 56(13): 7983-7987. 
+ 
+
+73. 
+Velez G, Yuan P, Sung C, Tansey G, Reed GF, Chan C-C, et al. Pharmacokinetics and 
+Toxicity of Intravitreal Chemotherapy for Primary Intraocular Lymphoma. Archives of 
+Ophthalmology 2001, 119(10): 1518-1524. 
+ 
+74. 
+Yang M, Wysocki A, Ripoll M. Hydrodynamic simulations of self-phoretic 
+microswimmers. Soft Matter 2014, 10(33): 6208-6218. 
+ 
+75. 
+Wang J, Xiong Z, Zhan X, Dai B, Zheng J, Liu J, et al. A Silicon Nanowire as a Spectrally 
+Tunable Light-Driven Nanomotor. Advanced Materials 2017, 29(30): 1701451. 
+ 
+76. 
+Vyas VS, Haase F, Stegbauer L, Savasci G, Podjaski F, Ochsenfeld C, et al. A tunable azine 
+covalent organic framework platform for visible light-induced hydrogen generation. 
+Nature communications 2015, 6(1): 1-9. 
+ 
+77. 
+Zhang G, Li X, Liao Q, Liu Y, Xi K, Huang W, et al. Water-dispersible PEG-curcumin/amine-
+functionalized covalent organic framework nanocomposites as smart carriers for in vivo 
+drug delivery. Nature communications 2018, 9(1): 2785. 
+ 
+78. 
+Benyettou F, Kaddour N, Prakasam T, Das G, Sharma SK, Thomas SA, et al. In vivo oral 
+insulin delivery via covalent organic frameworks. Chemical Science 2021, 12(17): 6037-
+6047. 
+ 
+79. 
+Pham TC, Nguyen V-N, Choi Y, Lee S, Yoon J. Recent Strategies to Develop Innovative 
+Photosensitizers for Enhanced Photodynamic Therapy. Chemical Reviews 2021, 121(21): 
+13454-13619. 
+ 
+
+80. 
+Chandra S, Kundu T, Kandambeth S, BabaRao R, Marathe Y, Kunjir SM, et al. Phosphoric 
+Acid Loaded Azo (−N═N−) Based Covalent Organic Framework for Proton Conduction. 
+Journal of the American Chemical Society 2014, 136(18): 6570-6573. 
+ 
+81. 
+Chen Z, Dinh HN, Miller E. Photoelectrochemical water splitting, vol. 344. Springer, 2013. 
+ 
+82. 
+Yan B, Vakulenko M, Min SH, Hauswirth WW, Nirenberg S. Maintaining ocular safety 
+with light exposure, focusing on devices for optogenetic stimulation. Vision Res 2016, 
+121: 57-71. 
+ 
+83. 
+Mohamed S, Claes C, Tsang CW. Review of small gauge vitrectomy: progress and 
+innovations. Journal of ophthalmology 2017, 2017. 
+ 
+ 
+ 
+ 
+
+Graphical Abstract: 
+ 
+ 
+ 
+ 
+Conceptual illustration of light-driven and light-steered COF microswimmers towards targeted 
+intraocular drug delivery and photothermal therapy applications under optical coherence 
+tomography-based real-time imaging. 
+ 
+
+Drug Loaded COF
+Microswimmers
+Light-driven propulsion
+Visible Light Laser
+Optical Trapping &
+Source
+Real-time Imaging
+Targeted Drug Release &
+Photothermal Therapy
+Central Retina
+Eye
+Optical Coherence
+Tomography 
+Figure 1: Structural properties of the two types of COF particles used as light-powered 
+microswimmers. a-c: Imine-linked TABP-PDA-COF nanoparticles. a: Precursors for synthesis and 
+molecular structure of the 2D covalent organic framework that stacks in the third dimension. b: 
+Calculated pore size distribution from nitrogen sorption isotherms at 77 K (see Fig. S1, S2 for 
+details), highlighting a fairly uniform pore diameter of 3.4 nm. c: SEM image of TABP-PDA COF 
+nanoparticles with a narrow diameter distribution around 450 nm. d-f: Azo-linked TpAzo-COF 
+microparticles. d: Precursors for synthesis and molecular structure of the 2D network that stacks 
+in the 3rd dimension. e: Calculated pore size distribution from nitrogen sorption isotherms at 77 
+K (see Fig. S3, S4 for details), highlighting a relatively uniform pore diameter of 2.6 nm. f: SEM 
+images of the TpAzo-COF microparticles with a sponge-like structure and high levels of textural 
+porosity, including macropores and heterogeneous size distribution (6.97 ± 17.62 µm, see Fig. 
+S3, S4). 
+ 
+ 
+
+685m²/g
+0.3
+3.41nm
+200nm
+0.25-
+NH2
+Y
+0.2
+(p)^p
+TAPB-PDA
+0.1
+3.4nm
+0.05-
+PDA
+H2N
+NH2
+10
+20
+30
+40
+TAPB
+Pore width (nm)
+d
+0.2
+TpAzo-COF
+2.54 mm
+500.mm
+OHI
+HO
+OH
+TpAzo
+00
+H2N
+OH
+Azo
+Tp
+80
+Pana width (nm) 
+ 
+Figure 2: Optical properties and propulsion of TABP-PDA-COF and TpAzo-COF microswimmers 
+in water and ionic media and their phototaxis behavior. a, f: Absorbance properties and optical 
+band gap extracted from UV-Vis diffuse reflectance spectra of TABP-PDA-COF (a) and TpAzo-COF 
+(f) particles, respectively, measured in the solid state. b, g: Mean speeds of the COF 
+microswimmers illuminated in distilled water at different wavelengths under the microscope. 
+The dashed line denotes the local Brownian motion speed. Density: 100 µg/ml, N = 50 particles. 
+Error bar = S.D. c, h: Propulsion in NaCl with increasing concentration and wavelength highlighting 
+strong ionic tolerance for light-driven propulsion.  d, i: Comparison of propulsion speed in 
+different commonly used biological media (dPBS, MEM) and MEM modified by removing glucose 
+or adding FBS. Density: 100 µg/ml, N = 50 particles (a-d). Mean ± S.D. e, j: Phototactic control of 
+diluted COF microswimmer particles following illumination from the side (S=start, E=end of 
+trajectory).  
+ 
+ 
+
+Mumination directbion 
+Figure 3: COF microswimmer biocompatibility, drug loading, and triggered release properties. 
+a-d: In vitro cell viability results for COF microswimmers a, c: cell viability percentages of HUVEC 
+cells in the presence of increasing TAPB-PDA-COF and TpAzo-COF microswimmer concentrations 
+with/without 470 nm and 630 nm illumination, respectively, for 30 minutes, mean ± S.D. b, d: 
+Corresponding fluorescence images of live cells (green) and dead cells (red) with 25 μg/ml, 30 
+minutes, 470 nm and 630 nm, respectively. e-h: DOX uptake & release results for COF 
+microswimmers. e: TAPB-PDA-COF loading and release capacity with Doxorubicin (DOX) in MEM 
+at different pH over time, reaching 138% for TABP-PDA-COF loaded in MEM. f: Corresponding 
+fluorescence image of DOX (red) loaded TAPB-PDA-COFs at 25 µg/ml concentration. g: TpAzo-
+COF with 75% loading and their subsequent stepwise release at different pH conditions; in 
+
+LiV
+:Dead
+Live.Dead
+DoX Loaded Particles
+DOX Loaded Particles
+nsulin Loaded Particle
+Insulin Loaded Particlesneutral pH (7.2), slightly acidic conditions (pH=5), and acidic (3.3) as encountered around cancer 
+cells. h: Corresponding fluorescence image of DOX (red) loaded TpAzo-COFs at 25 µg/ml 
+concentration.  i-l: Insulin uptake & release results for COF microswimmers. i: Insulin loading of 
+TAPB-PDA-COF with 60 % loading in MEM and release in different pH values over time. j: 
+Corresponding fluorescence images of FITC (green) labeled insulin-loaded TAPB-PDA COFs. k: 
+Insulin loading of TpAzo-COF with 40% loading in MEM and its release at different pH values over 
+time. l: Corresponding fluorescence images of FITC (green) labeled insulin-loaded TpAzo-COFs. 
+All scale bars are 100 μm. 
+ 
+ 
+
+ 
+Figure 4: Indocyanine green (ICG) loading, imaging, and hyperthermia functions of both COF 
+microswimmer types. a: ICG uptake into suitable structural pores (TAPB-PDA-COF) or texturally 
+porous structures (Tp-Azo-COF). b,c: NIR-based heating of 50% and 100% ICG-loaded COF 
+particles.  d: Intensity of photoacoustic signal vs. ICG loading, highlighting high sensitivity regimes 
+at low loading concentrations for TAPB-PDA-COF microswimmers. e: The photoacoustic signal 
+intensity vs. ICG loading highlights high sensitivity regimes at low loading concentrations for 
+TpAzo COF microswimmers.  
+ 
+
+2.9nm
+ structure
+texture 
+Figure 5: Real-time imaging of COF motion by photoacoustic and optical coherence tomography 
+imaging modalities. a-c: Photoacoustic imaging of focused light-driven actuation of ICG-loaded 
+COFs in both intraocular fluids. After 30 min, the accumulation of COF microswimmers in the 
+focus of the light with different wavelengths is visible. d: Mean speeds of COF microswimmer 
+particles illuminated with 470 nm light in intraocular fluids (Video S3). e: Optical coherence 
+images of COFs in aqueous humor (Video S4). The COF swimmers’ light-driven movement on the 
+tubing’s light-applied side is visible. The scale bar is 500 μm on each axis. 
+ 
+
+No Light
+Light
+No LighitSupporting Information 
+ 
+Designing Covalent Organic Framework-based Light-driven Microswimmers 
+towards Intraocular Theranostic Applications 
+ 
+ 
+ 
+ 
+Figure S1. TABP-PDA COF structural analysis. a: Powder XRD after washing. b: FT-IR of the 
+precursors and the COF. c: BET surface area measurement for overall surface area analysis.  
+ 
+ 
+ 
+ 
+Figure S2. TABP-PDA COF particle morphology and structure. a: SEM image illustrating uniform 
+size distribution of the washed COF microparticles. b: Particle size distribution showing high 
+uniformity. c: TEM image showing a single COF nanoparticle consisting of crystalline domains 
+with a lateral size of approx. 50 nm.  
+ 
+
+300
+Experimental
+Simulated
+N-H
+Intensity (a.u.)
+Transmittance
+mmn
+C-H
+=0
+C=N
+50)
+-Adsorption
+TAPB
+PDA
+IDesaption
+TAPB-PDA COF
+5
+10
+15
+20
+25
+30
+35
+40
+4000
+3500
+3000
+2500
+2000
+1500
+1000
+500
+0
+0.4.
+20 (Degrees)
+Wavenumber(cm-1)
+PAP3μm
+20-
+15-
+5.
+250300350400450500550
+100nm
+Particle size (nm) 
+Figure S3: TpAzo-COF structural analysis. a: Powder XRD after washing. b: FTIR of the COF. c: BET 
+surface area measurement for overall surface area analysis.  
+ 
+ 
+ 
+Figure S4. TpAzo-COF particle morphology and structure. a: SEM image illustrating the 
+agglomerated structure of TpAzo-COF microparticles. b: SEM image (zoomed in) showing sponge-
+like inner structure with macropores. c: Particle size distribution showing non-uniformity of the 
+particle agglomerates. The particle size is centered around 7 µm. d: TEM image showing the 
+interconnection of crystalline COF nanosheets with a domain size of approximately 50 nm or less. 
+
+b
+a
+TpAzo exp.
+100
+BET surface
+TpAzosim.
+600
+90
+Intensity (a.u.)
+area: 635 m2 g-1
+80
+?
+400
+300
+70
+200
+Volume
+09
+100-
+a-Adsorption
+Desorption
+50-
+0
+0.2
+0.4
+0.6
+4000350030002500200015001000500
+0.8
+5
+10
+15
+20
+25
+30
+35
+40
+P/Po
+2e(Degree)
+Wavenumber(cm-1)500nm
+20μm
+100nmSupporting Videos 
+ 
+Video S1. Light-driven propulsion of 100 µg/ml TABP-PDA and TpAzo COF microswimmers 
+inside distilled water with a 470-nm wavelength light source 
+ 
+Video S2. Phototaxis behavior of TABP-PDA and TpAzo COF microswimmers inside MEM using a 
+directional 470-nm wavelength light source 
+ 
+Video S3. TABP-PDA COF and TpAzo COF microswimmer propulsion inside the porcine aqueous 
+and porcine vitreous humor fluid 
+ 
+Video S4.  Optical coherence tomography (OCT) imaging and guided trapping of TABP-PDA and 
+TpAzo COF microswimmers inside the aqueous humor fluid 
+ 
+Video S5. Optical coherence tomography (OCT) imaging and guided propulsion of TABP-PDA 
+and TpAzo COF microswimmers inside the anterior chambers of the porcine eye 
+ 
+

3NE0T4oBgHgl3EQfuwHF/content/tmp_files/2301.02610v1.pdf.txt

@@ -0,0 +1,1236 @@
+Feedback-Gated Rectified Linear Units
+Marco Kemmerling 1
+Abstract
+Feedback connections play a prominent role in
+the human brain but have not received much
+attention in artificial neural network research.
+Here, a biologically inspired feedback mecha-
+nism which gates rectified linear units is pro-
+posed. On the MNIST dataset, autoencoders with
+feedback show faster convergence, better perfor-
+mance, and more robustness to noise compared
+to their counterparts without feedback.
+Some
+benefits, although less pronounced and less con-
+sistent, can be observed when networks with
+feedback are applied on the CIFAR-10 dataset.
+1. Introduction
+The brain has served as inspiration for artificial neural net-
+works (ANNs) for decades. While these models are usually
+heavily simplified compared to the brain, they have seen
+significant successes in areas such as image recognition
+(Krizhevsky et al., 2012), speech recognition (Hinton et al.,
+2012), and machine translation (Sutskever et al., 2014) in
+recent times.
+Despite successes, it is clear that the average human brain
+is vastly more powerful and versatile than any model used
+in practice today, and as such it may be useful to investigate
+how and where exactly the brain and ANNs differ.
+One such discrepancy between ANNs and the brain is the
+existence of feedback, or top-down connections.
+While
+there is clear evidence of prominent feedback connections
+in the brain, ANNs have overwhelmingly been designed
+based on the feedforward paradigm, although networks that
+do not work solely on the feedforward principle exist and
+are called recurrent neural networks (RNNs). Most RNNs
+used in practice today focus on recurrent connections from
+one layer to itself (e.g.
+LSTM networks (Hochreiter &
+Schmidhuber, 1997)), which, while recurrent, arguably do
+not constitute top-down connections. These networks are
+1University
+of
+Maastricht,
+Maastricht,
+The
+Nether-
+lands.
+Correspondence
+to:
+Marco
+Kemmerling
+<m.kemmerling@student.maastrichtuniversity.nl>.
+typically applied on problems where the input consists of
+sequence data, where the recurrence allows for memory of
+previously seen elements of the sequence.
+However, the usefulness of recurrent connections or feed-
+back is not necessarily restricted to sequence data. If the
+input is image data, a first look, or pass, at an image could
+be used to construct a rough idea of what the image con-
+tains, as well as to identify areas of interest, which can then
+be further examined on a second pass.
+While the network architectures considered in this paper
+feature real top-down connections, the focus is not on the
+network topology itself, but on how these top-down con-
+nections influence the behaviour of single neurons, i.e. a
+mechanism for incorporating feedback.
+This feedback mechanism is derived from neuroscience lit-
+erature and examined from two broad angles: (1) Whether
+the feedback mechanism can in any way improve on stan-
+dard methods. Relevant metrics include convergence speed
+and performance quality of the trained network. (2) If ex-
+amining the feedback’s properties and how it behaves un-
+der certain conditions (e.g. noisy signals) can offer any in-
+sights into what role the feedback might fulfil in the brain.
+Needless to say, care has to be taken when trying to infer
+functionality of mechanisms in the brain from simplified
+artificial networks. Nevertheless, experimentation on arti-
+ficial models offers an intriguing opportunity, as they are
+naturally easier to investigate and manipulate than the real
+brain.
+In the remainder of this paper, some neuroscientific back-
+ground is explored in section 2 to serve as context for the
+feedback mechanism, followed by a description of the feed-
+back mechanism itself as it occurs in the brain (section 2.1).
+In section 3 the mechanism is adapted for use in ANNs and
+some practical considerations on its use are given in sec-
+tion 3.1. The following sections describe a range of experi-
+ments with the intention to provide answers to the research
+questions posed above.
+2. Neuroscientific Background
+The neocortex, part of the cerebral cortex, is a part of the
+brain that evolved in mammals comparatively recently. It
+comprises around 80% of the human brain (Markram et al.,
+arXiv:2301.02610v1  [cs.NE]  6 Jan 2023
+
+Feedback-Gated Rectified Linear Units
+2004) and is therefore often speculated to be responsible
+for the emergence of higher intelligence.
+The most abundant type of neuron in the neocortex is the
+pyramidal neuron, constituting between 70-85% of cells.
+In contrast to the remaining neurons in the neocortex, so
+called interneurons, which are mostly inhibitory, pyramidal
+neurons are excitatory (DeFelipe & Fari˜nas, 1992).
+As the name suggests, pyramidal neurons have a cell body
+roughly shaped like a pyramid, with a base at the bottom
+and an apex at the top. Pyramidal neurons have two types
+of dendrites: basal dendrites, originating at the base, and
+one apical dendrite, originating at the apex. This apical
+dendrite terminates in what is called the apical tuft, where
+heavy branching of the apical dendrite occurs. (DeFelipe
+& Fari˜nas, 1992).
+These apical and basal dendrites are not just differently lo-
+cated, but also serve different functions. Basal dendrites
+receive regular feedforward input, while the apical tuft den-
+drites receive feedback input (Larkum, 2013).
+The neocortex appears to have a distinct structure which
+is characterised by its organisation into layers as well as
+columns. The columnar organisation is based on the ob-
+servation that neurons stacked on top of each other tend to
+be connected and have similar response properties, while
+only few connections exist between columns. Columns are
+hence hypothesised to be a basic functional unit in the cor-
+tex, although this is somewhat debated in the neuroscience
+community (Goodhill & Carreira-Perpi˜n´an, 2002).
+The further organisation into six layers was proposed by
+Brodman in 1909 (Brodmann, 1909). Layers 1 and 6 are
+of particular interest here. Layer 1 consists of almost no
+cell bodies, but mostly connections between axons and the
+apical dendrites of pyramidal neurons (Shipp, 2007), i.e. it
+serves as a connection hub for feedback signals. Layer 6
+sends signals to neurons in the thalamus which then in turn
+sends signals to layer 1 neurons in the same column (Shipp,
+2007), i.e. layers 1 and 6 create a loop where feedback is
+sent from layer 6 and received by layer 1.
+2.1. Distal Input to Pyramidal Neurons
+As described above, apical tuft dendrites receive feedback
+input, which appears to modulate the gain of the corre-
+sponding neuron (Larkum, 2004). It is hypothesised that
+this is a way for the cortex to combine an internal repre-
+sentation of the world with external input, i.e. feedback
+to a neuron may predict whether this particular neuron
+should be firing, and even small feedforward input may
+lead the neuron to fire as long as the feedback signal is
+strong (Larkum, 2013).
+Taking both feedforward and feedback input into account,
+the firing rate of a neuron can be modelled as follows
+(Larkum, 2004):
+f = g(µS + αµD + σ + fβ(µD) − θ)
+(1)
+where f is the firing rate of the neuron, g the gain, µS the
+average somatic current (i.e. feedforward input), µD the
+average distal current (i.e. feedback input), α is an atten-
+uation factor, σ represents fluctuations in the current, θ is
+the firing threshold, and β(µD) is an increasing function of
+the dendritic mean current which saturates for values above
+some current threshold.
+3. Feedback-Gated Rectified Linear Units
+The model described in the previous section serves as a
+basis to derive an activation function which can replace
+the common rectified linear unit (ReLU) (Nair & Hinton,
+2010), i.e. f(x) = max(0, x).
+To arrive at a more practical activation function, g and θ are
+dropped from equation 1, since the threshold is modelled
+through the bias unit and the gain (i.e. slope) of a ReLU is
+by definition 1 and can thus be safely dropped. Dropping
+the summands αµD and σ is less justifiable, but since they
+do not contribute to the core property of gain increase, they
+will be disregarded here, arriving at the following simpli-
+fied relationship:
+f = µS + fβ(µD)
+(2)
+Removing f from the right hand side:
+f =
+1
+1 − β(µD)µS
+(3)
+What remains is an exact definition of β(µD), which, ac-
+cording to (Larkum, 2004), is “an increasing function of the
+dendritic mean current µ which saturates for values above
+1000pA“. In other words, the function is bounded, i.e. the
+gain cannot be increased to arbitrarily high values. Accord-
+ingly, some maximum value βmax the function can produce
+and a threshold value η which describes when this maxi-
+mum is reached need to be defined. Assuming a piecewise
+linear model, β(µD) is thus defined as follows:
+β(µD) = min
+�βmax
+η
+µD, βmax
+�
+(4)
+As there are no obvious values to assign to βmax and η,
+they are treated as hyperparameters. Since setting βmax to
+1 results in a division by 0 and a value of βmax > 1 causes
+a negative slope, βmax should be smaller than 1.
+
+Feedback-Gated Rectified Linear Units
+Plugging equation 4 into equation 3 yields:
+f =
+1
+1 − min( βmax
+η
+µD, βmax)
+µS
+(5)
+Since negative values for µS are not taken into account in
+the above equations, µS is replaced with max(0, µS), i.e.
+the classic ReLU function:
+f =
+max(0, µS)
+1 − min( βmax
+η
+µD, βmax)
+(6)
+3.1. Feedback-Gated ReLUs in Practice
+The feedback path attempts to mimic the top-down path in
+the brain. As such, the origin of feedback terminating in a
+layer should be a layer that is higher in the (feedforward)
+hierarchy.
+Since feedback from higher layers can only be computed
+if these higher layers have priorly received feedforward in-
+put, at least two time steps are needed to incorporate the
+modified ReLUs into a network. Concretely, some data,
+e.g. an image is fed into the network twice, where the first
+pass enables the computation of feedback which can then
+be utilised in the second pass. Although more than two
+timesteps are not required, it is possible to use an arbitrary
+number of timesteps, which is examined in section 4.1.1.
+Any layer that receives feedback requires an additional set
+of weights to compute µD. Specifically, each layer hi with
+size n receiving feedback from layer hj with size m intro-
+duces n × m additional parameters.
+The resulting networks can then be unrolled to create a
+feedforward network, so that for t timesteps, each layer oc-
+curs t times, while using the same weights at each timestep
+(see figure 1). Since the unrolled network is purely feedfor-
+ward, the standard backpropagation is a suitable learning
+rule.
+In convolutional neural networks (LeCun, 1989), feedback
+is implemented on a filter-wise basis, i.e. each neuron does
+not receive its own unique feedback signal, but rather ev-
+ery filter receives a unique feedback signal that is shared
+between all units belonging to that filter.
+Dropout (Srivastava et al., 2014) should be used by drop-
+ping out the same units in all passes. Otherwise, if e.g.
+dropout is only applied on the last pass, the remaining units
+will still receive signals from dropped out units in previous
+passes, which defeats the purpose of dropout.
+4. Experimental Results
+The preceding sections describe a feedback mechanism and
+how it can be implemented in practice. Here, a range of ex-
+Figure 1. Left: autoencoder with (partial) feedback. Right: Un-
+rolled autoencoder.
+periments is performed to observe how this feedback mech-
+anism changes the behaviour of ANNs. Several networks
+are applied on two datasets, MNIST (LeCun et al., 2010)
+and CIFAR-10 (Krizhevsky et al., 2014). Specifically, the
+experiments are designed to answer the research questions
+posed in the introduction: (1) whether feedback can im-
+prove the performance of ANNs, (2) whether observing
+how the feedback works in artificial models can reveal any
+clues on what function feedback has in the brain. Sections
+4.1.3, 4.1.4, and 4.2.2 serve to answer the latter question,
+where section 4.1.3 is more of a general analysis of feed-
+back, while sections 4.1.4 and 4.2.2 test whether feedback
+might increase the networks robustness to noise. The re-
+maining sections are concerned primarily with question (1)
+in that they test convergence speed and performance quality
+in various configurations.
+4.1. MNIST
+The MNIST dataset is composed of 28 × 28 pixel binary
+images of handwritten digits, split into 60000 training and
+10000 test instances (LeCun et al., 2010). Each image is
+associated with one of ten classes representing the digits
+between 0 and 9.
+The models used in the following experiments are based
+on a (non-convolutional) autoencoder with two encoding
+and two decoding layers. The input layer has dimension
+(1 × 784), the first encoding layer (E1) outputs data of di-
+mension (1×392), the second (E2) of dimension (1×196),
+the first decoding layer (D1) of dimension (1 × 392) and
+the second decoding layer (D2) restores the data back to its
+original dimension. Except for the final layer, each layer is
+followed by a ReLU activation. The final layer makes use
+of a sigmoid activation function.
+First experiments were performed with only a single feed-
+back connection between the first decoder and the first en-
+coder (see figure 1).
+
+D2
+不
+D1
+E2
+不
+E1
+个
+InputD2
+个
+个个个个
+个
+>E1
+InputFeedback-Gated Rectified Linear Units
+Figure 2. Test set loss of autoencoders with and without feedback.
+The dimension of the second encoding layer is 196.
+Figure 3. Test set loss of autoencoders with and without feedback.
+The dimension of the second encoding layer is 10.
+Optimal values for η and βmax were determined by a grid
+search (βmax = 0.95, η = 5).
+Figure 2 shows the loss curves for the autoencoder with
+and without feedback. While the autoencoder with feed-
+back converges noticeably faster, the difference is relatively
+small. It is conceivable that feedback might have a greater
+effect if the difficulty of the task is increased. While diffi-
+culty is not a well defined term, reducing the dimension of
+the second encoding layer (i.e. the bottleneck) can arguably
+be seen as an increase in difficulty.
+The dimension of the second encoding layer is thus reduced
+to 10 (this modification will persist in all subsequent ex-
+periments) and the experiment is repeated. Indeed, figure
+3 shows a much larger gap between the autoencoder with
+feedback and the one without it, supporting the hypothe-
+sis that feedback may be more beneficial on more difficult
+tasks.
+Figure 4. Autoencoder performance with varying numbers of
+timesteps. Each configuration was trained and evaluated 10 times.
+The curves shown are the averaged losses on the test set.
+4.1.1. MORE THAN TWO TIMESTEPS
+While at least two timesteps are required to incorporate
+feedback, it is not clear whether exactly two timesteps
+should be used or whether > 2 timesteps can be benefi-
+cial. To examine this, autoencoders with 1, 2, 4, 6, and 8
+timesteps are trained.
+The results, depicted in figure 4, show that more than two
+timesteps yield no or negligible improvement. This may
+of course be data and/or task dependent. Since MNIST is
+a fairly simple dataset (binary images, clear separation of
+background and foreground, etc.), it is not inconceivable
+that tasks on other datasets may benefit from more than two
+timesteps.
+4.1.2. COMPREHENSIVE FEEDBACK
+In the previous experiments, feedback is only sent from
+one decoding layer to one encoding layer. Naturally, there
+are many more possible configurations that incorporate fur-
+ther feedback connections. In the following experiment,
+each layer receives feedback from every layer above it, i.e.
+every possible top-down connection is present in the net-
+work. This will be referred to as comprehensive feedback,
+whereas the previous approach will be referred to as partial
+feedback.
+As shown in figure 5, the configuration explained above
+does not only converge faster than a standard autoencoder,
+but also settles to a smaller loss value, which was not the
+case when only partial feedback was applied.
+4.1.3. FEEDBACK VS CONSTANT GAIN
+In an effort to gain some understanding on how exactly
+feedback helps to improve performance, the frequency of
+different feedback values is examined.
+A distinction is
+made between feedback and gain, where feedback refers
+
+0.250
+Without feedback
+With feedback
+0.225
+0.200
+0.175
+B80
+0.150
+0.125
+0.D75
+DOEZ
+4000
+00
+8400
+ babches0.26 -
+Without feedback
+0.24 -
+With feedback
+0.22
+0.20
+ 0.18 -
+0.16
+0.14
+0.12
+0.10
+0
+DOZ
+4000
+00
+DOt8
+ babches0.26
+1 timestep
+2 timesteps
+0.24
+4 timesteps
+0.22
+6 timesteps
+8 timesteps
+0.20
+ 0.18
+0.16
+0.14
+0.12
+0
+DOEZ
+4000
+00
+DOt8
+ babchesFeedback-Gated Rectified Linear Units
+Figure 5. Loss on the test set of autoencoders without feedback,
+partial feedback, and comprehensive feedback. Note that the hori-
+zontal axis is different from previous figures, i.e. the training time
+is longer.
+to µD and gain refers to
+1
+1−min( βmax
+η
+(µD),βmax).
+Figure 6 shows the data as collected in a network with a
+single feedback connection.
+While there are some smaller gain values, the overwhelm-
+ing majority of values are the maximum gain the network
+can produce. This raises the question whether there is much
+benefit to learning feedback or whether it might be simi-
+larly beneficial to simply multiply all activation values by
+a constant.
+This is easily tested by setting the gain of every ReLU in
+the affected layer to a constant value of 10.
+As can be seen in figure 7, this does lead to a steeper loss
+curve than the standard autoencoder, although not quite as
+steep as that of the autoencoder with actual learned feed-
+back. Further, the performance after training is completed
+is worse than that of the standard autoencoder.
+Repeating this same experiment for more than one feed-
+back connection, i.e. for an autoencoder with comprehen-
+sive feedback, yields results as illustrated in figure 8.
+In this setup, the simple multiplication by a constant ini-
+tially converges even faster than the autoencoder with
+learned feedback. While it does not achieve the same per-
+formance as the feedback autoencoder in later stages of
+training, it is on par with the standard autoencoder’s per-
+formance.
+Clearly, the effects of feedback cannot be fully explained
+by this constant gain, but the idea of a constant gain seems
+to have some merit.
+Figure 6. Distribution of feedback (top) and gain (bottom) values
+collected in a network with partial feedback over the complete
+MNIST test set.
+Figure 7. Comparison of a standard autoencoder, an autoencoder
+with partial feedback, and an autoencoder with partial constant
+gain (the gain of all units in the second encoding layer is set to
+10)
+
+0.250
+Without feedback
+Partial feedback
+0.225
+Comprehensive feedback
+0.200
+0.150
+0.125
+0
+2500
+5000
+DOSE
+# betchesFeedbadk Distribution
+840000
+ODOM
+500000
+400000
+ODODE
+240000
+0
+40
+20
+0
+21
+40Gain Distribution
+COADO
+CODOST
+0
+2
+t
+8
+1f0.26
+Without feedback
+0.24 -
+Partial feedback
+Partial constant gain
+0.22
+0.20
+0.16
+0.14
+0.12
+0.10
+0
+2400
+4000
+00
+00
+ betchesFeedback-Gated Rectified Linear Units
+Figure 8. Comparison of a standard autoencoder, an autoencoder
+with comprehensive feedback, and an autoencoder with compre-
+hensive gain (the gain of all layers is set to 10).
+4.1.4. NOISY ACTIVATIONS
+While noisy signals are usually not an issue in artificial net-
+works, noise in the brain is very prevalent (Faisal et al.,
+2008). To see whether feedback makes the model more
+robust to noise, gaussian noise with zero mean and vari-
+ous standard deviations is added to the (pre-)activations of
+both the network with feedback and the one without it. The
+networks are only evaluated with added noise, training is
+performed without noise. Note that in the network with
+feedback, noise is added to the activations in both passes.
+h = f(W T x + b + N(0, σ2) )
+(7)
+As figure 9 shows, the use of feedback significantly in-
+creases the network’s robustness to noise. While this is not
+especially useful for machine learning models, it may be
+part of the reason why the feedback path exists in the brain.
+Figure 9. Gaussian noise with zero mean and standard deviation
+σ = 2.0 is added to networks with and without feedback. The
+top row shows input instances to the network, the middle and bot-
+tom row show reconstructions of the network without and with
+feedback (respectively).
+4.2. CIFAR-10
+The CIFAR-10 dataset is composed of 32×32 pixel colour
+images of various objects, split into 50000 training and
+10000 test instances. Each image belongs to one of the fol-
+Figure 10. Gaussian noise with zero mean and varying standard
+deviations (horizontal) is added to networks with and without
+feedback. The quality of the reconstruction, as measured by the
+loss function (vertical axis), with respect to the magnitude of the
+standard deviation is shown for both networks.
+lowing classes: airplane, automobile, bird, cat, deer, dog,
+frog, horse, ship, truck (Krizhevsky et al., 2014).
+4.2.1. AUTOENCODER
+Similarly to the MNIST experiments, an autoencoder is
+trained on the CIFAR-10 dataset. Again, the architecture
+consists of two encoding and two decoding layers. Con-
+trary to MNIST, the encoding/decoding layers used here
+are convolutional/transposed convolutional layers with 16
+5 × 5 filters.
+As figure 11 shows, the autoencoder with feedback clearly
+performs better than the one without it, although the differ-
+ence between the two is not as pronounced as it is in the
+MNIST experiments.
+Curiously, if batch normalisation (Ioffe & Szegedy, 2015)
+is used after the activation functions, feedback cannot im-
+prove on the performance of the standard autoencoder. This
+may suggest that somehow feedback and batch normalisa-
+tion are interacting in such a way that the feedback is ren-
+dered ineffective.
+4.2.2. NOISY ACTIVATIONS
+The experiment from section 4.1.4 is repeated on the
+CIFAR-10 dataset. The network employed is the autoen-
+coder without batch normalisation from the previous ex-
+periment.
+Since feedback increased the robustness to noise in the
+MNIST autoencoder, the same behaviour would be ex-
+pected here. However, as apparent in figure 13, the net-
+work with feedback is much more sensitive to (even small
+amounts of) noise than the one without feedback.
+This may be an indication that the feedback learned by
+
+0.250
+Without feedback
+Comprehensive feedback
+0.225
+Comprehensive constant gain
+0.200
+ 0.175
+0.150
+0.125
+0
+2400
+4000
+00
+00
+# betches7210414a5965ahthoez225
+Without Feedback
+With Feedback
+2D -
+15
+B80]
+LD 
+0.5 -
+0.D
+i
+2
+3
+4
+5
+6
+8
+standard deviationFeedback-Gated Rectified Linear Units
+Figure 11. Test set loss of autoencoders with and without feed-
+back on the CIFAR-10 dataset. Neither model makes use of batch
+normalisation.
+Figure 12. Test set loss of autoencoders with and without feed-
+back on the CIFAR-10 dataset. Both models make use of batch
+normalisation.
+Figure 13. Gaussian noise with zero mean and varying standard
+deviations is added to the CIFAR-10 autoencoders with and with-
+out feedback. Although this is not apparent due to the scale of the
+plot, the data for the network without feedback follows a similar
+shape to the one with feedback.
+the network is fundamentally different from the feedback
+learned in the MNIST experiments, such that it has a com-
+pounding effect on noise, rather than a rectifying one.
+4.2.3. CLASSIFICATION
+Classification on the CIFAR-10 dataset is performed using
+a convolutional neural network. The network consists of
+two convolutional layers with 64 filters of size 5 × 5, each
+followed by a max pooling (Zhou & Chellappa, 1988) layer
+with a 2×2 window and a stride of 2. The convolution and
+pooling layers are followed by a fully connected layer (200
+units) and a softmax (Bridle, 1990) layer. Batch normali-
+sation is applied after the pooling layers and dropout with
+a rate of 0.5 is applied after the pooling and the fully con-
+nected layers.
+To test whether feedback can improve classification per-
+formance, the network is trained with (comprehensive) and
+without feedback. Figure 14 shows only a marginal per-
+formance difference between the two networks, with the
+feedback network being slightly better. At the end of train-
+ing, the classification accuracy over the complete test set is
+about 0.7% higher for the network with feedback.
+Note that the network employed here makes use of batch
+normalisation, which, as shown in the previous sec-
+tion, may be problematic in combination with feedback.
+Whether this is the case here is not clear, since this particu-
+lar network does not converge when batch normalisation is
+disabled (be it with or without feedback).
+5. Conclusion
+The feedback mechanism presented here is able to im-
+prove performance of conventional networks both in terms
+
+Without feedback
+0.10
+With feedback
+0.08
+0.04
+0.02
+0
+1400
+2400
+3000
+4000
+50i00
+# betchesWithout feedback
+0.10 -
+With feedback
+0.08
+0.D2
+0.D0
+DOZ
+4000
+8000
+1000
+# babches0.35
+Without feedback
+0.30
+With feedback
+0.25
+0.20
+B80]
+0.15
+0.10
+0.05
+0.00
+0.00
+0.02
+0.04
+0.05
+0.08
+0.10
+standard deviationFeedback-Gated Rectified Linear Units
+Figure 14. Classification loss on the CIFAR-10 test set. The train-
+ing time of 200000 batches corresponds to 512 epochs.
+of convergence speed and performance of the trained net-
+work when applied on the MNIST dataset. The benefits
+of feedback are less clear, however, when applied on the
+CIFAR-10 dataset. In principle, an autoencoder with feed-
+back can outperform a corresponding autoencoder without
+feedback to a small degree, but this positive effect of feed-
+back is negated when batch normalisation is utilised in the
+autoencoders. Understanding this unfavourable interaction
+between feedback and batch normalisation may be an op-
+portunity to gain a deeper understanding on how feedback
+works and what role it fulfils.
+Feedback appears to have some positive effect when per-
+forming classification on CIFAR-10, although this effect
+is so small that drawing any firm conclusions seems ill-
+advised.
+When investigating the networks robustness to noise, an
+even larger divide between performance on MNIST and
+CIFAR-10 can be observed. On CIFAR-10, feedback is not
+only not beneficial, it actually heavily increases the net-
+work’s sensitivity to noise, while the MNIST autoencoder
+becomes more robust when feedback is present.
+A possible explanation for this difference across datasets
+could be that the effectiveness of the feedback mechanism
+is data-dependent, i.e. it may be leveraging the highly regu-
+lar structure of the MNIST dataset and is thus not as useful
+on the less regularly structured CIFAR-10 dataset.
+A further general difference between the experiments on
+the two different datasets is the use of convolutional lay-
+ers, which were used in all of the CIFAR-10 experiments,
+but not in any of the MNIST experiments. It may be that
+providing feedback on a filter-wise basis is too simplistic,
+or that some other aspect related to convolution is not con-
+ducive to the feedback mechanism. Further research on the
+combination of feedback and convolutional networks may
+lead to some configuration that allows for more clear bene-
+fits of feedback.
+Naturally, it might also be the case that the results on
+MNIST are merely an outlier, which somehow defies a
+more fundamental problem with the usage of feedback in
+current ANNs, e.g. it may be that backpropagation is not
+an ideal learning algorithm for feedback, or that feedback
+relies on more realistic models such as spiking neural net-
+works (Ghosh-Dastidar & Adeli, 2009).
+Should clear evidence arise that feedback is useful beyond
+MNIST, an interesting avenue of future research would be
+the creation of feedback based multi-modal models, where
+sensory inputs from multiple different sources are com-
+bined to perform e.g. a classification task. For instance,
+if a network receives both visual and auditory input, the
+barking of a dog may result (mediated by feedback) in a
+higher expectation to observe a dog in the visual input.
+Acknowledgements
+I want to thank Kurt Driessens, Mario Senden, and Alexan-
+der Kroner for their supervision during this project.
+References
+Bridle, John S. Probabilistic interpretation of feedforward
+classification network outputs, with relationships to sta-
+tistical pattern recognition. In Neurocomputing, pp. 227–
+236. Springer, 1990.
+Brodmann, Korbinian.
+Vergleichende Lokalisationslehre
+der Grosshirnrinde in ihren Prinzipien dargestellt auf
+Grund des Zellenbaues. Barth, 1909.
+DeFelipe, Javier and Fari˜nas, Isabel. The pyramidal neu-
+ron of the cerebral cortex: Morphological and chemical
+characteristics of the synaptic inputs. Progress in Neu-
+robiology, 39(6):563–607, 1992.
+Faisal, A. Aldo, Selen, Luc P. J., and Wolpert, Daniel M.
+Noise in the nervous system.
+Nature Reviews Neuro-
+science, 9(4):292–303, 2008.
+Ghosh-Dastidar, Samanwoy and Adeli, Hojjat.
+Spiking
+neural networks. International journal of neural systems,
+19(04):295–308, 2009.
+Goodhill, Geoffrey J and Carreira-Perpi˜n´an, Miguel ´A.
+Cortical columns.
+Encyclopedia of cognitive science,
+2002.
+Hinton, Geoffrey, Deng, Li, Yu, Dong, Dahl, George E,
+Mohamed, Abdel-rahman, Jaitly, Navdeep, Senior, An-
+drew, Vanhoucke, Vincent, Nguyen, Patrick, Sainath,
+Tara N, et al. Deep neural networks for acoustic mod-
+eling in speech recognition: The shared views of four
+research groups. IEEE Signal Processing Magazine, 29
+(6):82–97, 2012.
+
+225
+Without feedback
+With feedback
+200
+175
+150 -
+125
+1D0
+0.75
+0.50
+0
+25000 50000 75000 140000125000150000175000 240000
+# betchesFeedback-Gated Rectified Linear Units
+Hochreiter, Sepp and Schmidhuber, J¨urgen. Long short-
+term memory.
+Neural computation, 9(8):1735–1780,
+1997.
+Ioffe, Sergey and Szegedy, Christian. Batch normalization:
+Accelerating deep network training by reducing internal
+covariate shift. In International Conference on Machine
+Learning, pp. 448–456, 2015.
+Krizhevsky, Alex, Sutskever, Ilya, and Hinton, Geoffrey E.
+Imagenet classification with deep convolutional neural
+networks. In Advances in neural information processing
+systems, pp. 1097–1105, 2012.
+Krizhevsky, Alex, Nair, Vinod, and Hinton, Geoffrey.
+The cifar-10 dataset.
+online: http://www. cs. toronto.
+edu/kriz/cifar. html, 2014.
+Larkum, M. E. Top-down dendritic input increases the gain
+of layer 5 pyramidal neurons. Cerebral Cortex, 14(10):
+1059–1070, 2004.
+Larkum, Matthew. A cellular mechanism for cortical asso-
+ciations: an organizing principle for the cerebral cortex.
+Trends in neurosciences, 36(3):141–151, 2013.
+LeCun, Yann. Generalization and network design strate-
+gies. Connectionism in perspective, pp. 143–155, 1989.
+LeCun, Yann, Cortes, Corinna, and Burges, Christo-
+pher JC. Mnist handwritten digit database. AT&T Labs
+[Online]. Available: http://yann. lecun. com/exdb/mnist,
+2, 2010.
+Markram, Henry, Toledo-Rodriguez, Maria, Wang, Yun,
+Gupta, Anirudh, Silberberg, Gilad, and Wu, Caizhi. In-
+terneurons of the neocortical inhibitory system. Nature
+Reviews Neuroscience, 5(10):793–807, 2004.
+Nair, Vinod and Hinton, Geoffrey E. Rectified linear units
+improve restricted boltzmann machines. In Proceedings
+of the 27th international conference on machine learning
+(ICML-10), pp. 807–814, 2010.
+Shipp, Stewart. Structure and function of the cerebral cor-
+tex. Current Biology, 17(12):R443–R449, 2007.
+Srivastava, Nitish, Hinton, Geoffrey E, Krizhevsky, Alex,
+Sutskever, Ilya, and Salakhutdinov, Ruslan. Dropout: a
+simple way to prevent neural networks from overfitting.
+Journal of machine learning research, 15(1):1929–1958,
+2014.
+Sutskever, Ilya, Vinyals, Oriol, and Le, Quoc V.
+Se-
+quence to sequence learning with neural networks. In
+Advances in neural information processing systems, pp.
+3104–3112, 2014.
+Zhou, YT and Chellappa, R. Computation of optical flow
+using a neural network.
+In IEEE International Con-
+ference on Neural Networks, volume 1998, pp. 71–78,
+1988.
+
+Feedback-Gated Rectified Linear Units
+6. Appendix
+6.1. Hyperparameter Tuning
+As mention in section 4.1, optimal values for βmax and η
+are determined by a grid search. The initial grid is defined
+by η = [5, 10, 15, . . . , 50] and βmax = [0.1, 0.2, . . . , 0.8].
+The highest value for βmax (0.8) consistently shows the
+best performance regardless of η’s values, as exemplified
+by figure 15. Note that a high constant value of η with
+varying values of βmax will generally lead to less spread
+between the loss curves, since the activation function will
+be more sensitive to βmax when η is low.
+Figure 15. Autoencoder performance with varying hyperparame-
+ters. Top: η is fixed at 5 and βmax is varied, bottom: η is fixed at
+50 and βmax is varied.
+While higher values of βmax lead to better performance,
+the inverse relationship can be seen with η, i.e. lower values
+of η lead to better performance. This is illustrated in figure
+16.
+Figure 16. Autoencoder performance when βmax is fixed at 0.8
+and η is varied.
+A second grid search with η = [1, 2, 3, 4, 5], βmax =
+[0.8, 0.85, 0.9, 0.95] is performed to determine whether
+even lower/higher values can further improve performance.
+Indeed, increasing βmax to 0.95 leads to better perfor-
+mance, but further decreasing η is not advantegeous.
+6.2. Feedback-Controlled Threshold
+Equation 1 describes not only gain modulation through
+feedback, but also an adjustment of the activation functions
+threshold, i.e. αµD is one of the terms in the summation.
+While gain modulation is the main property of interest in
+this paper, it is conceivable that the change in threshold
+plays a significant part in this mechanism as well.
+Incorporating this threshold mechanism into equation 6
+leads to:
+f =
+max(0, µS + αµD)
+1 − min( βmax
+η
+µD, βmax)
+(8)
+where α is a parameter to be learned by the network. While
+α could also be set to a constant (tuned) value, prior exper-
+iments suggest that it is beneficial to let the network adjust
+alpha during the course of training.
+As can be seen in figure 17, the added threshold mecha-
+nism is not able to improve upon the network implementing
+the gain mechanism. Although the models with feedback-
+controlled threshold both perform better than the standard
+autoencoder, the model with only gain and no threshold
+mechanism still has the overall best performance.
+
+Eta: 5
+0.26 -
+betamax: 0.1
+0.24
+betamax: 0.2
+betamax: 0.3
+0.22
+-betamax:0.4
+betamax: 0.5
+0.20
+betamax:0.6
+betamax: 0.7
+betamax: 0.8
+0.16
+0.14
+0.12
+OT'O
+DOEZ
+4000
+DOt8
+# betchesEta: 50
+0.26 -
+betamax: 0.1
+0.24 -
+betamax: 0.2
+betamax: 0.3
+0.22
+betamax: 0.4
+betamax: 0.5
+0.20
+betamax: 0.6
+betamax: 0.7
+betamax: 0.8
+0.16
+0.14 -
+0.12
+O1O
+0
+DOEZ
+4000
+00
+DOt8
+# betchesBeta max: 0.B
+0.26 
+Eta: 5
+Eta: 10
+0.24
+Eta: 15
+0.22
+Eta: 20
+Eta: 25
+0.20
+Eta: 30
+Eta: 35
+Eta: 40
+0.16
+Eta: 45
+Eta: 50
+0.14
+0.12
+OT'O
+0
+DOZ
+4000
+00
+DOt8
+# bebchesFeedback-Gated Rectified Linear Units
+Figure 17. Performance of the standard autoencoder, an autoen-
+coder with feedback-controlled threshold, an autoencoder with
+feedback-controlled gain, and an autoencoder with both feedback-
+controlled threshold and gain on the MNIST test set.
+6.3. Input With Reduced Contrast
+Images with reduced contrast are presented to the trained
+(on regular contrast images) network, to see if the second
+pass can reconstruct an image that is more akin to a regular
+contrast image. To reduce the contrast, each pixel of the
+image is multiplied by some contrast factor 0 ≤ c ≤ 1.
+Figure 19 shows the absolute difference in mean pixel value
+between the first and second pass reconstructions for a
+number of different contrast factors. A high contrast in-
+put image leads to a larger difference in mean pixel value,
+while a low contrast image leads to a smaller difference
+between first and second pass reconstructions.
+Figure 18. Absolute difference in mean pixel value between first
+and second pass reconstructions as a function of different contrast
+factors (from 0.0 to 1.0 in 0.1 increments). A contrast factor of
+1.0 corresponds to no reduction in contrast, while a contrast factor
+of 0.0 means the input images are entirely black.
+Figure 19. From top to bottom: original image, contrast reduced
+image, first pass reconstruction, second pass reconstruction. The
+contrast reduced image was produced by multiplying the original
+image with a contrast factor of 0.5, i.e. each pixel in the con-
+trast reduced image has values in the range [0.0, 0.5] instead of
+[0.0, 1.0]
+6.4. Additional Figures
+The following figures contain additional data that was col-
+lected as part of the experiments in section 4.
+
+Test loss
+Standard AE
+0.250
+Only threshold
+Only Gain
+0.225
+Threshold + Gain
+0.200
+0.150
+0.125
+0.100
+0
+2500
+5000
+7500
+10000 1250015000 17500 24000
+# bebches0.030
+2nd pa
+0.025
+1st &
+0.020
+betw.
+0.D15
+Difference
+0.D10
+0.D05
+0.0O0
+0.D
+t0
+0.6
+0.B
+1D
+conbrast fectarFeedback-Gated Rectified Linear Units
+Figure 20. Visualisation of activations in the MNIST autoencoder
+for one particular test instance. The leftmost column corresponds
+to the input layer and the remaining columns correspond to the
+first encoding layer, the second encoding layer, the first decoding
+layer, and the second decoding layer, respectively. The number of
+rectangles in each column corresponds to the number of units in
+that layer. Larger values are represented by green coloured rect-
+angles, and smaller values by white ones. Top: first pass, bottom:
+second pass.
+Figure 21. T-SNE visualisation of the second encoding layer of
+the autoencoder over the whole MNIST test set. From top to bot-
+tom: first pass, second pass, first pass with noise (as described in
+section 4.1.4), second pass with noise.
+
+50
+25
+0
+25
+50
+75
+100
+75
+50
+-25
+0
+25
+5050
+25
+0
+25
+50
+75
+60
+40
+-20
+0
+21
+4025
++
+25
+50
+75
+60
+40
+-20
+0
+24
+40
+824
+0
+-20
+40
+80
+80
+60
+40
+-20
+0
+2
+40
+84Feedback-Gated Rectified Linear Units
+Figure 22. Histograms as seen in section 4.1.3, but for the autoencoder with comprehensive feedback.
+
+1
+Ho
+1 - min(e μo.βmax
+COIDODE
+1250001
+2500000
+Encoder
+OADOSE
+150000
+ODOS
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+2
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+6
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+7500
+40000
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+2
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+4
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+7
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+DOIDOEL
+0
+500
+400
+DOE-
+200
+DOL-
+0
+0.
+0
+4
+6
+8
+1Feedback-Gated Rectified Linear Units
+Figure 23. Different gain values are manually fed into the second encoding layer of the network and the resulting reconstruction is
+visualised. In each of the above images, one specific input image is presented to the network, but the gain is varied. In row i of each
+image, every unit of the second encoding layer receives a gain of 10, except for unit i, which receives a gain between 0 and 10, depending
+on the column it is in. When using the MNIST autoencoder with comprehensive feedback, it can be observed that only one unit in the
+second encoding layer has any variation in gain (the remaining ones have a constant gain of 10 regardless of the input). This one unit
+corresponds to the fourth row from the bottom of each image and seems to be responsible for setting the ‘intensity‘ of the reconstruction.
+
+77777777
+7777777
+777777
+777777
+7
+7
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+7
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+G6666665555Feedback-Gated Rectified Linear Units
+Figure 24. CIFAR-10 classification as seen in section 4.2.3. From
+top to bottom: test set accuracy, training set loss, training set accu-
+racy, training set loss after applying a moving average filter (win-
+dow size 100).
+
+0.B
+0.7
+0.6 -
+0.4
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+0.2
+Without feedback
+0.1
+With feedback
+0
+25000 50000 75000 140000125000150000175000240000
+#betches225
+Without feedback
+With feedback
+200
+175
+150
+LDO -
+0.75
+0.50
+0
+25000 50000 75000 140000125000150000175000240000
+# babches0.9 
+0.B
+0.7 -
+0.6 
+0.4
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+With feedback
+0
+25000 50000 75000 140000125000150000175000240000
+# babches13
+Without feedback
+12
+With feedback
+11 
+LD
+0.9
+0.B 
+0.7
+0.6 
+0
+75000 140000 125000 150000 175000
+# betches

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+Extending Source Code Pre-Trained Language
+Models to Summarise Decompiled Binaries
+Ali Al-Kaswan
+Delft University of Technology
+Delft, The Netherlands
+a.al-kaswan@tudelft.nl
+Toufique Ahmed
+University of California, Davis
+Davis, California, USA
+tfahmed@ucdavis.edu
+Maliheh Izadi
+Delft University of Technology
+Delft, The Netherlands
+m.izadi@tudelft.nl
+Anand Ashok Sawant
+University of California, Davis
+Davis, California, USA
+asawant@ucdavis.edu
+Prem Devanbu
+University of California, Davis
+Davis, California, USA
+ptdevanbu@ucdavis.edu
+Arie van Deursen
+Delft University of Technology
+Delft, The Netherlands
+arie.vandeursen@tudelft.nl
+Abstract—Binary reverse engineering is used to understand
+and analyse programs for which the source code is unavailable.
+Decompilers can help, transforming opaque binaries into a
+more readable source code-like representation. Still, reverse
+engineering is difficult and costly, involving considering effort
+in labelling code with helpful summaries. While the automated
+summarisation of decompiled code can help reverse engineers
+understand and analyse binaries, current work mainly focuses on
+summarising source code, and no suitable dataset exists for this
+task. In this work, we extend large pre-trained language models of
+source code to summarise de-compiled binary functions. Further-
+more, we investigate the impact of input and data properties on the
+performance of such models. Our approach consists of two main
+components; the data and the model. We first build CAPYBARA,
+a dataset of 214K decompiled function-documentation pairs
+across various compiler optimisations. We extend CAPYBARA
+further by removing identifiers, and deduplicating the data.
+Next, we fine-tune the CodeT5 base model with CAPYBARA to
+create BinT5. BinT5 achieves the state-of-the-art BLEU-4 score
+of 60.83, 58.82 and, 44.21 for summarising source, decompiled,
+and obfuscated decompiled code, respectively. This indicates that
+these models can be extended to decompiled binaries successfully.
+Finally, we found that the performance of BinT5 is not heavily
+dependent on the dataset size and compiler optimisation level.
+We recommend future research to further investigate transferring
+knowledge when working with less expressive input formats such
+as stripped binaries.
+Index Terms—Decompilation, Binary, Reverse Engineering,
+Summarization, Deep Learning, Pre-trained Language Models,
+CodeT5, Transformers
+I. INTRODUCTION
+Reverse engineering binary programs has many applica-
+tions, in particular, software security [1]. Binary reverse
+engineering is a hard task, requiring highly skilled reverse
+engineers [1, 2]. Disassemblers and decompilers can help
+in this process. Disassemblers transform the binary into a
+low-level intermediate representation, and decompilers lift
+the representation to a high-level programming language-like
+representation. But the output of decompilers is still difficult
+to read and understand [1, 3]. Much of the work that goes
+into reverse engineering a binary is spent labelling functions
+with semantic descriptions [1]. Current approaches [4–10]
+mainly focus on recovering aspects lost in the compilation
+and decompilation process, such as names and types. Existing
+works fail to address the inherent difficulties in binary code
+comprehensibility, namely, the need for a high-level overview
+of the code.
+For source code, methods exist to automatically generate
+summaries from code [11, 12]. Source code summarisation
+is used to automatically generate short natural language de-
+scriptions of code, which support program comprehension
+and aid maintenance [12, 13]. While these methods have
+been successfully applied to programming languages such as
+Python, Java and PHP [14–16], using pre-trained language
+models [14–16], none of these methods has been applied to
+the relatively syntactically-poor output of decompilers (see
+Figures
+1a and
+1b). Being able to quickly determine the
+context and application of a function, can save valuable
+analysis time, and greatly benefit reverse engineers. Function
+and variable names alone, are inadequate representations of the
+source code [12], which is why having descriptive summaries
+of binaries is desirable.
+Following [17], source code can be described as having
+two information channels: the algorithmic channel and the
+natural language channel. The algorithmic channel specifies
+the execution of a program (semantics), while the natural
+language channel explains the purpose and context of the
+program to humans [17]. The natural channel includes function
+and variable names, code comments and the specific human-
+readable structure of programs. Processors only consider the
+algorithmic channel to execute a program, while humans use
+both the algorithmic channel and the natural channel to under-
+stand a piece of code [17]. Furthermore, code is very regular
+and predictable, even more so than natural languages [18].
+The compilation process, which transforms readable code
+into executable binaries, removes much of the information
+contained in the natural channel. Especially stripped binaries
+— binaries of which the symbol table is removed — are
+challenging, since they have almost no identifiers at all as
+arXiv:2301.01701v1  [cs.CR]  4 Jan 2023
+
+can be observed in Figure 1c.
+The goal of this paper is to advance the field of binary
+reverse engineering by exploring the application of code
+summarisation to decompiled binaries by taking advantage of
+source code pre-trained language models.
+However, there exists no dataset of aligned binaries and
+source code summaries since this is a new and unexplored
+task. As pointed out by LeClair and McMillan, the lack of
+standardised datasets is a major barrier to ongoing research,
+which we will address for this task [19]. In this paper, we
+create a dataset containing pairs of decompiled and stripped-
+decompiled functions and summaries of these functions. Dur-
+ing the creation of this dataset, we conform to the current best
+practices for dataset construction [19, 20].
+We apply this dataset to an existing pre-trained language
+model using transfer learning, by fine-tuning this pre-trained
+model on our dataset. For this task, we selected a pre-trained
+CodeT5 model, which was only trained on source code [14].
+We perform experiments on this model to explore the
+impact of decompilation, and the importance of identifiers.
+Furthermore, we explore the impact of compiler optimisation
+levels, the dataset size and the level of duplication.
+Our findings are that the decompilation and alignment
+of stripped functions has a very high failure rate; and the
+resulting stripped model has low performance. But, we found
+that the model shows state-of-the-art performance with both
+decompiled code as well as demi-stripped stripped code, code
+of which the identifiers were removed after decompilation. Our
+experiments on data duplication and dataset size further show
+that these models can be trained with few data, and that while
+duplicates have a high impact on performance, their presence
+is not paramount to model performance.
+Our key result: language models pre-trained on source code
+can be fine-tuned on binaries, opening up a range of new
+possibilities for the automated analysis of binaries.
+To summarise, the main contributions of this paper are:
+• CAPYBARA1, a dataset of Combined Aligned de-
+comPiled BinarY code And Related Annotations. A novel
+dataset of aligned, C, decompiled, stripped-decompiled
+and demi-stripped summary pairs2 (Section III);
+• BinT53, a Binary summarisation CodeT5 model, a simple
+and straightforward adaptation of a source code trained
+code summarisation model to decompiled code using
+CAPYBARA (Section IV);
+• An empirical investigation on the impact of the properties
+of decompiled code and the properties of CAPYBARA
+(Sections V and VI);
+The materials, including the processed and raw data, the
+trained model checkpoints and steps to replicate our exper-
+iments, are openly available in our replication package4.
+1CAPYBARA: https://doi.org/10.5281/zenodo.7229809
+2Decompiled code with strip-like obfuscation applied
+3BinT5: https://doi.org/10.5281/zenodo.7229913
+4Replication package: https://github.com/AISE-TUDelft/Capybara-BinT5
+II. BACKGROUND
+In this section, we introduce the background of compilers,
+binary reverse engineering, transfer learning and the code
+summarisation task.
+A. Compilers and Optimisation Levels
+Compilers are programs that convert source code from one
+programming language to another, but generally, and in the
+context of this work, the term is used to refer to programs
+that translate high-level code, like C, to a lower-level language
+such as machine code or bytecode. For our work, we focus
+on the GNU Compiler Collection (GCC)5 and Clang/LLVM
+(Clang).6
+Compilers feature optimisation levels. Generally, the goal of
+optimisations is the improvement of runtime performance or
+program size at the expense of compilation time and the ability
+to debug [21]. Compilers use optimisation flags, grouped into
+optimisation levels, where each level uses a different set of
+optimisation flags.
+By default, if GCC is invoked without any optimisation
+options, the program will be compiled with -O0. -O1, -O2
+and -O3 incrementally apply more optimisation to the binary
+at the expense of a higher compilation time [22]. Optimisations
+can restructure and transform the program in relation to the
+source code, by changing the control flow or the data of the
+program [23]. This obfuscation can complicate the reverse
+engineering process by reducing the accuracy of tools [23].
+B. Ghidra
+Ghidra7 is a free and open-source reverse engineering
+toolkit developed by the US National Security Agency. Ghidra
+contains many separate analysis modules that allow a reverse
+engineer to analyse binaries. Ghidra features a disassembler,
+which assembles binaries back into an intermediate represen-
+tation. In the case of x86-x64 binaries like the binaries this
+work focuses on, the intermediate representation will be the
+Assembly language. The decompiler, on the other hand, is a
+processor language-agnostic transformation engine that takes
+the disassembled code and creates a source code representa-
+tion, namely pseudo-C. Pseudo-C follows the general language
+conventions of C, but it cannot be compiled.
+Observe the relatively simple rtp sess ssrc function from
+creytiv/re8 shown in Figure 1a. We compile the project using
+the -O3 compiler level as defined in the project. We decompile
+the binaries using Ghidra’s decompiler using the standard
+configuration, the resulting pseudo-code is shown in Figure 1b.
+We observe that aside from the function name, almost the
+entire natural channel has been destroyed by the compilation
+and decompilation process. The parameter and variable names
+are gone, any documentation is removed and the relatively
+simple logic has been unrolled to a much more difficult-
+to-understand representation. Ghidra also incorrectly labelled
+5GCC: https://gcc.gnu.org/
+6Clang: https://clang.llvm.org/
+7Ghidra: https://ghidra-sre.org/
+8re: https://github.com/creytiv/re
+2
+
+/**
+* Get the Synchronizing source for an RTP/RTCP
+Socket
+�→
+* @param rs RTP Socket
+* @return Synchronizing source
+*/
+uint32_t rtp_sess_ssrc(const struct rtp_sock *rs){
+return rs ? rs -> enc.ssrc : 0;}
+(a) Source rtp sess ssrc function
+ulong rtp_sess_ssrc(long param_1){
+uint local_14 ;
+if (param_1 == 0){
+local_14 = 0;
+} else {
+local_14 = * (uint *) (param_1 + 4);}
+return (ulong) local_14;
+}
+(b) Decompiled rtp sess ssrc function
+ulong FUN_00100d30 ( long param_1 ){
+uint local_14 ;
+if (param_1 == 0) {
+local_14 = 0 ;
+} else {
+local_14 = * (uint *) (param_1 + 4);}
+return ( ulong ) local_14 ;}
+(c) Stripped decompiled rtp sess ssrc function
+Fig. 1: Example source, decompiled and stripped code snippet
+many of the variable types and failed to identify the struct
+datatype.
+Using our trained BinT5 model we can summarise the
+decompiled code and generate the following summary: Get
+the source for an RTP/RTCP Socket. This summary gives us
+an indication of the purpose of the function. Integrating this
+generated summary into Ghidra increases the readability of
+the entire binary. Keep in mind that a reverse engineer has
+to understand not just this function, but hundreds of different
+functions in a single binary.
+C. Stripping
+Aside from compiling with higher optimisation levels, bi-
+naries can also be stripped to obfuscate the underlying code
+and to resist analysis [24]. Commercial off-the-shelf software
+is often stripped to reduce the memory and storage footprint
+of the binaries, and to resist analysis to protect the intellectual
+property of the creator. Many vulnerable and malicious bina-
+ries are, unfortunately, also stripped to resist security analysis
+and hide their faults [5].
+Unix and Unix-like operating systems include a strip utility.
+The strip utility removes any operands that are not nec-
+essary for the execution of the binary while ensuring that
+the execution of the binary remains unchanged. The exact
+implementation and what constitutes unnecessary operands are
+left to the implementor.9 The strip utility as implemented in
+GNU/Linux removes the symbol table from the binary. The
+symbol table contains each symbol’s location, type and name.
+Like higher optimisation levels, the use of stripping can
+greatly complicate the efforts to reverse engineer a binary,
+as well as reduce the accuracy and effectiveness of reverse
+engineering tools [24].
+For example, we compile, strip and decompile the function
+in Figure 1a, and the resulting stripped decompiled function
+is shown in Figure 1c. In addition to the details lost by the
+decompilation process, the stripper removed all symbols, like
+the function names.
+D. Code Summarisation Task:
+Code summarisation (also referred to as source code sum-
+marisation) is the task of writing short descriptions from
+source code, usually a single-sentence summary of the source
+code. The main use is for software documentation, like the
+one-sentence JavaDoc description used in Java [19]. This
+documentation is important for program comprehension and
+maintenance. But the process of writing and maintaining
+these descriptions is a labour-intensive and time-consuming
+task, which is where the benefits of automating that process
+arise. Automatic code summarisation is an active and popular
+research problem in the field of software engineering [19].
+E. Transformer-based Models
+Transformers were originally proposed by Vaswani et al.
+as a sequence-to-sequence architecture [25]. Unlike the Re-
+current Neural Networks [26] (RNN), the Long Short-Term
+Memory [27] (LSTM) variant of RNNs [26] and Convolutional
+Neural Networks [28] (CNN), Transformers only use a mecha-
+nism called self-attention to capture dependencies between the
+input and output. The current state-of-the-art NLP models for
+programming languages such as CodeT5 [14], CodeBERT [15]
+and PolyGlotCodeBERT [16] are all based on the Transformer
+architecture [25].
+F. Transfer Learning
+Pre-trained Transformers-based language models, such as
+RoBERTa [29], CodeBERT [15] and CodeT5 [14] utilise
+a pre-train then fine-tune paradigm. The bespoke paradigm
+was initially introduced by Kenton and Toutanova. In this
+paradigm, the models are first trained in an unsupervised
+manner on a large unlabelled dataset. These pre-trained models
+can then be fine-tuned to perform a more specialised task,
+such as summarisation. Transfer learning uses the knowledge
+that is obtained in one task to solve a different task. It
+allows the creation of general models that are trained once
+on massive datasets. These general models, which contain
+general domain knowledge can then be fine-tuned for a specific
+downstream task. This approach is quicker and requires less
+training data than training a model on the downstream task
+from scratch [30].
+9strip: https://pubs.opengroup.org/onlinepubs/7908799/xcu/strip.html
+3
+
+Source 
+Code
+Compilation
+Decompilation
+Decompiled
+</>
+Stripping
+Decompilation
+Function 
+Extraction
+</>
+Comment 
+Alignment
+Comment 
+Alignment
+Stripped
+Comment 
+Extraction
+Demi-Stripped
+Comment 
+Alignment
+Demi
+Stripping
+Fig. 2: Data Collection Pipeline
+III. CAPYBARA DATASET
+We require a dataset of decompiled functions labelled with
+a descriptive summary to create and assess our solution. This
+dataset should be relatively large to suit the ‘data-hungry’
+nature of deep-learning models. Furthermore, the dataset needs
+to feature a diverse set of data representative of our solution’s
+actual real-life use case.
+A. Data Collection
+To create such a large and diverse dataset we made use
+of BinSwarm [7], an existing dataset of aligned decompiled
+and stripped decompiled functions10. BinSwarm collects C-
+based projects from Github. The projects are filtered to only
+include those that are actively being developed, using Travis
+CI and built for Ubuntu Linux. The projects are built using
+Docker. The resulting binaries are then copied and stripped,
+and both the stripped and unstripped binaries are decompiled
+using Ghidra. The functions are extracted from the stripped
+and unstripped decompiled code and aligned with the source
+code. The BinSwarm dataset only contains aligned tuples of
+source code and (stripped-) decompiled functions. We extract
+documentation from the original source code files to add
+descriptive comments to this dataset. To that end, we depend
+on the documentation included in the source code by the
+original authors in the form of single and multiline comments.
+We locate the functions in the unbuilt project files and align the
+decompiled functions with the comments in the source code
+using srcML11 to extract any documentation located directly
+before a function signature. A high-level overview of the entire
+process is shown in Figure 2.
+A function’s documentation often also contains other details
+besides the descriptive summary. We found that C projects
+do not follow a single documentation standard. For example,
+Javadoc for Java has a short one-line description or summary
+for each method at the beginning of the multiline comment
+10BinSwarm: https://hub.docker.com/r/binswarm/cbuilds
+11srcML: https://www.srcml.org/
+/** @brief Select the source of Microcontroller
+Clock Output
+�→
+* Exact sources available depend on your target.
+* On devices with multiple MCO pins, this function
+controls MCO1
+�→
+* @param[in] mcosrc the unshifted source bits
+*/
+Fig. 3: Example of documentation from jeanthom/ DirtyJTAG:
+rcc set mco
+block. In C, there is no singular documentation standard, so
+there might not be a single-line summary, and we will need
+to locate it in the comment block automatically.
+a) Summary Extraction Rules: We observe that the ma-
+jority of single-line data are descriptive summaries, so we
+extract the first sentence. We identify many documentation
+styles in our multi-line data, we define some automated rules
+to extract summaries from the documentation:
+• @brief or @purpose: If the documentation contains a
+‘@brief’ or ‘@purpose’ tag, we extract the first sentence
+after the tag. The ‘brief‘ tag is part of the Doxygen docu-
+mentation standard12, an example is shown in Figure 313.
+• Description: If the documentation contains a line with
+‘Description:‘, we extract the following sentence.
+• @param or @v: Documentation that contains an ‘@v’
+or ‘@param’ tag, usually has a summary in the sentence
+before the tag. We extract that sentence.
+b) Filtering Rules: To improve the quality of the dataset
+we filter out samples based on the rules used by the Code-
+SearchNet dataset [20] included in the CodeXGlue benchmark
+for the summarisation task [31]:
+• Documentation length: We remove any summaries that
+are too long or too short and remove anything shorter
+than 3 or longer than 256 tokens.
+• Special tokens: We follow the example of the Code-
+SearchNet [20] and remove all documentation that con-
+tains special tokens. We scan for web tokens (like
+‘http://’), HTML tokens (like ‘<head>’), paths (like
+‘C://Users/..’), since this documentation usually refers to
+external resources. We additionally filter any developer
+tokens (like ‘FIXME:’), as these documents do not pro-
+vide meaningful information about the function itself, but
+contain comments about the development process.
+• Language: We filter out any documentation that was not
+written in English using the FastText language identifica-
+tion algorithm [32]. Around 92.19% of the documentation
+is in English.
+• Empty documentation: We find that a large number of
+functions did not have any documentation associated with
+them at all. We simply remove these samples from the
+dataset.
+12Doxygen:https://doxygen.nl/manual/docblocks.html
+13jeanthom/DirtyJTAG:rcc set mco:https://gitlab.com/
+insane-adding-machines/unicore-mx/-/blob/master/lib/stm32/common/
+rcc common all.c#L192
+4
+
+GCC000• Abstract Syntax Tree: The authors of the CodeSearch-
+Net dataset [20] additionally, remove any samples that do
+not parse into an AST. We choose to omit this step since
+all of our samples have been successfully compiled and
+have thus at one point been parsed into an AST by the
+compiler.
+B. Dataset Preparation
+a) Synthesis of Demi-stripped Code: From the dataset
+of decompiled functions, we also create another dataset. We
+emulate the process of stripping by removing all the iden-
+tifiers from the decompiled code and replacing them with
+placeholders. For clarity, we call this demi-stripped data. Like
+the stripped dataset, the identifiers are all removed, but this is
+only done after the decompilation process. The decompiler still
+had access to the identifiers and could use the symbol table
+during decompilation. Most importantly, this demi-stripped
+dataset still has the same structure and control flow as the
+unstripped decompiled dataset and avoids any decompilation
+issues arising from stripping.
+b) Data Split: The dataset is split into a train, test and
+validation set. These sets constitute approximately, 80%, 10%
+and 10%
+[19] of the complete dataset. As recommended
+by Shi et al. and LeClair and McMillan, we prevent leakage
+of vocabulary and code patterns between the sets, by sampling
+the sets in a cross-project manner [13, 19]. This means that an
+entire project gets assigned to one of the sets, and functions
+from the same project cannot be assigned to different sets. The
+projects in the test and validation set are the same across all
+datasets.
+c) Duplication: Large corpora of code, like the cor-
+pus gathered by BinSwarm, tend to have a high degree
+of duplication [19]. As a result, snippets of code that are
+relatively unchanged appear in multiple parts of the corpus.
+This can be in the form of copied, generic or auto-generated
+functions. These functions will appear in multiple repositories
+and might be duplicated across the training and testing data.
+Besides exact duplicates, near-duplicates can also occur. Near-
+duplicates differ in a few minor aspects like additional code
+comments or different function names. While removing exact
+duplicates is relatively fast and straightforward, removing
+near-duplicates is much more challenging and computationally
+intensive [33]. The issue with code duplication in classical
+code summarisation is that the models and tools are supposed
+to be used to generate summaries for new and unseen code.
+The evaluation metrics should therefore measure the gener-
+alisation of the tool on new samples [33]. Duplicates and
+near-duplicates are not defined as new samples. A user of
+such a tool could simply look these samples up. Furthermore,
+large, high-capacity models like CodeT5 with 220M [14] or
+CodeBERT with 128M [15] parameters, have a large capacity
+to memorise duplicated code [33].
+However, the use case outlined in this work is more akin
+to deobfuscation. As explained by Allamanis, deobfuscation
+could be a use case where duplicates are valid and part of the
+true distribution of the problem [33]. Compiled code contains
+0
+200
+400
+600
+800
+1000
+1200
+1400
+Token count
+0.000
+0.001
+0.002
+0.003
+0.004
+0.005
+Density
+Source code
+Decompiled code
+Fig. 4: Tokens in source C and decompiled code
+a lot of duplicate code, and understanding this code is still
+difficult and essential for understanding the binary. While
+regular source code allows the reader to look up code snippets,
+decompiled binaries have an additional obfuscation applied.
+We, therefore, focus on the model’s performance on code
+with duplicates as we believe duplicates to be part of the true
+distribution of the data, but we also report the deduplicated
+results.
+C. Dataset Properties
+Table I shows the size of the processed dataset. Of the 2.1M
+aligned decompiled functions, we extract documentation for
+215k of them, and we found that the majority of samples, 1.5M
+did not have any documentation at all. Furthermore, BinSwarm
+only provided us with 415k aligned stripped samples, and we
+can extract documentation for only 14k of these samples.
+Dataset
+Including duplicates
+Deduplicated
+C/Demi/Decom
+214,587
+79,673
+Stripped
+14,245
+7,826
+TABLE I: Number of functions in dataset
+The vast majority of documentation is in the form of
+multi-line comments as opposed to single-line or double-slash
+comments. We found that the documentation and comments
+had a mean length of 42.60 and 8.14 tokens, respectively.
+Figure 4 shows the distribution of the number of tokens in
+source code and decompiled code. The source and decompiled
+code have a mean length of 399 and 779 tokens, respectively.
+Figure 5, shows that decompiled code also has close to double
+the LOC of source code, with means of 30.77 and 53.42 lines
+for source and decompiled, respectively.
+The majority of decompiled functions are compiled with
+optimisation level -O2, with a similar number of -O1 and -
+O3 samples and relatively few -O0 samples. Stripped data has
+a very even distribution of optimisation levels, with only -
+O0 having significantly fewer samples. Note that there are
+more optimisation levels than shown in Figure 6, for brevity
+the different levels are grouped into their base optimisation
+level. -Oa is grouped with -O0, -Of and -Og are grouped
+with -O1, -Os is grouped with -O2. We also observe some
+5
+
+0
+25
+50
+75
+100
+125
+150
+175
+200
+LOC
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Density
+Source code
+Decompiled code
+Fig. 5: LOC in source C and decompiled code
+0
+1
+2
+3
+Optimisation level
+0
+20000
+40000
+60000
+80000
+100000
+120000
+Decompiled
+0
+1000
+2000
+3000
+4000
+Stripped
+Decompiled
+Stripped
+Fig. 6: Distribution of optimisation levels in decompiled (left)
+and stripped (right)
+samples with an optimisation level higher than -O3 (-O8 and
+-O7), as specified by the GCC documentation, these levels are
+equivalent to -O314.
+IV. BINT5
+We select CodeT5 [14] as the base-model for our experi-
+ments since it is the highest-scoring publicly-available model
+on the CodeXGLUE [31] Code Summarisation benchmark15.
+CodeT5 is a programming language model built on the T5
+(Text-to-text Transfer Transformer) architecture [34] and pre-
+trained on a mix of supervised and unsupervised tasks. CodeT5
+employs an encoder-decoder architecture. In contrast to other
+models, CodeT5 is trained using both unimodal (PL only) and
+bimodal (NL-to-PL) tasks in eight programming languages.
+This bimodal training allows CodeT5 to perform strong cross-
+modal tasks such as code summarisation and code generation
+(PL-to-NL). Many other models only use the data and lan-
+guages included in the CodeXGlue dataset [15, 16, 31], while
+CodeT5 also uses a mined dataset of C and C++ code for
+its pre-training objectives [14]. The inclusion of C training
+data should help the model with the CAPYBARA dataset.
+There could be some overlap in the training data between
+CAPYBARA and the dataset used by Wang et al. which would
+cause leakage, we address these concerns in Section VII.
+14GCC optimisation levels: https://gcc.gnu.org/onlinedocs/gcc-4.4.2/gcc/
+Optimize-Options.html#Optimize-Options
+15CodeXGLUE benchmark: https://microsoft.github.io/CodeXGLUE/
+Fine-Tuning
+Base 
+Model
+BinT5
+Evaluation
+Results
+CAPYBARA 
+training & validation data
+CAPYBARA
+test data
+Fig. 7: BinT5 fine-tuning pipeline
+CodeT5 also utilises the transfer learning paradigm, which
+allows us to train the model with relatively little data. In
+this case, we make use of the CodeT5-base model, which
+was trained on mixed upstream tasks by the authors [14]. An
+overview of how we applied the model to create BinT5 is
+provided in Figure 7.
+V. EXPERIMENTAL SETUP
+To assess the effectiveness of our approach, we first evaluate
+the performance of the model, we then identify the aspects of
+the data that make this task inherently difficult, and we finally
+investigate aspects of the datasets and their influence on the
+complexity of the task.
+A. Research Questions
+In the context of the study, we thereby formulate the
+Research Questions (RQ) as follows.
+RQ1: How effective are fine-tuned Transformer-based models
+at decompiled code summarisation? To investigate the
+application of existing models to binaries using CAPY-
+BARA, we set a baseline by training a model on the code
+summarisation task on the source C-code dataset. We then
+train a summarisation model on both the decompiled and
+the stripped dataset. We use the evaluation metrics to
+compare the performance of the different models.
+RQ2: Which aspects of the input contribute most to model per-
+formance? We investigate which aspects of decompiled
+code increase the difficulty of the task. We, therefore,
+look at the impact of the symbol table on decompilation,
+for this, we fine-tune a model on the demi-stripped dataset
+and compare it to the other models. We also investigate
+the importance of the function name by removing just the
+function name from the decompiled code. Furthermore,
+we investigate the impact of the optimisation level by
+exploring the performance per optimisation level.
+RQ3: What is the impact of dataset properties on model per-
+formance? We finally investigate how the construction
+of CAPYBARA influences the models. To answer the
+final research question we remove the duplicates from
+the datasets and retrain the models, after which we
+compare the performance to the baselines. Furthermore,
+we investigate the impact of dataset size, by incrementally
+reducing the size of the training sets.
+6
+
+"Summarize Python: def inc_value(x):
+"increment value"
+"Generate Python: increment value'
+'def inc_value(x)...
+CodeT5
+"Defect: if x=0: x += 1
+"true"
+"Refine: if x=0: x += 1'
+"if x == 0: x += 1"
+"Translate Python to C: if x==O: x += 1
+"if (x==0) {x += 1;}"B. Baselines
+To first establish a performance baseline, we train a CodeT5-
+base model on the summarisation task on source C. Note
+that only samples which are aligned with decompiled code
+are included in the source C dataset. The baseline is used to
+compare the decompiled C, stripped decompiled C and the
+demi-stripped datasets to the source code.
+C. Evaluation Metrics
+We evaluate the performance between the reference sum-
+mary from CAPYBARA and the candidate summary produced
+by BinT5 using the EM, BLEU-4 [35], ROUGE-L [36] and,
+METEOR [37] metrics.
+a) Exact Match (EM): The simplest metric is the EM
+which scores a prediction one if it matches its reference exactly
+and zero otherwise.
+b) BLEU-4: The most widely used metric in the code
+summarisation task is the Bilingual Evaluation Understudy
+Score (BLEU) [13]. BLEU-4 produces a percentage number
+between 0 and 100, which defines the similarity between a
+candidate and a set of reference sentences. BLEU-4 calculates
+the cumulative 4-gram precision scores, the number of match-
+ing 4-grams divided by the total number of 4-grams in the
+candidate sentence [35]. The unigrams and bigrams account
+for the adequacy of the candidate while the longer three and
+4-grams account for fluency. To prevent short sentences the
+result is multiplied by a brevity penalty as well. A smoothing
+function is applied to prevent sequences with no matching 4-
+grams to score zero [38]. While Shi et al. recommend BLEU-4
+with smoothing method 4 [13], we opted to use the Moses [39]
+implementation of BLEU-4 which uses smoothing method 2
+since this is also utilised by CodeSearchNet, CodeXGlue and
+CodeT5 [14, 20, 31].
+c) ROUGE-L: ROUGE or Recall-Oriented Understudy
+for Gisting Evaluation, is a package which includes several
+metrics, the most popular among them is ROUGE-L [36].
+ROUGE-L is more recall oriented than BLEU-4. ROUGE-L
+simply finds the longest common subsequence (LCS) between
+the reference and the candidate. Note that the words do not
+need to be consecutive but they have to be in order.
+d) METEOR: METEOR or Metric for Evaluation for
+Translation with Explicit Ordering [37] uses word lists and
+stemming to also take synonyms into account and calculates
+the harmonic mean of the unigram precision and recall. Similar
+to ROUGE-L, METEOR is more recall-focused. METEOR has
+a higher correlation with human judgement than BLEU-4 [19]
+at the sentence level.
+D. CodeT5 finetuning and testing
+The concept of transfer learning, which is utilised in BinT5,
+depends on the use of a fine-tuning step to train the pre-trained
+model on the downstream task. We fine-tune a pre-trained
+CodeT5-base model on the constructed dataset. The model is
+trained on the summarisation task as defined in the model. We
+train the model on the train set, then evaluate it after every
+epoch on the validation set and finally test on the test set.
+During training, we measure the model performance using the
+BLEU-4 metric.
+E. Data deduplication
+To create a deduplicated version of the CAPYBARA
+dataset we make use of a fork16 of the near-duplicate-code-
+detector [33]. We use this tool to compare all the datasets’
+functions and find clusters of near-duplicate functions. We
+randomly select one function per cluster and discard the rest
+from the dataset. We use the standard tool configuration as
+recommended by Allamanis. Of the removed duplicates, we
+observe that a relatively large number originates from common
+libraries, such as SQLite17, that are packaged with binary
+programs. Thus a certain amount of duplication is also likely
+to occur “in the wild”.
+F. Configuration
+We process and visualise the data with Pandas 1.4.3 and
+Ghidra 10.0.418. FastText 1.0.3 with the largest lid.176.bin
+model is used to detect languages. We train the model using
+Transformers version 4.16.2 running on Torch 1.9.0+cu111 in
+the nvidia/cuda:11.4.0-base docker container image. We share
+a Docker image with all the libraries required to run BinT5
+pre-installed on DockerHub19.
+A grid search of the optimal settings was infeasible from a
+time perspective, so we performed training mainly using the
+recommended settings from the CodeT5-base model [14]. We
+double the source length for the decompiled, stripped, and
+demi-stripped code to 512 tokens instead of the standard 256
+tokens used for the source code to compensate for the fact
+that the average length of decompiled code is almost twice as
+long as the source code. We trained the model on a machine
+with an NVIDIA GeForce RTX3080 with 10GB of VRAM
+and an AMD Ryzen Threadripper 3990X 64-Core Processor
+with 192GB of RAM running Ubuntu 20.04.4 LTS. The GPU
+is running Nvidia driver version 510.60.02 with Cuda 11.6.
+The authors of CodeT5 used an NVIDIA A100 GPU with
+40GB of VRAM for fine-tuning [14]. To compensate for the
+lack of memory, we reduced the batch size to 2, which was the
+maximum length that could still fit in the VRAM, we increase
+the ‘gradient accumulation steps’ to 24 to still achieve the
+effective standard batch size of 48.
+VI. RESULTS
+We present the results of our experiments to answer the
+research questions, results are grouped per research question.
+The metrics are calculated for each sample from the test set,
+and the average scores are presented.
+A. RQ1: Model Effectiveness
+The performance of the CodeT5-base model on each of the
+datasets is presented in table II.
+16Near
+Duplicate
+Code
+Detector:
+https://github.com/SERG-Delft/
+near-duplicate-code-remover
+17SQLite: https://www.sqlite.org/index.html
+18It is not recommended to use Ghidra versions before 10.1 since these
+versions have not been patched against a Log4J RCE
+19BinT5 Docker Image: https://hub.docker.com/r/aalkaswan/bint5/tags
+7
+
+BLEU-4
+EM
+METEOR
+ROUGE-L
+C
+60.83
+52.19
+65.33
+66.51
+DecomC
+58.82
+48.92
+63.14
+64.51
+Stripped
+11.26
+1.85
+14.50
+17.25
+TABLE II: Result of fine-tuning CodeT5-base on mined
+datasets
+BLEU-4
+EM
+METEOR
+ROUGE-L
+DecomC
+58.82
+48.92
+58.4
+60.32
+Demi
+44.21
+35.10
+47.89
+49.59
+NoFunName
+46.99
+37.12
+45.92
+48.07
+TABLE III: Result of fine-tuning CodeT5-base on synthetic
+data
+We found that the decompiled code model generally pro-
+duced good summaries, evidenced by the BLEU-4 score of
+58.82, which is slightly lower than the baseline set by the
+source code. The stripped model mainly produced unusable
+summaries, as evidenced by the BLEU-4 score of 11. The
+high EM score could be an indication of a high duplication
+factor.
+Initial experiments with GraphCodeBERT [40] and Poly-
+glotGraphCodeBERT [16] base models fine-tuned on CAPY-
+BARA show performance around 5 and 3 BLEU-4 lower,
+respectively. This is a relatively small difference, especially
+considering the model size. This shows that the performance of
+BinT5 does not heavily depend on the additional pre-training
+on C and C# performed by Wang et al.. Furthermore, this result
+shows that it is improbable that significant dataset leakage has
+taken place.
+We found a relatively large difference between the number
+of recovered decompiled and stripped decompiled functions.
+This can likely be attributed to the fact that Ghidra struggles
+a lot more with recovering stripped functions. Recall that the
+symbol table commonly contains information regarding the
+location and name of functions. When this table is dropped,
+the start- and endpoints of functions are hard to infer by
+automatic tools, especially since many functions get inlined,
+and JUMP instructions replace CALL instructions. Aside from
+difficulties in demarcating functions, it is also difficult to
+align the associated source code function with the decompiled
+function. With unstripped code, the function name remains,
+meaning the functions can be aligned using the name. We
+attempted to utilise an existing solution by Alves-Foss and
+Song called Jima [41] to find function boundaries. Jima is the
+current state-of-the-art tool for function boundary detection
+in stripped binaries. The tool is implemented as a plugin for
+Ghidra, but in our experiments, we find no statistical difference
+between the base performance of Ghidra and Jima on our
+dataset. The difficulties in extracting stripped functions, make
+training and applying a model to stripped binaries challenging.
+Opt level
+BLEU-4
+EM
+METEOR
+ROUGE-L
+-O0
+72.88
+34.18
+73.19
+74.84
+-O1
+50.30
+59.84
+55.36
+54.84
+-O2
+62.31
+46.23
+64.50
+66.05
+-O3
+54.68
+54.99
+58.25
+59.28
+TABLE IV: Average BLEU-4 score of decompiled code per
+optimisation level
+B. RQ2: Input Properties
+As can be observed in Table III, the summaries produced
+by the demi-stripped model were substantially worse than the
+decompiled model, but most were still very usable, evident
+by the BLEU-4 score above 44. Just removing the function
+name gave quite similar results to demi-stripping. We find that
+the loss of identifiers significantly lowers the performance of
+the model, but stripped code also suffers from decompilation
+faults, which seem to have a much larger impact on the model
+performance. Hence, the performance of BinT5 on demi-
+stripped code can be viewed as more representative of the
+actual model and not impacted by faults introduced by Ghidra.
+Table IV shows the average score per optimisation level. We
+can observe that -O0 and -O2 perform better than -O1 and -
+O3. Recall that -O0 is completely unoptimised, and that the
+vast majority of our decompiled dataset is compiled with -O2,
+which would explain why those optimisation levels perform
+better.
+C. RQ3: Dataset Properties
+The performance of the base model on each of the dedupli-
+cated datasets is presented in table V:
+BLEU-4
+EM
+METEOR
+ROUGE-L
+∆BLEU-4
+C
+45.86
+32.87
+46.06
+47.53
+14.97
+DecomC
+42.48
+28.08
+25.23
+27.66
+16.34
+Demi
+25.38
+14.51
+42.47
+44.47
+18.83
+Stripped
+7.19
+0.00
+4.75
+5.50
+4.07
+TABLE V: Result of fine-tuning CodeT5-base on the dedupli-
+cated datasets and the difference with the baseline
+We find that the influence of deduplication on our model’s
+performance is relatively small on source code, at only 24%.
+Duplicates have a relatively large impact on the decompiled
+(28%) and demi-stripped (43%) code. Deduplication also
+greatly decreases the EM rate across the board. Duplicates
+have a relatively large impact on performance, but even with
+the duplicates removed the model still produces many high-
+quality summaries. The experiments on deduplication show
+that the model seems to have a deeper understanding of the
+data and is not simply reproducing previously seen samples.
+As can be seen in Figure 8, the dataset size does not
+have much of an impact, the model can be trained with
+half or a quarter of the training samples without suffering
+a considerable hit to performance. This could be attributed
+to the high duplication factor of our dataset. It could also be
+8
+
+0
+20
+40
+60
+80
+100
+Fraction of train set
+25
+30
+35
+40
+45
+50
+55
+60
+BLEU4
+Decompiled
+Deduplicated
+Fig. 8: BLEU-4 per trainset size for decompiled code and
+deduplicated decompiled code
+because the model was already pre-trained well by Wang et al.
+and requires very little data for fine-tuning. This is a testament
+to the relative ease with which these models could be extended
+to decompiled code.
+We also performed experiments where we did not apply the
+filtering rules provided by CodeXGlue and where we always
+mined the first sentence of any type of documentation. While
+we were able to collect around 480K decompiled samples, the
+model performed substantially worse, only scoring 36.97 and
+33.26 BLEU-4 on C and decompiled code, respectively. These
+results show that the dataset quality also heavily impacts the
+model performance.
+VII. DISCUSSION
+In the previous section, we found that BinT5 shows con-
+siderable performance for decompiled code and demi-stripped
+code on both regular as well as deduplicated data. While
+this is a promising result, we conduct a small investigation
+of the decompiled samples. We will put our observations on
+identifiers into the context of the extreme summarisation task.
+Based on this we discuss the implications of our work. Finally,
+we will close this section by discussing the threats to validity.
+A. Exploration of Results
+To explore the results of BinT5 we pick 25 high and 25
+low-scoring samples from the test set of the deduplicated
+decompiled dataset. High samples have a BLEU-4 score higher
+than 75 while low-scoring samples have a score lower than 25.
+a) High Samples: With the high-performing samples
+BinT5 tends to produce summaries which are very close to
+the references. For instance, BinT5 produced Print description
+of a datatype in XML against the baseline Dump description of
+a datatype in XML. Of the 25 high-scoring samples we found
+that all have counterparts with a similar function summary
+in the training set. These functions also tend to have similar
+names, but their decompiled function body was significantly
+different, which is likely why deduplication didn’t remove
+these functions.
+b) Low Samples: From the low-performing samples we
+observe that many summaries produced by BinT5 are seman-
+tically very similar to the reference. For instance, the function
+vl set simd enabled20, has the reference Toggle usage of
+SIMD instructions while BinT5 produced Enable or Disable
+the Simd Channel. This sample scores a BLEU-4 score of 0.0,
+because of the limitations around the BLEU-4 metric, while
+for a human evaluator the output is still very usable. Similarly,
+for some samples, BinT5 produces shorter summaries contain-
+ing shorthands. The reference Check if the given nickname is
+blocked for ”normal client” use against Check whether nick is
+blocked, also scores poorly. Of the 25 low-scoring samples
+we observe that around 11 are semantically similar to the
+reference and likely very useful for understanding the function.
+B. Identifiers and Extreme Summarisation
+We find a relatively small difference in performance be-
+tween source code and decompiled code. This indicates that
+in-function comments and variable names are relatively unim-
+portant for the model performance. Although Ahmed and
+Devanbu observed that identifiers might be more important
+than syntax in the code-summarisation task [16], we can
+further conclude that the function name is explicitly essential
+for model performance. Removing just the function name from
+the decompiled samples, as opposed to removing all identifiers
+in demi-stripping, results in slightly higher performance than
+demi-stripped code, which indicates a very high dependence
+on the name of the function in the code summarisation task,
+which is a logical finding in the context of the extreme code
+summarisation task.
+The extreme code summarisation task, as proposed by Al-
+lamanis et al. aims to reproduce the function name given
+a function body [16, 42]. It is framed as a summarisation
+problem where the output is around 3 tokens in length, instead
+of the 10+ tokens that regular code summarisation targets.
+We found similar results when performing this task with
+our dataset, namely, high performance on regular decompiled
+code (with function names removed) and low performance on
+stripped code.
+A manual assessment of the stripped data shows that many
+of the aligned functions were not decompiled properly. We
+find that many functions are cut-off after a few instructions
+because the decompiler did not recover the full control flow.
+Other functions are missing side effects, like changes to global
+variables.
+C. Implications
+We propose a novel solution to aid reverse engineers in
+their work. Among many use cases, this solution could help
+malware analysts to understand novel malware and its weak-
+nesses quickly. The software can be analysed to find possible
+vulnerabilities and malicious payloads. The source code can
+20Colmap/Colmap:vl set simd enabled:
+https://github.com/colmap/
+colmap/blob/87b3aa325bd8e5fb913788e29e9ac1e085e28b67/lib/VLFeat/
+generic.c#L1070
+9
+
+be reconstructed for old binaries for which the source code is
+lost.
+If the application of NLP to binaries gets significantly better,
+and the limitations around stripping and other obfuscation
+techniques get resolved, it would have serious implications
+for the cybersecurity domain. On one hand, it would assist
+defenders, but on the other hand, attackers can leverage
+these same methods to find and exploit vulnerabilities, build
+malicious payloads and lift intellectual property from binaries.
+CAPYBARA itself could be used to create and assess
+neural decompilation, to perform a deeper investigation into
+the extreme summarisation task, or to simply train a code
+summarisation model on C code. CAPYBARA consists of a
+large corpus of C and decompiled C code, which could be used
+to pre-train language models, such that these models could
+support decompiled code out-of-the-box.
+While our work focused on decompiled code, our observa-
+tions show some limits of transformer-based models and their
+applicability to different data. Our dataset can help and inspire
+other researchers to improve upon our work. We hope other
+researchers use this dataset to train and evaluate their own
+models. Furthermore, the process outlined in Chapter III could
+help others construct standardised datasets for other tasks. The
+steps outlined for the creation of this dataset can be followed
+to create other datasets for other languages as well.
+D. Threats to Validity
+Internal Validity questions if other factors could have
+affected the outcome. The training and evaluation data contains
+a significant amount of noise, either in the form of badly de-
+compiled functions or incorrect documentation. We carefully
+collect and process the data, but we are unable to know to
+which extent the documentation matches the original code.
+While machine learning models (and specifically NLP models)
+should be able to handle noisy data, this might introduce some
+bias into the models. CodeT5 was also pre-trained on a C and
+C# dataset, this dataset is unpublished and we were unable to
+reach the authors. Some data leakage might have taken place,
+but it is unlikely that it had much of an impact. The data
+was only used for pre-training and would only have included
+source code. To prevent this threat from arising in any future
+studies, we make CAPYBARA publicly available.
+External Validity refers to the generalisability of our
+results. This work only focuses on stripping and compiler
+optimisations as a means of resisting binary analysis, other
+techniques like control flow obfuscation and packing are
+also used to prevent reverse engineering. Other works focus
+on unpacking and deobfuscation, so we consider our work
+orthogonal to theirs. The data gathered for CAPYBARA were
+exclusively from open-source projects. Decompiling closed-
+source projects is explicitly forbidden by some EULAs and the
+lack of source code documentation makes it difficult to evalu-
+ate using reference summaries. However, reverse engineering
+open-source software is not very useful in practice, since the
+source code is readily available. Closed-source software might
+have different data distribution and will present other chal-
+lenges like obfuscation. Finally, only functions that decompile
+(Ghidra produces any output) and that are documented, are
+represented in CAPYBARA. This is most apparent in the
+stripped dataset, where we can only recover a small fraction
+of the total number of functions. A deeper investigation into
+new decompilation techniques for stripped code, specifically
+into the aspect of function boundary detection is left as future
+work.
+Construct Validity relates to the adequacy of the theoretical
+constructs and the use of appropriate evaluation metrics. The
+leading metric in our evaluations does not capture semantic
+meaning. While BLEU-4 is the most popular metric for this
+task, its reliability has been called into question [43, 44]. We,
+therefore, included other metrics, which do take semantics into
+account, in our evaluation. Finally, our entire approach hinges
+on the assumption that function summaries, as they are used
+for source code, are useful for binary analysis. Whether or not
+this is actually the case, should be further investigated with a
+qualitative study.
+VIII. RELATED WORK
+Binary reverse engineering and the use of NLP for software
+engineering are vast and active fields, so we select and discuss
+the closest state-of-the-art works in the field. We categorise the
+studies into identifier recovery and binary translation. Finally,
+we will discuss the open challenges and the relation of our
+own work to these challenges.
+a) Recovering Identifiers from Stripped Binaries: De-
+bin [5] aims to recover debug information from stripped
+binaries. The authors use a tree-based classification and a
+probabilistic graph-based model. All the variable names and
+types are jointly recovered using a maximum a posteriori
+probability inference. VarBERT [45] uses a Transformer-
+based NLP model for the task of variable name recovery. The
+authors pre-trained a BERT model which is then fine-tuned to
+predict the names and types from unstripped binaries.
+FUNCRE
+[7]
+uses
+a
+pre-trained
+and
+fine-tuned
+ROBERTA [29] model to predict usages of inlined library
+functions. Recall that compilers with optimisations enabled
+can inline functions in the binary (Chapter II). The authors
+use indelible markers, which do not get destroyed by the
+compiler, to mark usages of library functions and to construct
+a dataset and train a model.
+b) Binary Translation: Neutron [10] frames decompila-
+tion as a neural machine translation problem and utilises an
+Attention-LSTM-based neural translation network to translate
+disassembled binaries back to C source code. The binaries
+are not stripped and do not have any optimisations enabled.
+The translations created by Neutron can contain syntax errors,
+so the authors apply regular expressions to create a tailor-
+made syntax checker. Neutron achieves high accuracy on the
+translation task, but only on unstripped and non-optimised
+code.
+10
+
+c) Our Novelty: Several aspects have not been properly
+addressed and investigated. The application of code summari-
+sation methods to decompiled code has not been addressed
+by any work at all. Furthermore, some works on binary code
+fail to take compiler optimisations into account [10]. We,
+therefore, investigate the application of code summarisation
+methods to decompiled code and we enable compiler optimi-
+sations.
+IX. CONCLUSION
+In this paper, we proposed a new automatic binary code
+summarisation task. With this new task, we also introduce
+CAPYBARA, a novel dataset to train and evaluate models
+on this task, with both mined as well as synthetic data.
+Paired with this dataset, we train BinT5, a Transformer-
+based code summarisation model to show the effectiveness of
+CAPYBARA. We used BinT5 to further explore the datasets,
+outlining the inherent difficulties in the data.
+We found that while BinT5 shows considerable performance
+on regular decompiled code, but its performance is being
+hampered by the decompiler on stripped code, evidenced by
+BinT5s strong performance on demi-stripped code. Further-
+more, we found that while duplicates have a large impact
+on the model, their presence is not paramount to the model’s
+performance. Finally, we observe that BinT5 could be trained
+with just a fraction of the samples in CAPYBARA.
+Our work has shown that a well-known and well-studied
+task from the source code domain [13], namely source code
+summarisation, can be applied to binary code. This is only one
+of the many different applications of NLP for code. Our paper
+constitutes the first step in the application of source code NLP
+methods to such tasks on binary code.
+REFERENCES
+[1] D. Votipka, S. Rabin, K. Micinski, J. S. Foster,
+and M. L. Mazurek, “An observational investigation
+of reverse engineers’ process and mental models,”
+in Extended Abstracts of the 2019 CHI Conference
+on Human Factors in Computing Systems, ser. CHI
+EA
+’19.
+New
+York,
+NY,
+USA:
+Association
+for
+Computing Machinery, 2019, p. 1–6. [Online]. Available:
+https://doi.org/10.1145/3290607.3313040
+[2] Y. David, U. Alon, and E. Yahav, “Neural reverse
+engineering of stripped binaries using augmented control
+flow graphs,” Proceedings of the ACM on Programming
+Languages, vol. 4, no. OOPSLA, nov 2020. [Online].
+Available: https://doi.org/10.1145/3428293
+[3] J. Caballero and Z. Lin, “Type inference on executables,”
+ACM Comput. Surv., vol. 48, no. 4, May 2016. [Online].
+Available: https://doi.org/10.1145/2896499
+[4] L. Chen, Z. He, and B. Mao, “Cati: Context-assisted type
+inference from stripped binaries,” in 2020 50th Annual
+IEEE/IFIP International Conference on Dependable Sys-
+tems and Networks (DSN), 2020, pp. 88–98.
+[5] J. He, P. Ivanov, P. Tsankov, V. Raychev, and M. Vechev,
+“Debin: Predicting debug information in stripped bina-
+ries,” in Proceedings of the 2018 ACM SIGSAC Confer-
+ence on Computer and Communications Security, 2018,
+pp. 1667–1680.
+[6] J.
+Lacomis,
+P.
+Yin,
+E.
+Schwartz,
+M.
+Allamanis,
+C. Le Goues, G. Neubig, and B. Vasilescu, “Dire: A neu-
+ral approach to decompiled identifier naming,” in 2019
+34th IEEE/ACM International Conference on Automated
+Software Engineering (ASE).
+IEEE, 2019, pp. 628–639.
+[7] T. Ahmed, P. Devanbu, and A. A. Sawant, “Learning to
+find usage of library functions in optimized binaries,”
+IEEE Transactions on Software Engineering, pp. 1–1,
+2021.
+[8] X. Jin, K. Pei, J. Y. Won, and Z. Lin, “Symlm: Predicting
+function names in stripped binaries via context-sensitive
+execution-aware code embeddings,” 2022.
+[9] D. Lehmann and M. Pradel, “Finding the dwarf: Recov-
+ering precise types from webassembly binaries,” 2022.
+[10] R. Liang, Y. Cao, P. Hu, and K. Chen, “Neutron: an
+attention-based neural decompiler,” Cybersecurity, vol. 4,
+p. 5, 03 2021.
+[11] C. Zhang, J. Wang, Q. Zhou, T. Xu, K. Tang, H. Gui,
+and F. Liu, “A survey of automatic source code summa-
+rization,” Symmetry, vol. 14, no. 3, p. 471, 2022.
+[12] G. Sridhara, E. Hill, D. Muppaneni, L. Pollock, and
+K. Vijay-Shanker, “Towards automatically generating
+summary comments for java methods,” ser. ASE ’10.
+New York, NY, USA: Association for Computing
+Machinery, 2010, p. 43–52. [Online]. Available: https:
+//doi.org/10.1145/1858996.1859006
+[13] Shi, E. Wang, Y. Du, L. Chen, J. Han, S. Zhang,
+H. Zhang, D. Sun, and H. Sun, “On the evaluation of
+neural code summarization.”
+ICSE, 2022.
+[14] Y.
+Wang,
+W.
+Wang,
+S.
+Joty,
+and
+S.
+C.
+Hoi,
+“Codet5: Identifier-aware unified pre-trained encoder-
+decoder models for code understanding and generation,”
+in Proceedings of the 2021 Conference on Empirical
+Methods in Natural Language Processing, 2021, pp.
+8696–8708.
+[15] Z. Feng, D. Guo, D. Tang, N. Duan, X. Feng, M. Gong,
+L. Shou, B. Qin, T. Liu, D. Jiang et al., “Codebert: A pre-
+trained model for programming and natural languages,”
+in Findings of the Association for Computational Lin-
+guistics: EMNLP 2020, 2020, pp. 1536–1547.
+[16] T. Ahmed and P. Devanbu, “Multilingual training for
+software engineering,” in 2022 IEEE/ACM 44th Inter-
+national Conference on Software Engineering (ICSE).
+IEEE, 2022, pp. 1443–1455.
+[17] C. Casalnuovo, E. T. Barr, S. K. Dash, P. Devanbu,
+and E. Morgan, “A theory of dual channel constraints,”
+in Proceedings of the ACM/IEEE 42nd International
+Conference on Software Engineering: New Ideas and
+Emerging Results, 2020, pp. 25–28.
+[18] A. Hindle, E. T. Barr, M. Gabel, Z. Su, and P. Devanbu,
+“On the naturalness of software,” Commun. ACM,
+vol.
+59,
+no.
+5,
+p.
+122–131,
+apr
+2016.
+[Online].
+Available: https://doi.org/10.1145/2902362
+11
+
+[19] A. LeClair and C. McMillan, “Recommendations for
+datasets for source code summarization,” in Proceedings
+of the 2019 Conference of the North American Chapter
+of
+the
+Association
+for
+Computational
+Linguistics:
+Human Language Technologies, Volume 1 (Long and
+Short Papers). Minneapolis, Minnesota: Association for
+Computational Linguistics, Jun. 2019, pp. 3931–3937.
+[Online]. Available: https://aclanthology.org/N19-1394
+[20] H. Husain, H.-H. Wu, T. Gazit, M. Allamanis, and
+M. Brockschmidt, “CodeSearchNet challenge: Evaluat-
+ing the state of semantic code search,” arXiv preprint
+arXiv:1909.09436, 2019.
+[21] K. Hoste and L. Eeckhout, “Cole: compiler optimization
+level exploration,” in Proceedings of the 6th annual
+IEEE/ACM international symposium on Code generation
+and optimization, 2008, pp. 165–174.
+[22] M. T. Jones, “Optimization in gcc,” Linux journal, vol.
+2005, no. 131, p. 11, 2005.
+[23] S. Blazy and S. Riaud, “Measuring the robustness
+of source program obfuscation: Studying the impact
+of compiler optimizations on the obfuscation of c
+programs,” in Proceedings of the 4th ACM Conference
+on Data and Application Security and Privacy, ser.
+CODASPY ’14.
+New York, NY, USA: Association
+for Computing Machinery, 2014, p. 123–126. [Online].
+Available: https://doi.org/10.1145/2557547.2557577
+[24] Z. Zhang, W. You, G. Tao, Y. Aafer, X. Liu, and
+X. Zhang, “Stochfuzz: Sound and cost-effective fuzzing
+of stripped binaries by incremental and stochastic rewrit-
+ing,” in 2021 IEEE Symposium on Security and Privacy
+(SP).
+IEEE, 2021, pp. 659–676.
+[25] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit,
+L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin,
+“Attention is all you need,” in Proceedings of the 31st
+International Conference on Neural Information Process-
+ing Systems, 2017, pp. 6000–6010.
+[26] D. E. Rumelhart, G. E. Hinton, and R. J. Williams,
+Learning Representations by Back-Propagating Errors.
+Cambridge, MA, USA: MIT Press, 1988, p. 696–699.
+[27] J. Schmidhuber, S. Hochreiter et al., “Long short-term
+memory,” Neural Comput, vol. 9, no. 8, pp. 1735–1780,
+1997.
+[28] Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E.
+Howard, W. Hubbard, and L. D. Jackel, “Backpropaga-
+tion applied to handwritten zip code recognition,” Neural
+computation, vol. 1, no. 4, pp. 541–551, 1989.
+[29] L. Zhuang, L. Wayne, S. Ya, and Z. Jun, “A robustly
+optimized
+BERT
+pre-training
+approach
+with
+post-
+training,” in Proceedings of the 20th Chinese National
+Conference on Computational Linguistics.
+Huhhot,
+China:
+Chinese
+Information
+Processing
+Society
+of
+China, Aug. 2021, pp. 1218–1227. [Online]. Available:
+https://aclanthology.org/2021.ccl-1.108
+[30] J. D. M.-W. C. Kenton and L. K. Toutanova, “Bert: Pre-
+training of deep bidirectional transformers for language
+understanding,” in Proceedings of NAACL-HLT, 2019,
+pp. 4171–4186.
+[31] S. Lu, D. Guo, S. Ren, J. Huang, A. Svyatkovskiy,
+A. Blanco, C. Clement, D. Drain, D. Jiang, D. Tang,
+G. Li, L. Zhou, L. Shou, L. Zhou, M. Tufano, M. Gong,
+M. Zhou, N. Duan, N. Sundaresan, S. K. Deng, S. Fu,
+and S. Liu, “Codexglue: A machine learning benchmark
+dataset for code understanding and generation,” 2021.
+[Online]. Available: https://arxiv.org/abs/2102.04664
+[32] A. Joulin, E. Grave, P. Bojanowski, M. Douze, H. J´egou,
+and T. Mikolov, “Fasttext.zip: Compressing text classifi-
+cation models,” arXiv preprint arXiv:1612.03651, 2016.
+[33] M. Allamanis, “The adverse effects of code duplication
+in machine learning models of code,” in Proceedings
+of the 2019 ACM SIGPLAN International Symposium
+on
+New
+Ideas,
+New
+Paradigms,
+and
+Reflections
+on
+Programming
+and
+Software.
+Athens
+Greece:
+ACM, Oct. 2019, pp. 143–153. [Online]. Available:
+https://dl.acm.org/doi/10.1145/3359591.3359735
+[34] C. Raffel, N. Shazeer, A. Roberts, K. Lee, S. Narang,
+M. Matena, Y. Zhou, W. Li, and P. J. Liu, “Exploring
+the limits of transfer learning with a unified text-to-text
+transformer,” Journal of Machine Learning Research,
+vol. 21, no. 140, pp. 1–67, 2020. [Online]. Available:
+http://jmlr.org/papers/v21/20-074.html
+[35] K. Papineni, S. Roukos, T. Ward, and W.-J. Zhu, “Bleu: a
+method for automatic evaluation of machine translation,”
+in Proceedings of the 40th Annual Meeting of the
+Association for Computational Linguistics. Philadelphia,
+Pennsylvania,
+USA:
+Association
+for
+Computational
+Linguistics, Jul. 2002, pp. 311–318. [Online]. Available:
+https://aclanthology.org/P02-1040
+[36] C.-Y. Lin, “ROUGE: A package for automatic evaluation
+of
+summaries,”
+in
+Text
+Summarization
+Branches
+Out.
+Barcelona, Spain: Association for Computational
+Linguistics, Jul. 2004, pp. 74–81. [Online]. Available:
+https://aclanthology.org/W04-1013
+[37] A. Lavie and M. J. Denkowski, “The meteor metric for
+automatic evaluation of machine translation,” Machine
+translation, vol. 23, no. 2, pp. 105–115, 2009.
+[38] B. Chen and C. Cherry, “A systematic comparison
+of smoothing techniques for sentence-level BLEU,”
+in Proceedings of the Ninth Workshop on Statistical
+Machine
+Translation.
+Baltimore,
+Maryland,
+USA:
+Association for Computational Linguistics, Jun. 2014,
+pp. 362–367. [Online]. Available: https://aclanthology.
+org/W14-3346
+[39] H. Hoang and P. Koehn, “Design of the moses de-
+coder for statistical machine translation,” in Software
+Engineering, Testing, and Quality Assurance for Natural
+Language Processing, 2008, pp. 58–65.
+[40] D. Guo, S. Ren, S. Lu, Z. Feng, D. Tang, S. Liu, L. Zhou,
+N. Duan, A. Svyatkovskiy, S. Fu et al., “Graphcodebert:
+Pre-training code representations with data flow,” in
+ICLR, 2021.
+[41] J. Alves-Foss and J. Song, “Function boundary detection
+in
+stripped
+binaries,”
+in
+Proceedings
+of
+the
+35th
+12
+
+Annual Computer Security Applications Conference,
+ser. ACSAC ’19.
+New York, NY, USA: Association
+for Computing Machinery, 2019, p. 84–96. [Online].
+Available: https://doi.org/10.1145/3359789.3359825
+[42] M. Allamanis, H. Peng, and C. Sutton, “A convolutional
+attention network for extreme summarization of source
+code,”
+in
+Proceedings
+of
+The
+33rd
+International
+Conference on Machine Learning, ser. Proceedings of
+Machine Learning Research, M. F. Balcan and K. Q.
+Weinberger, Eds., vol. 48.
+New York, New York, USA:
+PMLR, 20–22 Jun 2016, pp. 2091–2100. [Online].
+Available: https://proceedings.mlr.press/v48/allamanis16.
+html
+[43] D. Roy, S. Fakhoury, and V. Arnaoudova, Reassessing
+Automatic Evaluation Metrics for Code Summarization
+Tasks. New York, NY, USA: Association for Computing
+Machinery, 2021, p. 1105–1116. [Online]. Available:
+https://doi.org/10.1145/3468264.3468588
+[44] S. Haque, Z. Eberhart, A. Bansal, and C. McMillan,
+“Semantic similarity metrics for evaluating source code
+summarization,” arXiv e-prints, pp. arXiv–2204, 2022.
+[45] P. Banerjee, K. K. Pal, F. Wang, and C. Baral, “Vari-
+able name recovery in decompiled binary code using
+constrained masked language modeling,” arXiv preprint
+arXiv:2103.12801, 2021.
+13
+

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+Aleatoric and Epistemic Discrimination in Classification
+Hao Wang 1 Luxi He 2 Rui Gao 3 Flavio P. Calmon 4
+Abstract
+Machine learning (ML) models can underperform
+on certain population groups due to choices made
+during model development and bias inherent in
+the data. We categorize sources of discrimina-
+tion in the ML pipeline into two classes: aleatoric
+discrimination, which is inherent in the data distri-
+bution, and epistemic discrimination, which is due
+to decisions during model development. We quan-
+tify aleatoric discrimination by determining the
+performance limits of a model under fairness con-
+straints, assuming perfect knowledge of the data
+distribution. We demonstrate how to characterize
+aleatoric discrimination by applying Blackwell’s
+results on comparing statistical experiments. We
+then quantify epistemic discrimination as the gap
+between a model’s accuracy given fairness con-
+straints and the limit posed by aleatoric discrimi-
+nation. We apply this approach to benchmark ex-
+isting interventions and investigate fairness risks
+in data with missing values. Our results indi-
+cate that state-of-the-art fairness interventions are
+effective at removing epistemic discrimination.
+However, when data has missing values, there is
+still significant room for improvement in handling
+aleatoric discrimination.
+1. Introduction
+Algorithmic discrimination may occur in different stages of
+the machine learning (ML) pipeline. For example, histori-
+cal biases in the data-generating process can propagate to
+downstream tasks; human biases can influence a ML model
+through inductive bias; optimizing solely for accuracy can
+lead to disparate model performance across groups in the
+data (Suresh & Guttag, 2019; Mayson, 2019). The past
+years have seen a rapid increase in algorithmic interventions
+that aim to mitigate biases in ML models (see e.g., Zemel
+1MIT-IBM
+Watson
+AI
+Lab
+2Harvard
+College
+3UT-
+Austin
+4Harvard University.
+Hao Wang <hao@ibm.com>,
+Luxi
+He
+<luxihe@college.harvard.edu>,
+Rui
+Gao
+<rui.gao@mccombs.utexas.edu>,
+Flavio
+P.
+Calmon
+<flavio@seas.harvard.edu>.
+et al., 2013; Feldman et al., 2015; Calmon et al., 2017;
+Menon & Williamson, 2018; Zhang et al., 2018; Zafar et al.,
+2019; Friedler et al., 2019; Bellamy et al., 2019; Kim et al.,
+2019; Celis et al., 2019; Yang et al., 2020; Jiang & Nachum,
+2020; Jiang et al., 2020; Martinez et al., 2020; Lowy et al.,
+2021; Alghamdi et al., 2022). A recent survey (Hort et al.,
+2022) found nearly 400 fairness-intervention algorithms,
+including 123 pre-processing, 212 in-processing, and 56
+post-processing algorithms introduced in the past decade.
+Which sources of biases are (the hundreds of) existing fair-
+ness interventions trying to control? In order to create effec-
+tive strategies for reducing algorithmic discrimination, it is
+critical to disentangle where biases in model performance
+originate. For instance, if the training set contains few sam-
+ples from a given population group, then increasing sample
+diversity is a more effective strategy than selecting a more
+complex model class or training strategy. Conversely, if a
+model class does not accurately represent the underlying
+distribution of a certain population group, then increasing
+sample size for that group will not resolve performance
+disparities.
+We divide algorithmic discrimination into two categories:
+aleatoric and epistemic discrimination.1 Aleatoric discrimi-
+nation captures inherent biases in the data distribution that
+can lead to unfair decisions in downstream tasks. Epistemic
+discrimination, in turn, is due to algorithmic choices made
+during model development and lack of knowledge about the
+optimal “fair” predictive model.
+In this paper, we provide methods for measuring aleatoric
+and epistemic discrimination in classification task for group
+fairness metrics. Since aleatoric discrimination only de-
+pends on properties of the data distribution and the fairness
+measure of choice, we quantify it by asking a fundamental
+question:
+For a given data distribution, what would be the best achiev-
+able performance (e.g., accuracy) under a group fairness
+constraint?
+1We borrow this notion from ML uncertainty literature (see
+H¨ullermeier & Waegeman, 2021, for a survey). Therein, aleatoric
+uncertainty refers to the variability in the outcome of an experiment
+resulting from inherently random effects; epistemic uncertainty
+refers to uncertainty caused by a lack of knowledge about the best
+predictive model.
+arXiv:2301.11781v1  [cs.LG]  27 Jan 2023
+
+Aleatoric and Epistemic Discrimination in Classification
+We refer to the answer as the fairness Pareto frontier. This
+frontier delineates the optimal performance achievable by
+a classifier when unlimited data and computing power are
+available. For a fixed data distribution, the fairness Pareto
+frontier represents the ultimate, information-theoretic limit
+for accuracy and group fairness beyond which no model can
+achieve. Characterizing this limit enables us to (i) separate
+sources of discrimination and create strategies to control
+them accordingly; (ii) evaluate the effectiveness of existing
+fairness interventions for reducing epistemic discrimination;
+and (iii) inform the development of data collection methods
+that promote fairness in downstream tasks.
+At first, computing the fairness Pareto frontier can appear to
+be an intractable problem since it requires searching over all
+possible classifiers—even if the data distribution is known
+exactly. Our main technical contribution is to provide a pre-
+cise characterization of this frontier by solving a sequence of
+optimization problems. Our main proof technique is based
+on Blackwell’s seminal results (Blackwell, 1953), which
+proposed the notion of comparisons of statistical experi-
+ments and inspired a line of works introducing alternative
+comparison criteria (see e.g., Shannon, 1958; Cam, 1964;
+Torgersen, 1991; Cohen et al., 1998; Raginsky, 2011). Here,
+we apply these results to develop an algorithm that itera-
+tively refines the achievable fairness Pareto frontier. We also
+prove convergence guarantees for our algorithm and demon-
+strate how it can be used to benchmark existing fairness
+interventions.
+We quantify epistemic discrimination by comparing a clas-
+sifier’s performance with the information-theoretic optimal
+given by the fairness Pareto frontier. Our experiments indi-
+cate that given sufficient data, state-of-the-art fairness inter-
+ventions are effective at reducing epistemic discrimination
+as their gap to the information-theoretic limit is small (see
+Figure 1). Existing interventions do not eliminate aleatoric
+discrimination as this type of discrimination is not caused
+by choice of learning algorithm or model class, and is due
+to the data distribution.
+We further analyze the fairness Pareto frontier to show that
+factors such as data missing values can significantly con-
+tribute to aleatoric discrimination. We observe that when
+population groups have disparate missing patterns, aleatoric
+discrimination escalates, leading to a sharp decline in the
+effectiveness of fairness intervention algorithms (see Fig-
+ure 2).
+Related Work
+There is significant work analyzing the tension between
+group fairness measures and model performance metrics
+(Kleinberg et al., 2016; Chouldechova, 2017; Corbett-
+Davies et al., 2017; Chen et al., 2018; Dutta et al., 2020;
+Wang et al., 2021). We recast the fairness Pareto frontier
+in terms of the conditional distribution PˆY|Y,S of predicted
+outcome ˆY given true label Y and group attributes S. This
+conditional distribution is related to confusion matrices2
+conditioned on each subgroup (see Remark 1 for detailed
+discussion). In this regard, our work is related to Verma &
+Rubin (2018); Alghamdi et al. (2020); Kim et al. (2020);
+Yang et al. (2020); Berk et al. (2021), which observed that
+many group fairness metrics can be written in terms of the
+confusion matrices for each subgroup. Among them, the
+closest work to ours is Kim et al. (2020), which optimized
+accuracy and fairness objectives over these confusion matri-
+ces and proposed a post-processing technique for training
+fair classifiers. However, they only imposed marginal sum
+constraints for the confusion matrices. We demonstrate
+that the feasible region of confusion matrices can be much
+smaller (see Remark 2 for an example), leading to a tighter
+approximation of the fairness Pareto frontier.
+Recently, many strategies have been proposed to reduce
+the tension between group fairness and model performance
+by investigating properties of the data distribution. For ex-
+ample, Blum & Stangl (2019); Suresh & Guttag (2019);
+Fogliato et al. (2020); Wang et al. (2020); Mehrotra &
+Celis (2021); Fernando et al. (2021); Wang & Singh (2021);
+Zhang & Long (2021); Kallus et al. (2022); Jeong et al.
+(2022) studied how noisy or missing data affect fairness and
+model accuracy. Dwork et al. (2018); Ustun et al. (2019);
+Wang et al. (2021) considered training a separate classifier
+for each subgroup when their data distributions are differ-
+ent. Another line of research introduces data pre-processing
+techniques that manipulate data distribution for reducing its
+bias (e.g., Calmon et al., 2017; Kamiran & Calders, 2012).
+Among all these works, the closest one to ours is Chen
+et al. (2018), which decomposed group fairness measures
+into bias, variance, and noise (see their Theorem 1) and pro-
+posed strategies for reducing each term. The main difference
+compared with Chen et al. (2018) is that we characterize
+a fairness Pareto frontier that depends on not only fairness
+metrics but also on a performance measure. This effort gives
+a complete picture of how data distribution influences the
+fairness-accuracy tension and is more technically involved.
+Also, the fairness Pareto frontier only depends on the data
+distribution and fairness metrics of choice so it cannot be
+improved by adding more data or altering the model class.
+2. Preliminaries
+Next, we introduce notation, overview the key results in
+Blackwell (1953) on comparisons of experiments, and out-
+line the fair classification setup considered in this paper.
+2A confusion matrix (Kulkarni et al., 2020) is a table that
+measures the performance of a given ML model. In binary classi-
+fication, a confusion matrix reports the number of true positives,
+false negatives, false positives, and true negatives.
+
+Aleatoric and Epistemic Discrimination in Classification
+Notation.
+For a positive integer n, let [n] ≜ {1, · · · , n}.
+We denote all probability distributions on the set X by
+P(X). Moreover, we define the probability simplex ∆m ≜
+P([m]). When random variables A, X, Z form a Markov
+chain, we write A → X → Z. We write the mutual infor-
+mation between A, X as I(A; X) ≜ EPA,X
+�
+log
+PA,X(A,X)
+PA(A)PX(X)
+�
+.
+Since I(A; X) is determined by the marginal distribution
+PA and the conditional distribution PX|A, we also write
+I(A; X) as I(PA; PX|A). When A, X are independent, we
+write A
+|=
+X.
+If a random variable A ∈ [n] has finite support, the condi-
+tional distribution PX|A : [n] → P(X) can be equivalently
+written as P ≜ (P1, · · · , Pn) where each Pi = PX|A=i ∈
+P(X). Additionally, if X is a finite set [m], then PX|A
+can be fully characterized by a transition matrix. We use
+T (m|n) to denote all transition matrices from [n] to [m]:
+�
+�
+�P ∈ Rn×m ��� 0 ≤ Pi,j ≤ 1,
+m
+�
+j=1
+Pi,j = 1, ∀i ∈ [n]
+�
+�
+� .
+Comparisons of Experiments
+Given two statistical experiments (i.e., conditional distribu-
+tions) P and Q, is there a way to decide which one is more
+informative? Here P and Q have the common input alpha-
+bet [n] and potentially different output spaces. Blackwell
+gave an answer in his seminal work (Blackwell, 1953) from
+a decision-theoretic perspective. We review these results
+next.
+Let A be a closed, bounded, convex subset of Rn. A deci-
+sion function f(x) = (a1(x), · · · , an(x)) is any mapping
+from X to A. It is associated a loss vector:
+v(f) =
+��
+a1(x)dP1(x), · · · ,
+�
+an(x)dPn(x)
+�
+. (1)
+The collection of all v(f) is denoted by B(P , A). Black-
+well defined that P is more informative than Q if for every
+A, B(P , A) ⊇ B(Q, A). Intuitively, this result means any
+risk achievable with Q is also achievable with P . Moreover,
+Blackwell considered the standard measure P ∗ which is the
+probability distribution of p(¯X) where p(x) : X → ∆n is a
+function defined as
+�
+dP1
+dP1 + · · · + dPn
+, · · · ,
+dPn
+dP1 + · · · + dPn
+�
+.
+(2)
+and ¯X follows the probability distribution P1+···+Pn
+n
+. One
+of the most important findings by Blackwell in his paper is
+to discover the following equivalent conditions.
+Lemma 1 (Blackwell (1951; 1953)). The following three
+conditions are equivalent:
+• P is more informative than Q;
+• for any continuous and convex function φ : ∆n → R,
+�
+φ(p)dP ∗(p) ≥
+�
+φ(p)dQ∗(p);
+• there is a stochastic transformation T such that TPi =
+Qi. In other words, there exists a Markov chain A →
+X → Z for any distributions on A such that P = PX|A
+and Q = PZ|A.
+Additionally, if P and Q can be characterized by transition
+matrices, the above conditions are also equivalent to:
+• there exists a transition matrix M such that Q =
+P M.
+If P = PX|A is more informative than Q = PZ|A, by the
+third condition of Lemma 1 and the data processing inequal-
+ity,
+I(PA; PX|A) ≥ I(PA; PZ|A)
+(3)
+holds for any marginal distribution PA. However, the con-
+verse does not hold in general—even if (3) holds for any PA,
+P is not necessarily more informative than Q (see Rauh et al.
+(2017) for a counter-example). In this regard, Blackwell’s
+conditions are “stronger” than mutual information-based
+form of the data processing inequality.
+Fair Classification
+Consider a multi-class classification task, where the goal is
+to train a probabilistic classifier h : X → ∆C that uses input
+features X to predict their true label Y ∈ [C]. Additionally,
+assume the classifier produces a predicted outcome ˆY ∈ [C]
+and let S ∈ [A] represent group attributes (e.g., race and
+sex). Our framework can be easily extended to the setting
+where multiple subgroups overlap (Kearns et al., 2018).
+Throughout this paper, we focus on three standard group
+fairness measures: statistical parity (SP) (Feldman et al.,
+2015), equalized odds (EO) (Hardt et al., 2016; Pleiss et al.,
+2017), and overall accuracy equality (OAE) (Berk et al.,
+2021) (see Table 1 for their definitions) but our analysis can
+be extended to many other group fairness metrics, including
+the ones in Table 1 of Kim et al. (2020).
+We use Blackwell’s conditions to provide a precise charac-
+terization of fairness Pareto frontier in multi-class classifica-
+tion. In brief, Blackwell’s conditions allow us to approxi-
+mate the set of achievable joint distributions PˆY|S,Y across
+all classifiers h. Since both accuracy and the group fairness
+criteria in Table 1 can be cast in terms of PˆY|S,Y, this ap-
+proximation can then be used to bound the best achievable
+accuracy under a group fairness constraint. We will develop
+this procedure in detail in the next section.
+
+Aleatoric and Epistemic Discrimination in Classification
+FAIRNESS METRIC
+ABBR.
+DEFINITION
+EXPRESSION W.R.T. P
+Statistical Parity
+SP ≤ αSP
+| Pr(ˆY = ˆy|S = s) − Pr(ˆY = ˆy|S = s′)| ≤ αSP
+���
+�C
+y=1
+�
+µs,y
+µs P(s,y),ˆy −
+µs′,y
+µs′ P(s′,y),ˆy
+���� ≤ αSP
+Equalized Odds
+EO ≤ αEO
+| Pr(ˆY = ˆy|S = s, Y = y) − Pr(ˆY = ˆy|S = s′, Y = y)| ≤ αEO
+��P(s,y),ˆy − P(s′,y),ˆy
+�� ≤ αEO
+Overall Accuracy Equality
+OAE ≤ αOAE
+| Pr(ˆY = Y |S = s) − Pr(ˆY = Y |S = s′)| ≤ αOAE
+����C
+y=1
+�
+µs,y
+µs P(s,y),y −
+µs′,y
+µs′ P(s′,y),y
+���� ≤ αOAE
+Table 1. Standard group fairness metrics under multi-group and multi-class classification task. Here αSP, αEO, αOAE, ∈ [0, 1] are threshold
+parameters, ˆy, y ∈ [C], s, s′ ∈ [A], and µs,y, µs are defined in Proposition 1. Our analysis can be extended to many other group fairness
+metrics (see e.g., Table 1 in Kim et al., 2020).
+3. Fairness Pareto Frontier
+In this section, we introduce our main concept—fairness
+Pareto frontier (FairFront). We use it to measure aleatoric
+discrimination and quantify epistemic discrimination by
+comparing a classifier’s performance to the FairFront. We
+recast FairFront in terms of the conditional distribution
+PˆY|S,Y and apply Blackwell’s conditions to characterize the
+feasible region of this conditional distribution. This effort
+converts a functional optimization problem into a convex
+program with a small number of variables. However, this
+convex program may involve infinitely many constraints.
+Hence, we introduce a greedy improvement algorithm that
+iteratively computes FairFront and tightens the feasible re-
+gion of PˆY|S,Y. We end this section by establishing a con-
+vergence guarantee for our algorithm.
+Recall that we refer to aleatoric discrimination as the inher-
+ent biases of the data distribution that can lead to an unfair
+or inaccurate classifier. As its definition suggests, aleatoric
+discrimination only relies on properties of the data distri-
+bution and fairness metric of choice—it does not depend
+on the hypothesis class nor optimization method. Below
+we introduce FairFront that delineates a curve of optimal
+accuracy over all probabilistic classifiers under certain fair-
+ness constraints. We use FairFront to quantify aleatoric
+discrimination.
+Definition 1. For αSP, αEO, αOAE
+∈
+[0, 1], we define
+FairFront(αSP, αEO, αOAE) as the solution of the following
+optimization problem:
+max
+h
+E
+�
+IˆY=Y
+�
+(4a)
+s.t. SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE
+(4b)
+where ˆY is produced by applying the classifier h to X; the
+maximum is taken over all measurable h; and the definitions
+of SP, EO, and OAE are in Table 1.
+Solving this functional optimization problem is difficult
+since it optimizes over all measurable classifiers. There is a
+line of works that proposed different fairness-intervention al-
+gorithms for training group-fair classifiers (see e.g., Menon
+& Williamson, 2018; Zhang et al., 2018; Zafar et al., 2019;
+Celis et al., 2019; Yang et al., 2020; Wei et al., 2021; Al-
+ghamdi et al., 2022). They restrict the model class and
+vary loss functions and optimizers to find classifiers that
+approach FairFront as close as possible. However, these
+algorithms only describe a lower bound for FairFront. They
+do not determine what is the best achievable accuracy for a
+given fairness constraint.
+We circumvent the above-mentioned challenges by rewrit-
+ing FairFront in terms of the conditional distribution PˆY|S,Y.
+The caveat is that although each classifier yields a PˆY|S,Y,
+not every conditional distribution corresponds to a valid
+classifier (see an example in Remark 2). Hence, we intro-
+duce the following definition which characterizes all feasible
+PˆY|S,Y.
+Definition 2. Given PX|S,Y, we define C as the set of all
+conditional distributions PˆY|S,Y where ˆY is produced by
+some probabilistic classifier h. In other words,
+C ≜
+�
+PˆY|S,Y | (S, Y) → X → ˆY
+�
+.
+(5)
+Throughout this paper, we write PˆY|S,Y or its corresponding
+transition matrix P interchangeably.
+Remark 1. We demonstrate the connection between the
+conditional distribution PˆY|S,Y and confusion matrices in the
+setting of binary classification with binary subgroups. We
+define ˆC as the counterpart of C when we replace PX|S,Y with
+an empirical distribution ˆPX|S,Y computed from a dataset.
+The confusion matrix for group s ∈ {0, 1} consists of four
+numbers: True Positive (TPs), False Positive (FPs), False
+Negative (FNs), True Negative (TNs). Assume that the num-
+ber of positive-label data n+
+s = TPs + FNs and negative-
+label data n−
+s = TNs + FPs are given—these numbers do
+not depend on the classifier. Then there is a one-to-one
+
+Aleatoric and Epistemic Discrimination in Classification
+mapping from each element in ˆC to a confusion matrix:
+ˆPˆY|S=s,Y=+(+) = 1
+n+
+s
+TPs,
+ˆPˆY|S=s,Y=−(+) = 1
+n−
+s
+FPs,
+ˆPˆY|S=s,Y=+(−) = 1
+n+
+s
+FNs,
+ˆPˆY|S=s,Y=−(−) = 1
+n−
+s
+TNs.
+Hence, ˆC essentially characterizes all feasible confusion
+matrices and C is the population counterpart of ˆC. Note
+that C is determined by the data distribution while ˆC (and
+confusion matrices) are tailored to a specific dataset.
+We
+establish
+basic
+properties
+of
+C
+and
+FairFront(αSP, αEO, αOAE) in the following lemma. Then we
+demonstrate how to use C for characterizing the fairness
+Pareto frontier.
+Lemma 2. C is a convex subset of T (C|AC) and
+FairFront(αSP, αEO, αOAE) is a concave function w.r.t.
+αSP, αEO, αOAE.
+Proposition 1. FairFront(αSP, αEO, αOAE) in (4) is equal to
+the solution of the following convex optimization:
+max
+P ∈RAC×C
+A
+�
+s=1
+C
+�
+y=1
+µs,yP(s,y),y
+(6a)
+s.t. SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE
+(6b)
+P ∈ C.
+(6c)
+Here the constants µs,y ≜ Pr(S = s, Y = y) and µs ≜
+Pr(S = s) for s ∈ [A], y ∈ [A] and P(s,y),ˆy denotes the
+(C(s − 1) + y)-th row, ˆy-th column of P .
+For example, in the setting of binary classification with
+binary group attribute, the above optimization only has 8
+variables, 14 linear constraints + a single convex constraint
+P ∈ C. Hence, its optimal value can be directly computed
+by standard convex optimization solvers as long as we know
+how to characterize C. Next, we discuss two special cases—
+X is independent of (S, Y) or X is discrete—under which C
+has a simple characterization.
+Remark 2. Note that Kim et al. (2020) investigated fairness
+Pareto frontiers via confusion matrices. The main difference
+is that Definition 1 in Kim et al. (2020) relaxed the constraint
+(6c) to P ∈ T (C|AC) where T (C|AC) represents all
+transition matrices from [AC] to [C]. This leads to a loose
+approximation of the frontier because C is often a strict
+subset of T (C|AC). To demonstrate this point, consider
+the scenario where X
+|=
+(S, Y). Then ˆY
+|=
+(S, Y) by data
+processing inequality so
+C = {P ∈ T (C|AC) | each row of P is the same} . (7)
+Optimizing over C rather than T (C|AC) can significantly
+tighten the fairness Pareto frontier.
+Remark 3. If X is a discrete variable with a finite sup-
+port [D], we can write PX|S,Y as a transition matrix Φ ∈
+T (D|AC). By introducing an auxiliary variable M ∈
+T (C|D), we can write P ∈ C equivalently as linear con-
+straints: P = ΦM by using the last condition of Lemma 1.
+Consequently, Proposition 1 boils down to a linear program.
+However, this characterization fails to generalize to continu-
+ous data because Φ and M will have an infinite dimension;
+for categorical data, this characterization suffers from the
+curse of dimensionality since the support size of X grows
+exponentially fast w.r.t. the number of features.
+The above two remarks provide precise characterizations of
+C under specific assumptions. In what follows, we consider
+a more general setting by leveraging Blackwell’s conditions
+(Section 2). Before diving into the analysis, we first intro-
+duce a function g : X → ∆AC defined as
+g(x) =
+�
+PS,Y|X(1, 1|x), · · · , PS,Y|X(A, C|x)
+�
+.
+(8)
+To obtain this function in practice, one can train a probabilis-
+tic classifier that uses input features X to predict (S, Y). We
+use this classifiers’ output probability as an approximation
+of the function g.
+The following theorem is the main theoretical result in this
+paper. It provides a precise characterization of the set C
+through a series of convex constraints.
+Theorem 1. The set C is the collection of all transition
+matrices P ∈ T (C|AC) such that the following condition
+holds:
+For any k ∈ N and any {ai | ai ∈ [−1, 1]AC, i ∈ [k]},
+C
+�
+ˆy=1
+max
+i∈[k]
+�
+aT
+i ΛΛΛµpˆy
+�
+≤ E
+�
+max
+i∈[k]{aT
+i g(X)}
+�
+,
+(9)
+where pˆy
+is the
+ˆy-th column of P
+and ΛΛΛµ
+=
+diag(µ1,1, · · · , µA,C).
+Intuitively, (9) uses piece-wise linear functions to approx-
+imate the boundary of C where k represents the num-
+ber of linear pieces.
+Unfortunately, replacing P ∈ C
+with this series of constraints in (6) may result in an in-
+tractable problem since standard duality-based approaches
+will lead to infinitely many dual variables.
+To resolve
+this issue, we first fix k and let Ck be the set of P such
+that (9) holds under this fixed k. Accordingly, we define
+FairFrontk(αSP, αEO, αOAE) as the optimal value of (6) when
+replacing C with Ck. Since C1 ⊇ C2 ⊇ · · · ⊇ C, we have
+FairFront1(αSP, αEO, αOAE) ≥ FairFront2(αSP, αEO, αOAE) ≥
+· · ·
+≥
+FairFront(αSP, αEO, αOAE). However, computing
+FairFrontk(αSP, αEO, αOAE) still involves infinitely many con-
+straints.
+Next, we introduce a greedy improvement algorithm that
+consists of solving a sequence of tractable optimization
+
+Aleatoric and Epistemic Discrimination in Classification
+Algorithm 1 Approximate the fairness Pareto frontier.
+Input: D = {(xi, yi, si)}N
+i=1, maximum number of iterations
+T; maximum pieces k; classifier g(x); threshold parameters
+αSP, αEO, αOAE.
+Initialize: set A = ∅; µs,y = |{i|si=s,yi=y}|
+N
+; t = 1.
+Repeat:
+Solve a convex program:
+max
+P
+A
+�
+s=1
+C
+�
+y=1
+µs,yP(s,y),y
+s.t. P ∈ T (C|AC)
+SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE
+C
+�
+ˆy=1
+max
+i∈[k]
+�
+aT
+i ΛΛΛµpˆy
+�
+≤ E
+�
+max
+i∈[k]{aT
+i g(X)}
+�
+∀(a1, · · · , ak) ∈ A.
+Let vt and P t be the optimal value and optimal solution.
+Solve a DC program:
+min
+ai∈[−1,1]AC
+i∈[k]
+E
+�
+max
+i∈[k]{aT
+i g(X)}
+�
+−
+C
+�
+ˆy=1
+max
+i∈[k]
+�
+aT
+i ΛΛΛµpt
+ˆy
+�
+.
+If the optimal value is ≥ 0 or t = T,
+stop;
+otherwise,
+add the optimal (a1, · · · , ak) to A and t = t + 1.
+return: vt, P t, A.
+problems for approximating FairFrontk(αSP, αEO, αOAE). We
+use A to collect the constraints of P and set A = ∅ initially.
+At each iteration, our algorithm solves a convex program
+to find an optimal P that maximizes the accuracy while
+satisfying the desired group fairness constraints and the con-
+straints in A; then we verify if this P is within the set Ck by
+solving a DC (difference of convex) program (Shen et al.,
+2016; Horst & Thoai, 1999). If P ∈ Ck, then the algo-
+rithm stops; otherwise, the algorithm will find the constraint
+that is mostly violated by P and add this constraint to A.
+We describe our algorithm in Algorithm 1 and establish a
+convergence guarantee in the following theorem.
+Theorem 2. Let T = ∞. If Algorithm 1 stops, its output P t
+is an optimal solution of FairFrontk(αSP, αEO, αOAE). Other-
+wise, any convergent sub-sequence of {P t}∞
+t=1 converges
+to an optimal solution of FairFrontk(αSP, αEO, αOAE).
+Note that the output vt from Algorithm 1 is always an up-
+per bound for FairFront(αSP, αEO, αOAE), assuming the es-
+timation error is sufficiently small. The tightness of this
+upper bound is determined by k (i.e., how well the piece-
+wise linear functions approximate the boundary of C), T
+(i.e., the total number of iterations). On the other hand,
+running off-the-shelf in-processing and post-processing
+fairness interventions can only yield lower bounds for
+FairFront(αSP, αEO, αOAE). In the next section, we compare
+our upper bound given by Algorithm 1 with the lower
+bounds given by some state-of-the-art methods to demon-
+strate the tightness of our algorithm.
+4. Numerical Experiments
+Recall that FairFront in Definition 1 characterizes the high-
+est achievable accuracy under a fairness constraint. We use
+it to quantify aleatoric discrimination and measure epistemic
+discrimination by comparing a classifier’s accuracy and fair-
+ness violation with FairFront. In this section, we apply
+FairFront to analyze the performance of existing fairness in-
+terventions and how data biases, specifically missing values,
+impact their effectiveness. We find that given sufficient data,
+state-of-the-art fairness interventions are successful at reduc-
+ing epistemic discrimination as their gap to the FairFront is
+small. However, we also discover that when different popu-
+lation groups have varying missing data patterns, aleatoric
+discrimination increases, which diminishes the performance
+of fairness intervention algorithms. We provide additional
+experimental results and details in Appendix C.
+4.1. Benchmark Fairness Interventions
+Setup.
+We evaluate our results on the UCI Adult dataset
+(Bache & Lichman, 2013), the ProPublica COMPAS dataset
+(Angwin et al., 2016), and the German Credit dataset (Bache
+& Lichman, 2013).3 We consider a binary classification
+problem with binary groups since most existing fairness
+interventions are designed for this scenario. For the Adult
+dataset, we choose sex (female or male) as the group at-
+tribute and income (> 50K or <= 50K) as the target for pre-
+diction; for the COMPAS dataset, we choose race (African-
+American or Caucasian) as the group attribute and is recid
+(recid. or no recid.) as the target for prediction. The details
+about how we pre-process these datasets are deferred to Ap-
+pendix C. We measure fairness violations via Max equalized
+odds:
+max | Pr(ˆY = ˆy|S = s, Y = y) − Pr(ˆY = ˆy|S = s′, Y = y)|
+where the max is taken over y, ˆy, s, s′. We run Algorithm 1
+with k = 6 pieces, 20 iterations, and varying αEO to get
+FairFront on each dataset. We compute the expectations
+and the g function from the empirical distributions and solve
+the DC program by using the DCCP package provided by
+Shen et al. (2016).
+Fairness
+interventions.
+We
+consider
+five
+existing
+fairness-intervention algorithms:
+Reduction (Agar-
+wal
+et
+al.,
+2018),
+EqOdds
+(Hardt
+et
+al.,
+2016),
+CalEqOdds (Pleiss et al., 2017), LevEqOpp (Chzhen
+3We defer the experimental results on the German Credit
+dataset to Appendix C.
+
+Aleatoric and Epistemic Discrimination in Classification
+Figure 1. Comparing existing fairness interventions with FairFront on the Adult (Left) and COMPAS (Right) datasets. We use FairFront
+to quantify aleatoric discrimination and measure epistemic discrimination by comparing a classifier’s accuracy and fairness violation
+with FairFront. As shown, SOTA fairness interventions are effective at reducing epistemic discrimination as their gap to the FairFront is
+small.
+et al., 2019), and FairProjection (Alghamdi et al.,
+2022).
+Among them, Reduction is an in-processing
+method and the rest are all post-processing methods. For
+the first three benchmarks, we use the implementations
+from IBM AIF360 library (Bellamy et al., 2018); for
+LevEqOpp and FairProjection, we use the Python
+implementations from the Github repo in Alghamdi et al.
+(2022). For Reduction and FairProjection, we
+can vary their tolerance of fairness violations to produce a
+fairness-accuracy curve; for EqOdds, CalEqOdds, and
+LevEqOpp, each of them produces a single point since
+they only allow hard equality constraint. We note that
+FairProjection is optimized for transforming prob-
+abilistic classifier outputs (see also Wei et al., 2021), but
+here we threshold the probabilistic outputs to generate bi-
+nary predictions which may limit its performance. Finally,
+we train a random forest as the Baseline classifier. In
+order to emulate the setting where the data distribution is
+known exactly, we train models using the entire dataset and
+resample 30% data as the test set.
+Results.
+We benchmark fairness interventions against
+FairFront in Figure 1. First, we observe that if we run
+Algorithm 1 for a single iteration, which is equivalent to
+solving Proposition 1 without (6c), its solution is very close
+to 1 for all αEO. This demonstrates the benefits of incorporat-
+ing Blackwell’s conditions into the fairness Pareto frontier.
+Second, we observe that fairness-accuracy curves given
+by state-of-the-art (SOTA) fairness interventions are very
+close to the fairness Pareto frontier. This result not only
+demonstrates the tightness of our approximation (recall
+that Algorithm 1 gives an upper bound of FairFront and
+benchmarks give a lower bound) but also shows that SOTA
+fairness interventions have already achieved near-optimal
+fairness-accuracy curves—their epistemic discrimination
+is small since they approach the FairFront limit. In what
+follows, we demonstrate how missing values in data can
+increase aleatoric discrimination and dramatically reduce
+the effectiveness of fairness interventions.
+4.2. Fairness Risks in Missing Values
+Real-world data often have missing values and the missing
+patterns can be different across different protected groups
+(see Jeong et al., 2022, for some examples). There is a grow-
+ing line of research (see e.g., Jeong et al., 2022; Fernando
+et al., 2021; Wang & Singh, 2021; Subramonian et al., 2022;
+Caton et al., 2022; Zhang & Long, 2021; Schelter et al.,
+2019) studying the fairness risks of data with missing val-
+ues. In this section, we apply our result to demonstrate how
+disparate missing patterns influence the fairness-accuracy
+curves.
+Setup.
+We choose sex (group 0: female, group 1: male) as
+the group attribute for the Adult dataset, and race (group 0:
+African-American, group 1: Caucasian) for the COMPAS
+dataset. To investigate the impact of disparate missing pat-
+terns on aleatoric discrimination, we artificially generate
+missing values in both datasets. This is necessary as the
+datasets do not contain sufficient missing data. The miss-
+ing values are generated according to different probabilities
+for different population groups. For each data point from
+group 0, we erase each input feature with a varying proba-
+bility p0 ∈ {10%, 50%, 70%}, while for group 1, we erase
+
+Adult
+Aleatoric
+84.5
+Epistemic
+discrimination,
+discrimination
+84.0
+(%)
+83.5
+Accuracy (
+83.0
+FairFront
+Baseline
+82.5
+FairProjection
+Reduction
+82.0
+LevEqOpp
+CalEqOdds
+EqOdds
+81.5
+0
+2.5
+5.0
+7.50
+10.0
+12.5
+Max equalized odds (%)COMPAS
+77.0
+76.0
+(%)
+Accuracy
+75.0
+74.0
+FairFront
+Baseline
+FairProjection
+Reduction
+73.0
+LevEqOpp
+CalEqOdds
+EqOdds
+72.0
+0
+5.0
+10.0
+15.0
+20.0
+Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification
+Figure 2. We demonstrate the fairness risks of disparate missing patterns. We vary missing probability of group 0 (female in Adult/African-
+American in COMPAS) among {10%, 50%, 70%} and let the missing probability of group 1 (male in Adult/Caucasian in COMPAS) be
+10%. We use mode imputation to pre-process missing data. We apply Reduction to the imputed data and plot its fairness-accuracy
+curve against the FairFront with the level of transparency representing the degree of disparity in the missing patterns.
+each input feature with a fixed probability p1 = 10%. We
+then apply mode imputation to the missing values, replacing
+them with the mode of non-missing values for each feature.
+Finally, we apply Algorithm 1 along with Reduction and
+Baseline to the imputed data. The experimental results
+are shown in Figure 2.
+Results.
+As we increase the missing probability of
+group 0, FairFront decreases since it becomes more difficult
+to accurately predict outcomes for group 0. This in turn
+affects the overall model performance, since the fairness
+constraint requires that the model performs similarly for
+both groups. We also observe the fairness-accuracy curves
+of Reduction decrease as the missing data for group 0
+become more prevalent. In other words, as the missing data
+for group 0 increase, it becomes more difficult to maintain
+both high accuracy and fairness in the model’s prediction.
+5. Final Remarks and Limitations
+The past years have witnessed a growing line of research in-
+troducing various fairness-intervention algorithms. Most of
+these interventions focus on optimizing model performance
+subject to group fairness constraints. Though comparing
+and benchmarking these methods on various datasets is valu-
+able (e.g., see benchmarks in Friedler et al., 2019; Bellamy
+et al., 2019; Wei et al., 2021), this does not reveal if there is
+still room for improvement in their fairness-accuracy curves,
+or if existing methods approach the information-theoretic
+optimal limit when infinite data is available. Our results
+address this gap by introducing the fairness Pareto frontier,
+which measures the highest possible accuracy under a group
+fairness constraint. We precisely characterize the fairness
+Pareto frontier using Blackwell’s conditions and present a
+greedy improvement algorithm that approximates it from
+data. Our results show that the fairness-accuracy curves
+produced by state-of-the-art fairness interventions are very
+close to the fairness Pareto frontier on standard datasets.
+Additionally, we demonstrate that when data are biased
+due to missing values, the fairness Pareto frontier degrades.
+Although existing fairness interventions can still reduce per-
+formance disparities, they come at the cost of significantly
+lowering overall model accuracy. The methods we present
+for computing the fairness Pareto frontier can also be ap-
+plied to analyze other sources of aleatoric discrimination,
+such as when individuals may misreport their data or when
+there are measurement errors. Overall, the fairness Pareto
+frontier can serve as a valuable framework for guiding data
+collection and cleaning.
+Our results indicate that existing fairness interventions can
+be effective in reducing epistemic discrimination, and there
+are diminishing returns in developing new fairness inter-
+ventions focused solely on optimizing accuracy for a given
+group fairness constraint on pristine data. However, existing
+fairness interventions have yet to effectively provide both
+fair and accurate classification when additional sources of
+aleatoric discrimination are present (such as missing values
+in data). This suggests that there is still significant need for
+research on handling aleatoric sources of discrimination that
+appear throughout the data collection process.
+
+Adult
+84.5
+84.0
+Accuracy (%)
+83.5
+83.0
+82.5
+82.0
+FairFront
+Baseline
+Reduction
+81.5
+0
+10.0
+20.0
+30.0
+40.0
+50.0
+Max equalized odds (%)COMPAS
+76.0
+74.0
+72.0
+%)
+70.0
+Accuracy (
+68.0
+65.9
+63.9
+62.0
+FairFront
+Baseline
+60.0
+Reduction
+0
+20.0
+40.0
+60.0
+Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification
+References
+Agarwal, A., Beygelzimer, A., Dud´ık, M., Langford, J., and
+Wallach, H. A reductions approach to fair classification.
+In International Conference on Machine Learning, pp.
+60–69. PMLR, 2018.
+Alghamdi, W., Asoodeh, S., Wang, H., Calmon, F. P., Wei,
+D., and Ramamurthy, K. N. Model projection: Theory
+and applications to fair machine learning. In 2020 IEEE
+International Symposium on Information Theory (ISIT),
+pp. 2711–2716. IEEE, 2020.
+Alghamdi, W., Hsu, H., Jeong, H., Wang, H., Michalak,
+P. W., Asoodeh, S., and Calmon, F. P. Beyond Adult and
+COMPAS: Fair multi-class prediction via information
+projection. In Advances in Neural Information Processing
+Systems, 2022.
+Angwin, J., Larson, J., Mattu, S., and Kirchner, L. Machine
+bias. ProPublica, 2016.
+Bache, K. and Lichman, M. UCI Machine Learning Reposi-
+tory, 2013.
+Bellamy, R. K., Dey, K., Hind, M., Hoffman, S. C., Houde,
+S., Kannan, K., Lohia, P., Martino, J., Mehta, S., Mo-
+jsilovic, A., et al. Ai fairness 360: An extensible toolkit
+for detecting, understanding, and mitigating unwanted
+algorithmic bias. arXiv preprint arXiv:1810.01943, 2018.
+Bellamy, R. K., Dey, K., Hind, M., Hoffman, S. C., Houde,
+S., Kannan, K., Lohia, P., Martino, J., Mehta, S., Mo-
+jsilovi´c, A., et al. Ai fairness 360: An extensible toolkit
+for detecting and mitigating algorithmic bias. IBM Jour-
+nal of Research and Development, 63(4/5):4–1, 2019.
+Berk, R., Heidari, H., Jabbari, S., Kearns, M., and Roth, A.
+Fairness in criminal justice risk assessments: The state of
+the art. Sociological Methods & Research, 50(1):3–44,
+2021.
+Blackwell, D. Comparison of experiments. Proceedings of
+the Second Berkeley Symposium on Mathematical Statis-
+tics and Probability, pp. 93–102, 1951.
+Blackwell, D. Equivalent comparisons of experiments. The
+annals of mathematical statistics, pp. 265–272, 1953.
+Blum, A. and Stangl, K. Recovering from biased data: Can
+fairness constraints improve accuracy? arXiv preprint
+arXiv:1912.01094, 2019.
+Calmon, F., Wei, D., Vinzamuri, B., Natesan Ramamurthy,
+K., and Varshney, K. R. Optimized pre-processing for dis-
+crimination prevention. Advances in neural information
+processing systems, 30, 2017.
+Cam, L. L. Sufficiency and approximate sufficiency. The
+Annals of Mathematical Statistics, pp. 1419–1455, 1964.
+Caton, S., Malisetty, S., and Haas, C. Impact of imputation
+strategies on fairness in machine learning. Journal of
+Artificial Intelligence Research, 74:1011–1035, 2022.
+Celis, L. E., Huang, L., Keswani, V., and Vishnoi, N. K.
+Classification with fairness constraints: A meta-algorithm
+with provable guarantees. In Proceedings of the confer-
+ence on fairness, accountability, and transparency, pp.
+319–328, 2019.
+Chen, I., Johansson, F. D., and Sontag, D. Why is my
+classifier discriminatory? Advances in neural information
+processing systems, 31, 2018.
+Chouldechova, A. Fair prediction with disparate impact: A
+study of bias in recidivism prediction instruments. Big
+data, 5(2):153–163, 2017.
+Chzhen, E., Denis, C., Hebiri, M., Oneto, L., and Pontil, M.
+Leveraging labeled and unlabeled data for consistent fair
+binary classification. Advances in Neural Information
+Processing Systems, 32, 2019.
+Cohen, J., Kempermann, J. H., and Zbaganu, G. Com-
+parisons of stochastic matrices with applications in in-
+formation theory, statistics, economics and population.
+Springer Science & Business Media, 1998.
+Corbett-Davies, S., Pierson, E., Feller, A., Goel, S., and
+Huq, A. Algorithmic decision making and the cost of fair-
+ness. In Proceedings of the 23rd acm sigkdd international
+conference on knowledge discovery and data mining, pp.
+797–806, 2017.
+Dutta, S., Wei, D., Yueksel, H., Chen, P.-Y., Liu, S., and
+Varshney, K. Is there a trade-off between fairness and
+accuracy? a perspective using mismatched hypothesis
+testing. In International Conference on Machine Learn-
+ing, pp. 2803–2813. PMLR, 2020.
+Dwork, C., Immorlica, N., Kalai, A. T., and Leiserson, M.
+Decoupled classifiers for group-fair and efficient machine
+learning. In Conference on fairness, accountability and
+transparency, pp. 119–133. PMLR, 2018.
+Feldman, M., Friedler, S. A., Moeller, J., Scheidegger, C.,
+and Venkatasubramanian, S. Certifying and removing dis-
+parate impact. In proceedings of the 21th ACM SIGKDD
+international conference on knowledge discovery and
+data mining, pp. 259–268, 2015.
+Fernando, M.-P., C`esar, F., David, N., and Jos´e, H.-O. Miss-
+ing the missing values: The ugly duckling of fairness in
+machine learning. International Journal of Intelligent
+Systems, 36(7):3217–3258, 2021.
+
+Aleatoric and Epistemic Discrimination in Classification
+Fogliato, R., Chouldechova, A., and G’Sell, M. Fairness
+evaluation in presence of biased noisy labels. In Interna-
+tional Conference on Artificial Intelligence and Statistics,
+pp. 2325–2336. PMLR, 2020.
+Friedler, S. A., Scheidegger, C., Venkatasubramanian, S.,
+Choudhary, S., Hamilton, E. P., and Roth, D. A compara-
+tive study of fairness-enhancing interventions in machine
+learning. In Proceedings of the conference on fairness,
+accountability, and transparency, pp. 329–338, 2019.
+Hardt, M., Price, E., and Srebro, N. Equality of opportunity
+in supervised learning. In Advances in Neural Informa-
+tion Processing Systems, volume 29, 2016.
+Horst, R. and Thoai, N. V. Dc programming: overview.
+Journal of Optimization Theory and Applications, 103(1):
+1–43, 1999.
+Hort, M., Chen, Z., Zhang, J. M., Sarro, F., and Harman,
+M. Bia mitigation for machine learning classifiers: A
+comprehensive survey. arXiv preprint arXiv:2207.07068,
+2022.
+H¨ullermeier, E. and Waegeman, W. Aleatoric and epistemic
+uncertainty in machine learning: An introduction to con-
+cepts and methods. Machine Learning, 110(3):457–506,
+2021.
+Jeong, H., Wang, H., and Calmon, F. P. Fairness without
+imputation: A decision tree approach for fair prediction
+with missing values. In Proceedings of the AAAI Confer-
+ence on Artificial Intelligence, volume 36, pp. 9558–9566,
+2022.
+Jiang, H. and Nachum, O. Identifying and correcting label
+bias in machine learning. In International Conference on
+Artificial Intelligence and Statistics, pp. 702–712. PMLR,
+2020.
+Jiang, R., Pacchiano, A., Stepleton, T., Jiang, H., and Chi-
+appa, S. Wasserstein fair classification. In Uncertainty in
+Artificial Intelligence, pp. 862–872. PMLR, 2020.
+Kallus, N., Mao, X., and Zhou, A. Assessing algorithmic
+fairness with unobserved protected class using data com-
+bination. Management Science, 68(3):1959–1981, 2022.
+Kamiran, F. and Calders, T. Data preprocessing techniques
+for classification without discrimination. Knowledge and
+information systems, 33(1):1–33, 2012.
+Kearns, M., Neel, S., Roth, A., and Wu, Z. S. Preventing
+fairness gerrymandering: Auditing and learning for sub-
+group fairness. In International Conference on Machine
+Learning, pp. 2564–2572. PMLR, 2018.
+Kim, J. S., Chen, J., and Talwalkar, A. Fact: A diagnostic
+for group fairness trade-offs. In International Conference
+on Machine Learning, pp. 5264–5274. PMLR, 2020.
+Kim, M. P., Ghorbani, A., and Zou, J. Multiaccuracy: Black-
+box post-processing for fairness in classification. In Pro-
+ceedings of the 2019 AAAI/ACM Conference on AI, Ethics,
+and Society, pp. 247–254, 2019.
+Kleinberg, J., Mullainathan, S., and Raghavan, M. Inherent
+trade-offs in the fair determination of risk scores. arXiv
+preprint arXiv:1609.05807, 2016.
+Kulkarni, A., Chong, D., and Batarseh, F. A. Foundations
+of data imbalance and solutions for a data democracy. In
+data democracy, pp. 83–106. Elsevier, 2020.
+Lowy, A., Pavan, R., Baharlouei, S., Razaviyayn, M., and
+Beirami, A. Fermi: Fair empirical risk minimization via
+exponential R´enyi mutual information. arXiv preprint
+arXiv:2102.12586, 2021.
+Martinez, N., Bertran, M., and Sapiro, G. Minimax pareto
+fairness: A multi objective perspective.
+In Interna-
+tional Conference on Machine Learning, pp. 6755–6764.
+PMLR, 2020.
+Mayson, S. G. Bias in, bias out. The Yale Law Journal, 128
+(8):2218–2300, 2019.
+Mehrotra, A. and Celis, L. E. Mitigating bias in set selection
+with noisy protected attributes. In Proceedings of the
+2021 ACM Conference on Fairness, Accountability, and
+Transparency, pp. 237–248, 2021.
+Menon, A. K. and Williamson, R. C. The cost of fairness in
+binary classification. In Conference on Fairness, Account-
+ability and Transparency, pp. 107–118. PMLR, 2018.
+Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V.,
+Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P.,
+Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cour-
+napeau, D., Brucher, M., Perrot, M., and Duchesnay, E.
+Scikit-learn: Machine learning in Python. Journal of
+Machine Learning Research, 12:2825–2830, 2011.
+Pleiss, G., Raghavan, M., Wu, F., Kleinberg, J., and Wein-
+berger, K. Q. On fairness and calibration. Advances in
+neural information processing systems, 30, 2017.
+Raginsky, M. Shannon meets blackwell and le cam: Chan-
+nels, codes, and statistical experiments. In 2011 IEEE
+International Symposium on Information Theory Proceed-
+ings, pp. 1220–1224. IEEE, 2011.
+Rauh, J., Banerjee, P. K., Olbrich, E., Jost, J., Bertschinger,
+N., and Wolpert, D. Coarse-graining and the blackwell
+order. Entropy, 19(10):527, 2017.
+
+Aleatoric and Epistemic Discrimination in Classification
+Schelter, S., He, Y., Khilnani, J., and Stoyanovich, J. Fair-
+prep: Promoting data to a first-class citizen in stud-
+ies on fairness-enhancing interventions. arXiv preprint
+arXiv:1911.12587, 2019.
+Shannon, C. E. A note on a partial ordering for communi-
+cation channels. Information and control, 1(4):390–397,
+1958.
+Shen, X., Diamond, S., Gu, Y., and Boyd, S. Disciplined
+convex-concave programming. In 2016 IEEE 55th Con-
+ference on Decision and Control (CDC), pp. 1009–1014.
+IEEE, 2016.
+Subramonian, A., Chang, K.-W., and Sun, Y. On the dis-
+crimination risk of mean aggregation feature imputation
+in graphs. Advances in Neural Information Processing
+Systems, 2022.
+Suresh, H. and Guttag, J. V. A framework for understanding
+unintended consequences of machine learning. arXiv
+preprint arXiv:1901.10002, 2:8, 2019.
+Torgersen, E. Comparison of statistical experiments, vol-
+ume 36. Cambridge University Press, 1991.
+Ustun, B., Liu, Y., and Parkes, D. Fairness without harm:
+Decoupled classifiers with preference guarantees. In In-
+ternational Conference on Machine Learning, pp. 6373–
+6382. PMLR, 2019.
+Verma, S. and Rubin, J. Fairness definitions explained.
+In 2018 ieee/acm international workshop on software
+fairness (fairware), pp. 1–7. IEEE, 2018.
+Wang, H., Hsu, H., Diaz, M., and Calmon, F. P. To split or
+not to split: The impact of disparate treatment in classi-
+fication. IEEE Transactions on Information Theory, 67
+(10):6733–6757, 2021.
+Wang, S., Guo, W., Narasimhan, H., Cotter, A., Gupta, M.,
+and Jordan, M. Robust optimization for fairness with
+noisy protected groups. Advances in Neural Information
+Processing Systems, 33:5190–5203, 2020.
+Wang, Y. and Singh, L. Analyzing the impact of missing val-
+ues and selection bias on fairness. International Journal
+of Data Science and Analytics, 12(2):101–119, 2021.
+Wei, D., Ramamurthy, K. N., and Calmon, F. P. Optimized
+score transformation for consistent fair classification. J.
+Mach. Learn. Res., 22:258–1, 2021.
+Yang, F., Cisse, M., and Koyejo, S. Fairness with overlap-
+ping groups; a probabilistic perspective. In Advances in
+Neural Information Processing Systems, volume 33, pp.
+4067–4078, 2020.
+Zafar, M. B., Valera, I., Gomez-Rodriguez, M., and Gum-
+madi, K. P. Fairness constraints: A flexible approach
+for fair classification. The Journal of Machine Learning
+Research, 20(1):2737–2778, 2019.
+Zemel, R., Wu, Y., Swersky, K., Pitassi, T., and Dwork, C.
+Learning fair representations. In International conference
+on machine learning, pp. 325–333. PMLR, 2013.
+Zhang, B. H., Lemoine, B., and Mitchell, M. Mitigating un-
+wanted biases with adversarial learning. In Proceedings
+of the 2018 AAAI/ACM Conference on AI, Ethics, and
+Society, pp. 335–340, 2018.
+Zhang, Y. and Long, Q. Assessing fairness in the presence of
+missing data. Advances in neural information processing
+systems, 34:16007–16019, 2021.
+
+Aleatoric and Epistemic Discrimination in Classification
+A. Technical Background
+In this section, we extend some results in Blackwell (1951; 1953) to our setting. For a random variable X, we denote
+its probability distribution by L(X). A conditional distribution PX|A : [n] → P(X) can be equivalently written as
+P ≜ (P1, · · · , Pn) where each Pi = PX|A=i ∈ P(X). Let A be a closed, bounded, convex subset of Rn. A decision
+function is a mapping f : X → A, which can also be written as f(x) = (a1(x), · · · , an(x)). A decision function is
+associated a loss vector:
+v(f) =
+��
+a1(x)dP1(x), · · · ,
+�
+an(x)dPn(x)
+�
+.
+(11)
+The collection of all v(f) is denoted by B(PX|A, A) or B(P , A).
+For a vector λλλ ∈ ∆n such that λλλ > 0, we define a function pλλλ(x) : X → ∆n:
+pλλλ(x) =
+�
+λ1dP1
+λ1dP1 + · · · + λndPn
+, · · · ,
+λndPn
+λ1dP1 + · · · + λndPn
+�
+.
+(12)
+Note that pλλλ(X) is a sufficient statistic for X, considering A as the parameter (it can be proved by Fisher-Neyman factorization
+theorem). In other words, two Markov chains hold: A → pλλλ(X) → X and A → X → pλλλ(X) for any distribution on A.
+Consider a new set of probability distributions P ∗
+λλλ ≜ (L(pλλλ(X1)), · · · , L(pλλλ(Xn))) where L(Xi) = Pi. Here P ∗
+λλλ can be
+viewed as a conditional distribution from [n] to P(∆n) since each L(pλλλ(Xi)) is a probability distribution over ∆n. The
+following lemma follows from the sufficiency of pλλλ(X).
+Lemma 3 (Adaptation of Theorem 3 in Blackwell (1951)). For any A, B(P , A) = B(P ∗
+λλλ, A).
+Proof. Suppose that f ∗(p) = (a∗
+1(p), · · · , a∗
+n(p)) is a decision function for (P ∗
+λλλ, A). Accordingly, we define f(x) =
+(a∗
+1(pλλλ(x)), · · · , an(pλλλ(x))) where the function pλλλ is defined in (12). Then it is clear that f is a decision function for
+(P , A). By the law of unconscious statistician, we have
+�
+a∗
+i (p)dP ∗
+λ
+λ
+λ,i(p) = E [a∗
+i (pλλλ(Xi))] =
+�
+a∗
+i (pλλλ(x))dPi(x).
+(13)
+Hence, v(f ∗) = v(f), which implies B(P ∗
+λλλ, A) ⊆ B(P , A). For the other direction, suppose f(x) = (a1(x), · · · , an(x))
+is a decision function for (P , A). Let f ∗(p) = (a∗
+1(p), · · · , a∗
+n(p)) where a∗
+i (p) ≜ E [ai(Xi) | pλλλ(Xi) = p]. Since pλλλ(X)
+is a sufficient statistics, for any i ∈ [n]
+L(Xi|pλλλ(Xi) = p) = L(X1|pλλλ(X1) = p).
+(14)
+Therefore, f ∗(p) = E [f(X1)|pλλλ(X1) = p]. Since A is a convex set, f ∗ is a decision function for (P ∗, A). By the law of
+total expectation, we have
+�
+a∗
+i (p)dP ∗
+λλλ,i(p) =
+�
+ai(x)dPi(x).
+(15)
+Hence, v(f) = v(f ∗), which implies B(P , A) ⊆ B(P ∗
+λλλ, A).
+For a vector λλλ ∈ ∆n such that λλλ > 0, the condition distribution PX|A induces a weighted standard measure P ∗
+λλλ ≜ L (pλλλ(¯X))
+where L(¯X) = λ1P1 + · · · + λnPn.
+Theorem 3 (Adaptation of Theorem 4 in Blackwell (1951)). For any two conditional distributions PX|A and QY|A, let
+P ∗
+λλλ and Q∗
+λλλ be their weighted standard measures, respectively. Then B(PX|A, A) ⊇ B(QY|A, A) for any closed, bounded,
+convex set A if and only if for any continuous convex φ : ∆n → R,
+�
+φ(p)dP ∗
+λλλ(p) ≥
+�
+φ(p)dQ∗
+λλλ(p)
+Proof. First, by Lemma 3, we know B(PX|A, A) = B(P ∗
+λλλ, A) and B(QY|A, A) = B(Q∗
+λλλ, A).
+We denote ΛΛΛ =
+diag(λ1, · · · , λn). Consider any A = conv(a1, · · · , ak). Let
+f ∗(p) = argmin
+a∈A
+pTΛΛΛ−1a.
+(16)
+
+Aleatoric and Epistemic Discrimination in Classification
+Note that f ∗(p) ∈ {a1, · · · , ak} since this set contains all the extreme points of A.4 By definition, for any decision function
+w.r.t. (P ∗
+λλλ, A), we have
+pTΛΛΛ−1f(p) ≥ pTΛΛΛ−1f ∗(p),
+∀p.
+(17)
+Let v = v(f). By the same reason with (13), we have
+vj =
+�
+aj(pλλλ(x))dPj(x)
+(18)
+= 1
+λj
+�
+aj(pλλλ(x))
+λjdPj
+λ1dP1 + · · · + λndPn
+(x)(λ1dP1 + · · · + λndPn)(x)
+(19)
+= 1
+λj
+�
+aj(pλλλ(x))[pλλλ(x)]j(λ1dP1 + · · · + λndPn)(x)
+(20)
+= 1
+λj
+E [aj(pλλλ(¯X))[pλλλ(¯X)]j]
+(21)
+= 1
+λj
+�
+aj(p)pjdP ∗
+λλλ(p),
+(22)
+where the last step is due to the law of unconscious statistician. Therefore,
+n
+�
+j=1
+vj =
+�
+pTΛΛΛ−1f(p)dP ∗
+λλλ(p)
+(23)
+≥
+�
+pTΛΛΛ−1f ∗(p)dP ∗
+λλλ(p)
+(24)
+=
+�
+min
+i {pTΛΛΛ−1ai}dP ∗
+λλλ(p).
+(25)
+The equality is achieved by v(f ∗). Hence, for any A = conv(a1, · · · , ak)
+min
+v∈B(PX|A,A)
+n
+�
+j=1
+vj =
+�
+min
+i {aT
+i ΛΛΛ−1p}dP ∗
+λλλ(p).
+(26)
+Recall that Theorem 2.(3) in Blackwell (1951) states
+B(PX|A, A) ⊇ B(PY|A, A)
+for every closed, bounded, convex A
+⇔
+min
+v∈B(PX|A,A)
+n
+�
+j=1
+vj ≤
+min
+v∈B(PY|A,A)
+n
+�
+j=1
+vj
+for every closed, bounded, convex A.
+By approximation theory, the second condition can be relaxed to any A that is a convex hull of a finite set. By (26), this
+relaxed condition is equivalent to
+�
+φ(p)dP ∗
+λλλ(p) ≥
+�
+φ(p)dQ∗
+λλλ(p)
+(27)
+for all φ(p) that are the maximum of finitely many linear functions. By approximation theory again, the above condition is
+equivalent to the one holding for any continuous convex function φ.
+B. Omitted Proofs
+B.1. Proof of Lemma 2
+Proof. Clearly, C is a subset of T (C|AC). Let λ ∈ (0, 1) and PˆY0|S,Y, PˆY1|S,Y ∈ C. Now we introduce a Bernoulli random
+variable B such that Pr(B = 0) = λ. Finally, we define ˆYλ = BˆY1 +(1−B)ˆY0. By definition, we have (S, Y) → X → ˆYλ
+4If (16) has multiple optimal solutions, we always choose the one from {a1, · · · , ak}.
+
+Aleatoric and Epistemic Discrimination in Classification
+so PˆYλ|S,Y ∈ C. Moreover,
+PˆYλ|S,Y = λPˆY0|S,Y + (1 − λ)PˆY1|S,Y.
+Hence, C is convex.
+Let λ ∈ (0, 1). Assume P and ¯P achieve the maximal values of Proposition 1 under (αSP, αEO, αOAE) and (¯αSP, ¯αEO, ¯αOAE),
+respectively. We define Pλ = λP + (1 − λ) ¯P , which satisfies the constraints of Proposition 1 with thresholds (λαSP + (1 −
+λ)¯αSP, λαEO + (1 − λ)¯αEO, λαOAE + (1 − λ)¯αOAE). Finally, since the objective function of Proposition 1 is a linear function, it
+is equal to λFairFront(αSP, αEO, αOAE) + (1 − λ)FairFront(¯αSP, ¯αEO, ¯αOAE) under Pλ.
+B.2. Proof of Theorem 1
+Proof. The proof relies on Theorem 3 and Lemma 1. For simplicity, we write the conditional PˆY|S,Y as its corresponding
+transition matrix P . Let µµµ = (Pr(S = 1, Y = 1), · · · , Pr(S = A, Y = C)). The function (12) in our setting can be written
+as:
+pµµµ(ˆy) =
+�
+µ1,1P(1,1),ˆy
+�
+s,y µs,yP(s,y),ˆy
+, · · · ,
+µA,CP(A,C),ˆy
+�
+s,y µs,yP(s,y),ˆy
+�
+.
+(28)
+pµµµ(x) =
+�
+µ1,1dPX|S=1,Y=1
+�
+s,y µs,ydPX|S=s,Y=y
+(x), · · · ,
+µA,CdPX|S=A,Y=C
+�
+s,y µs,ydPX|S=s,Y=y
+(x)
+�
+.
+(29)
+Note that pµµµ(x) = g(x) due to Bayes’ rule (see (8) for the definition of g). By Lemma 1, we can rewrite C in Definition 2 as
+C =
+�
+P | PˆX|S,Y is more informative than P
+�
+.
+(30)
+By Lemma 1 and Theorem 3, the above set is further equivalent to all transition matrices P ∈ T (C|AC) satisfying
+C
+�
+ˆy=1
+φ
+�
+µ1,1P(1,1),ˆy
+�
+s,y µs,yP(s,y),ˆy
+, · · · ,
+µA,CP(A,C),ˆy
+�
+s,y µs,yP(s,y),ˆy
+� �
+s,y
+µs,yP(s,y),ˆy ≤ E [φ(g(X))]
+(31)
+for any function φ : ∆AC → R which is the maximum of finitely many linear functions. Now we can write φ(p) =
+maxi∈[k]
+�
+aT
+i p
+�
+—we ignore the bias term because aT
+i p+bi = (ai +bi1)T p. Then the inequality in (31) can be simplified
+as
+C
+�
+ˆy=1
+max
+i∈[k]
+�
+aT
+i ΛΛΛµpˆy
+�
+≤ E
+�
+max
+i∈[k]{aT
+i g(X)}
+�
+,
+(32)
+where pˆy is the ˆy-th column of P and ΛΛΛµ = diag(µ1,1, · · · , µA,C). Finally, we can always normalize the above inequality
+so that each ai ∈ [−1, 1]AC.
+B.3. Proof of Theorem 2
+Proof. We denote
+f(P ) ≜
+A
+�
+s=1
+C
+�
+y=1
+µs,yP(s,y),y,
+g(P ; a1, · · · , ak) ≜
+C
+�
+ˆy=1
+max
+i∈[k]
+�
+aT
+i ΛΛΛµpˆy
+�
+− E
+�
+max
+i∈[k]{aT
+i g(X)}
+�
+,
+F ≜ Ck ∩ {P ∈ T (C|AC) | SP ≤ αSP, EO ≤ αEO, OAE ≤ αOAE} .
+Let Ft be the constraint set of P at the t-th iteration of our algorithm. Note that F ⊆ Ft by definition. If the algorithm
+stops at the t-th iteration, then for any {ai | ai ∈ [−1, 1]AC, i ∈ [k]}, P t satisfies
+g(P t; a1, · · · , ak) ≤ 0,
+
+Aleatoric and Epistemic Discrimination in Classification
+which implies P t ∈ F. Consequently,
+f(P t) = max
+P ∈Ft f(P ) ≥ max
+P ∈F f(P ) ≥ f(P t).
+As a result, f(P t) = maxP ∈F f(P ) so P t is an optimal solution of FairFrontk(αSP, αEO, αOAE).
+If the algorithm never stops, consider any convergent sub-sequence of P t that converges to a limit point P ∗ ∈ T (C|AC).
+To simplify our notation, we assume P t → P ∗ as t → ∞. Since {Ft}t≥1 is non-increasing and they all contain F, there
+exists a set F∗ such that
+lim
+t→∞ Ft = F∗,
+F ⊆ F∗.
+Therefore, we have
+f(P ∗) = lim
+t→∞ f(P t) = lim
+t→∞ max
+P ∈Ft f(P ) = max
+P ∈F∗ f(P ).
+Since F ⊆ F∗, we have
+f(P ∗) = max
+P ∈F∗ f(P ) ≥ max
+P ∈F f(P ).
+If P ∗ ̸∈ F, then there exists a (¯a1, · · · , ¯ak), such that g(P ∗; ¯a1, · · · , ¯ak) > 0. Let (a1,t, · · · , ak,t) be the output of Step
+2 at t-th iteration. Since P ∗ ∈ Ft for all t, we have
+g(P ∗; a1,t, · · · , ak,t) ≤ 0.
+(33)
+By the optimality of (a1,t, · · · , ak,t), we have
+g(P t; a1,t, · · · , ak,t) ≥ g(P t; ¯a1, · · · , ¯ak).
+(34)
+Suppose that some sub-sequence of (a1,t, · · · , ak,t) converges to a vector (a∗
+1, · · · , a∗
+k). For the sake of simplicity, we
+assume (a1,t, · · · , ak,t) → (a∗
+1, · · · , a∗
+k) as t → ∞. On the one hand, taking limit of t → ∞ on both sides of (34) leads to
+g(P ∗; a∗
+1, · · · , a∗
+k) ≥ g(P ∗; ¯a1, · · · , ¯ak).
+On the other hand, taking limit of t → ∞ on both sides of (33) leads to
+g(P ∗; a∗
+1, · · · , a∗
+k) ≤ 0.
+Therefore,
+0 ≥ g(P ∗; a∗
+1, · · · , a∗
+k) ≥ g(P ∗; ¯a1, · · · , ¯ak) > 0,
+which is impossible. Therefore, P ∗ ∈ F and, as a result, we have
+f(P ∗) = max
+P ∈F∗ f(P ) ≥ max
+P ∈F f(P ) ≥ f(P ∗) =⇒ max
+P ∈F f(P ) = f(P ∗).
+C. Details on the Experimental Results
+C.1. Additional Experiments
+In this section, we present additional experimental results to further support our findings. First, we reproduce our ex-
+perimental results on the German Credit dataset (Bache & Lichman, 2013). We compare existing fairness interventions
+with FairFront in Figure 3. Our observation is consistent with those on the previous two datasets—the fairness-
+accuracy curves given by SOTA fairness interventions, such as Reduction and FairProjection, are close to the
+information-theoretic limit.
+As previously demonstrated in Figure 1 and 3, the fairness-accuracy curves generated by Reduction and
+FairProjection are close to FairFront. To further evaluate the performance of these methods, we train a classifier that
+approximates the Bayes optimal and feed it to Reduction and FairProjection. The results are shown in Figure 4,
+which demonstrates that the (training) accuracy-fairness curves generated by Reduction and FairProjection can
+approach FairFront when using the Bayes optimal baseline classifier.
+
+Aleatoric and Epistemic Discrimination in Classification
+Figure 3. Comparing existing fairness interventions with FairFront on the German Credit dataset.
+C.2. Dataset
+Adult.
+We use sex (female or male) as the group attribute and income (> 50K or <= 50K) as the target for
+prediction.
+We use sex, hours-per-week, education-num, age, marital status, relationship status (husband or wife)
+as the input features—we include the group attribute as an input feature.
+We group age into a total of 12 dis-
+joint intervals: [0, 20), [20, 25), · · · , [65, 70), [70, ∞); we group hours-per-week into a total of 14 disjoint intervals:
+[0, 10), [10, 15), · · · , [65, 70), [70, ∞).
+COMPAS.
+We use race (African-American or Caucasian) as the group attribute and is recid (recid. or no recid.) as the
+target for prediction. We use race, age, c charge degree, sex, priors count, c jail in, c jail out as the input features—we
+include the group attribute as an input feature. We use the last two features by taking their difference to be their length of stay.
+We remove entries where COMPAS case could not be found (is recid = -1) and entries with inconsistent arrest information.
+We also binarize sex and remove traffic offenses. We quantize age the same way we do in the Adult dataset and quantize
+length of stay by every 30 days and let 0 be a separate category.
+German Credit.
+We use age (below or above 25 years old) as the group attribute and the credit column, which represents
+whether the loan was a good decision, as the target for prediction. We use loan duration in month, credit amount, age,
+number of existing credits at this bank, sex, credit history, savings, and length of present employment as input features. We
+include the group attribute age as an input feature. We group credit amount into three disjoint intervals: [0, 5000), [5000,
+10000),[10000,∞). We group duration of loan into two categories: under 36 months and over 36 months.
+C.3. Benchmark
+Each benchmark method’s hyper-parameter values are provided below. Each point in Figure 1 for Baseline, EqOdds,
+CalEqOdds, Reduction, LevEqOpp, and FairProjection is obtained by applying the obtained classifier to 10
+different test sets. For the Adult dataset, we use Random Forest with n estimators=15, min samples leaf=3, criterion =
+log loss, bootstrap = False as our baseline classifier; for the COMPAS dataset, we use Random Forest with n estimators = 17
+as our baseline classifier. For the German Credit dataset, we use Random Forest with n estimators=100,min samples split
+=2,min samples leaf=1 as our baseline classifier. They are all implemented by Scikit-learn (Pedregosa et al., 2011).
+EqOdds (Hardt et al., 2016).
+We use AIF360 implementation of EqOddsPostprocessing and the default hyper-
+parameter setup.
+CalEqOdds (Pleiss et al., 2017).
+We use AIF360 implementation of CalibratedEqOddsPostprocessing and
+the default hyper-parameter setup.
+
+German Credit
+86.0
+85.5
+85.0
+%)
+84.5
+Accuracy (
+84.0
+83.5
+FairFront
+Baseline
+FairProjection
+83.0
+Reduction
+LevEqOpp
+82.5
+CalEqOdds
+EqOdds
+0
+5.0
+10.0
+15.0
+20.0
+Max equalized odds (%)Aleatoric and Epistemic Discrimination in Classification
+Figure 4. Comparing Reduction and FairProjection with FairFront on the Adult (Left), COMPAS (Middle), and German Credit
+(Right) datasets. We train a baseline classifier that approximates the Bayes optimal and feed it into the two fairness interventions. As
+shown, their fairness-accuracy curves are very close to the FairFront in this case.
+Reduction (Agarwal et al., 2018).
+We use AIF360 implementation of ExponentiatedGradientReduction.
+We vary the allowed fairness constraint violation ϵ ∈ {0.001, 0.01, 0.2, 0.5, 1, 2, 5, 10, 15} for Adult dataset and ϵ ∈
+{0.001, 0.01, 0.2, 0.5, 1, 2, 5, 10, 15} for Adult with missing values. We vary ϵ ∈ {0.001, 2, 5, 10, 15, 20, 25, 30, 35, 40} for
+COMPAS to obtain a fairness-accuracy curve, and ϵ ∈ {0.001, 0.1, 0.5, 1, 2, 7, 8, 10, 15, 20, 25, 30} for COMPAS with 50%
+missing values in the minority group. We use ϵ ∈ {20, 50, 80, 95} for German Credit dataset and ϵ ∈ {5, 8, 10, 20, 23}
+when using Bayes Optimal classifier.
+LevEqOpp (Chzhen et al., 2019).
+We use the Python implementation of LevEqopp from the Github repo in Alghamdi
+et al. (2022). We follow the same hyperparameters setup as in the original method.
+FairProjection (Alghamdi et al., 2022).
+We use the implementation from the Github repo in Alghamdi et al. (2022)
+and set use protected = True. We use Random Forest with n estimators = 17 as the baseline classifier to predict S from
+(X, Y). We set the list of fairness violation tolerance to be {0.07, 0.075, 0.08, 0.085, 0.09, 0.095, 0.1, 0.5, 0.75, 1.0} for
+Adult dataset and {0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.1, 0.5, 1.0} for COMPAS dataset to obtain a fairness-accuracy
+curve. We set the list of fairness violation tolerance to be {0.005, 0.01, 0.02, 0.07, 0.1, 0.15} on the German Credit dataset
+experiment, and {0.0001, 0.001, 0.005, 0.01, 0.015, 0.02, 0.05} when using a Bayes optimal baseline classifier.
+
+Adult
+84.5
+84.2
+%)
+84.0
+Accuracy 
+83.7
+83.5
+83.2
+FairFront
+83.0
+Baseline
+FairProjection
+Reduction
+82.7
+0
+2.5
+5.00
+7.50
+10.0
+12.5
+Max equalized odds (%)COMPAS
+77.0
+76.8
+Accuracy
+76.6
+76.4
+FairFront
+Baseline
+FairProjection
+76.2
+Reduction
+0
+5.0
+10.0
+15.0
+20.0
+Max equalized odds (%)German Credit
+86.0
+85.9
+85.9
+(%)
+85.8
+Accuracy (
+85.8
+85.7
+85.7
+FairFront
+85.6
+Baseline
+FairProjection
+85.6
+Reduction
+0
+5.0
+10.0
+15.0
+20.0
+25.0
+Max equalized odds (%)

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+Two-link Staggered Quark Smearing in QUDA
+Steven Gottlieb,𝑎 Hwancheol Jeong𝑎,∗ and Alexei Strelchenko𝑏
+𝑎Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
+𝑏Scientific Computing Division, Fermi National Accelerator Laboratory, Batavia, Illinois, 60510, USA
+E-mail: sg@indiana.edu, sonchac@gmail.com, astrel@fnal.gov
+Gauge covariant smearing based on the 3D lattice Laplacian can be used to create extended op-
+erators that have better overlap with hadronic ground states. For staggered quarks, we make use
+of two-link parallel transport to preserve taste properties. We have implemented the procedure
+in QUDA. We present the performance of this code on the NVIDIA A100 GPUs in Indiana Uni-
+versity’s Big Red 200 supercomputer and on the AMD MI250X GPUs in Oak Ridge Leadership
+Computer Facility’s (OLCF’s) Crusher and discuss its scalability. We also study the performance
+improvement from using NVSHMEM on OLCF’s Summit. Reusing precomputed two-link prod-
+ucts for all sources and sinks, it reduces the total smearing time for a baryon correlator measurement
+by a factor of 100–120 as compared with the original MILC code and reduces the overall time by
+60–70%.
+The 39th International Symposium on Lattice Field Theory (Lattice2022),
+8-13 August, 2022
+Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.05518v1  [hep-lat]  13 Jan 2023
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+1.
+Introduction
+Lattice QCD calculations require operators that have a strong overlap with particular hadronic
+states. For example, lattice QCD calculations that study the low energy hadron spectrum benefit
+from extended operators that have better overlap with the ground state than local operators at a
+single lattice site. Decomposing a correlation function in terms of energy eigenstates, a two-point
+correlation function 𝐶(𝑡) = �
+x
+�
+O(𝑡, x) O†(0, 0)
+�
+can be expressed as
+𝐶(𝑡) =
+∑︁
+𝑛
+| ⟨0|O|𝑛⟩|2 𝑒−𝐸𝑛𝑡 ,
+(1)
+where 𝐸𝑛 is the energy of the 𝑛-th energy eigenstate. We can extract some low energy properties
+from it, including the mass from the ground state contribution. If the configuration is gauge-
+fixed, one can use extended operators without concern about parallel transport; however, by using
+gauge-covariant smearing of the source one can avoid having to fix the gauge.
+There are two popular kinds of gauge covariant smearing: Jacobi smearing and Gaussian
+smearing. Jacobi smearing is an iterative version of Wuppertal smearing which takes the three-
+dimensional scalar propagator as a smeared source [1–4]. The smeared source follows the ex-
+ponential distribution on a free gauge configuration. Alternatively, Gaussian smearing applies a
+hopping operator iteratively to the given source. The resulting smeared source follows the Gaussian
+distribution on a free gauge configuration [2, 3, 5].
+In this paper, we are interested in a variant of Gaussian smearing, which replaces the hopping
+operator with the three-dimensional lattice Laplacian operator for staggered quarks [6]. To preserve
+the taste symmetry of staggered quarks, the Laplacian should extend to the next-to-nearest-neighbor
+sites.
+We define the two-link products joining the next-to-nearest-neighbor sites and call this
+smearing method two-link staggered quark smearing.
+The MILC code with which we started was doing two-site parallel transport by applying
+single-site parallel transport twice in the same direction. This requires two communications and
+two matrix-vector multiplies per direction. We found that this smearing was taking an inordinate
+amount of time when done on the CPU. The situation is even worse when other parts of the
+calculation run on the GPU, because one allocates only one MPI rank per GPU, requiring multi-
+threading using OpenMP to use more than one CPU core per rank. Hence, we have implemented
+the procedure in QUDA [7–9]. The exascale computers in the U.S. all make use of GPUs, so more
+and more of our projects will make use of this timely addition to our codes.
+Section 2 briefly describes the two-link staggered quark smearing. Section 3 describes our GPU
+implementation and algorithmic improvement. In Secs. 4 and 5, we present benchmark results on
+some (recent or latest) NVIDIA and AMD GPUs, respectively. We also apply our QUDA smearing
+routine to a baryon correlator measurement in Sec. 6 to show how our code performs in a production
+job. We summarize our conclusions in Sec. 7.
+2.
+Two-link staggered quark smearing
+Let us consider a hopping operator 𝐻 defined by
+𝐻(𝑥, 𝑦) =
+3
+∑︁
+𝜇=1
+𝑈𝜇(𝑥) 𝛿𝑥+ ˆ𝜇, 𝑦 + 𝑈†
+𝜇(𝑥 − ˆ𝜇) 𝛿𝑥− ˆ𝜇, 𝑦 ,
+(2)
+2
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+x
+0
+5
+10
+15
+20
+y
+0
+5
+10
+15
+20
+|h(x, y)|z = 0, t = 0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+(a) Free gauge (𝑈 = 1)
+x
+0
+5
+10
+15
+20
+y
+0
+5
+10
+15
+20
+|h(x, y)|z = 0, t = 0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+(b) HISQ gauge
+Figure 1: Example distributions of a point source smeared by the two-link staggered quark smearing on a
+free gauge configuration (left) and a HISQ gauge configuration (right). Red data points represent norms of
+smeared quark field at (𝑥, 𝑦, 𝑧, 𝑡) = (𝑥, 𝑦, 0, 0). Contours are drawn by connecting these points.
+where 𝑥, 𝑦 are space-time coordinates and 𝑈𝜇(𝑥) is the gauge field. An iterative Gaussian smearing
+operation to a quark field 𝜓(𝑦) can be written as
+�𝜓 = 𝐶(1 + 𝛼𝐻)𝑛𝜓 ,
+(3)
+where 𝐶, 𝛼 ∈ R and 𝑛 ∈ Z are tuning parameters. For the unit gauge configuration 𝑈𝜇(𝑥) = 1, the
+smeared field �𝜓(𝑥) approaches the Gaussian distribution as the iteration count 𝑛 grows [2, 3, 5].
+The 3D lattice Laplacian ∇2 is defined by
+∇2(𝑥, 𝑦) =
+3
+∑︁
+𝜇=1
+�
+𝑈𝜇(𝑥) 𝛿𝑥+ ˆ𝜇, 𝑦 + 𝑈†
+𝜇(𝑥 − ˆ𝜇) 𝛿𝑥− ˆ𝜇, 𝑦
+�
+− 6 𝛿𝑥,𝑦
+= 𝐻(𝑥, 𝑦) − 6 𝛿𝑥,𝑦 .
+(4)
+Replacing 𝑈𝜇(𝑥) with the two-link product 𝑉𝜇(𝑥) ≡ 𝑈𝜇(𝑥)𝑈𝜇(𝑥 + ˆ𝜇) and adjusting the coordinate,
+we define (ignoring factors of 𝑎) the two-link 3D lattice Laplacian ∇2
+two:
+4∇2
+two(𝑥, 𝑦) ≡
+3
+∑︁
+𝜇=1
+�
+𝑉𝜇(𝑥) 𝛿𝑥+2 ˆ𝜇, 𝑦 + 𝑉†
+𝜇(𝑥 − 2 ˆ𝜇) 𝛿𝑥−2 ˆ𝜇, 𝑦
+�
+− 6 𝛿𝑥,𝑦
+(5)
+=
+3
+∑︁
+𝜇=1
+�
+𝑈𝜇(𝑥) 𝑈𝜇(𝑥 + ˆ𝜇) 𝛿𝑥+2 ˆ𝜇, 𝑦 + 𝑈†
+𝜇(𝑥 − ˆ𝜇) 𝑈†
+𝜇(𝑥 − 2 ˆ𝜇) 𝛿𝑥−2 ˆ𝜇, 𝑦
+�
+− 6 𝛿𝑥,𝑦 .
+(6)
+Note that ∇2
+two preserves taste properties for staggered quarks.
+If we rewrite Eq. (3) in terms of the Laplacian ∇2,
+�𝜓 = 𝐶(1 + 𝛼(∇2 + 6))𝑛𝜓 ≡ 𝐶′(1 + 𝛼′∇2)𝑛𝜓 ,
+(7)
+3
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+where 𝐶′ ≡ 𝐶(1 + 6𝛼)𝑛 and 𝛼′ ≡
+𝛼
+1 + 6𝛼. Now, we define the two-link staggered quark smearing
+as
+�𝜓 =
+�
+1 + 𝜎
+𝑛 ∇2
+two
+�𝑛
+𝜓 ,
+(8)
+where 𝜎 and 𝑛 are tuning parameters. This is a taste-preserving gauge covariant smearing for
+staggered quarks. Figure 1 shows the distributions of a point source after this smearing is applied
+on the free gauge configuration (Fig. 1a) and a HISQ gauge configuration (Fig. 1b). Although the
+latter is distorted by the existence of the gauge field, these results imply we can get a better overlap
+with hadronic ground states with a suitable choice of tuning parameters 𝜎 and 𝑛 [2, 4, 10]. Using a
+gauge-smeared link in the place of 𝑈𝜇 can relax the distortion as well as increase the overlap with
+the ground state [11]. There are also other approaches that enhance the overlap with some good
+properties [5, 12].
+3.
+GPU implementation
+Before this project began, the MILC code library [13] provided a CPU based routine for the
+two-link staggered quark smearing. In need of the GPU equivalent, we have implemented it in
+QUDA. We have also added an interface for this QUDA smearing in the MILC code, which runs
+all QUDA benchmarks in this paper.1
+We also improved upon the CPU algorithm in the MILC code. Instead of carrying out two
+consecutive parallel transports (as in Eq. (6)) at each smearing iteration, we precompute the two-link
+product 𝑉𝜇 in advance of smearing, and every iteration just loads it from the memory and uses
+Eq. (5). This two-link product can even be reused for different sources or sinks. In the GPU
+algorithm, we perform the smearing with significantly less memory traffic, fewer floating point
+operations, and less communication between MPI ranks.
+In Fig. 2, we compare the smearing time required by the MILC code to that of our QUDA
+code using all CPUs and GPUs, respectively, on Big Red 200 at Indiana University. Big Red 200 is
+similar to Perlmutter at NERSC in that it has a CPU partition in which each node contains two AMD
+64-core EPYC 7742 processors and a GPU partition in which each node contains an AMD processor
+and four NVIDIA A100 GPUs. Big Red 200 has a Slingshot 10 network, whereas Perlmutter was
+upgraded from Slingshot 10 to 11. We ran benchmarks for four different lattice volumes on either
+two CPU or two GPU nodes. For production jobs, we like to run with a local volume of at least 244
+per GPU and 324 or larger is preferred. Thus, these tests with a fixed number of nodes were not
+designed for maximum efficiency. In the case of QUDA runs, we also measure the second source
+smearing that reuses the precomputed two-link product. The first source smearing includes the
+computation of the two-link product and 𝑛 iterations of smearing, while the second source smearing
+does only 𝑛 iterations of smearing. Typical computations make use of multiple sources and sinks on
+the same gauge configuration, so the second source smearing result is quite pertinent. The results
+show that the first source smearing by QUDA on the GPU takes from 0.79 to 0.15 times as long as
+the MILC code smearing on the CPU, and the second source smearing by QUDA takes from 0.16
+1We are working on merging the QUDA code and the MILC code interface into the main (develop) branches of
+QUDA and the MILC code’s GitHub repositories, respectively[7, 13].
+4
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+243 × 64
+323 × 96
+403 × 96
+Lattice volume
+0
+2
+4
+6
+8
+Runtime (s)
+0.19
+0.80
+1.48
+0.15
+0.34
+0.49
+0.03
+0.09
+0.15
+MILC code
+(2-node×2×64-core AMD EPYC 7742)
+QUDA 1st source (2-node×4×A100)
+QUDA 2nd source (2-node×4×A100)
+643 × 96
+7.49
+1.15
+0.42
+Figure 2: Time taken by smearing a source quark field with 𝑛 = 50 two-link staggered quark smearing
+by the MILC code and QUDA. They are measured on two nodes of Big Red 200. The MILC code is run
+with one MPI rank per CPU core (total peak double precision FLOPS ≈ 14 TFLOPS), and QUDA is run
+with one MPI rank per GPU (total peak double precision FLOPS ≈ 80 TFLOPS). Lattices are divided by
+(𝑥, 𝑦, 𝑧, 𝑡) = (8, 8, 4, 1) for MILC code runs and (𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 2, 1) for QUDA runs. Note that splitting
+in 𝑡-dimension does not improve the performance of 3D Gaussian smearing applied to fixed time sources
+such as point sources or wall sources.
+to 0.06 times as long as the MILC code. The improvement increases for larger lattice volumes, as
+the smaller cases are too small for the GPU code to run efficiently.
+In this section, we have compared the maximum — from the point of view of using all available
+CPU/GPU resources — performances of the MILC code smearing and the QUDA smearing on
+two nodes of a computer with both CPU and GPU nodes. In practice, however, our primary goal
+is to implement the QUDA smearing in measurements running on GPU nodes. In this situation,
+smearing running on the CPU can become a severe bottleneck. Section 6 discusses an example.
+4.
+Performance on NVDIA GPU
+In this section, we report the performance of the two-link staggered quark smearing in QUDA
+(QUDA two-link smearing) on two NVIDIA GPU based systems. We smear three different color
+wall sources in succession. Only the first color source smearing computes the two-link product,
+while the others reuse it.
+In Fig. 3, we measure the performance and scalability of the QUDA two-link smearing on
+NVIDIA A100 GPUs in Big Red 200. The total runtime includes one run of two-link computation
+(unshaded) and three 𝑛 = 50 smearing iterations (shaded).
+We find that the single two-link
+computation takes around 10–40% of the time. This indicates the advantage of reusing the two-link
+product.
+Figure 3a presents the smearing time by varying the lattice volume. While the lattice volume
+increases geometrically by a factor of 16, the total smearing time increases by factors of 2.75, 5.45,
+and 7.53, respectively. This indicates the computation has not been saturated yet up to the largest
+lattice volume, so its performance (FLOPS) would be better for a bigger lattice. Still, for small
+lattices, it would be a fraction of time compared to typical lattice simulation scales. Figure 3b
+presents the smearing time by varying the number of nodes and GPUs. Note that the two-node run
+5
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+124
+244
+484
+964
+Lattice volume
+0
+2
+4
+Runtime (s)
+0.04
+0.11
+0.60
+4.52
+8 × A100
+(a) Volume scalability
+1(4)
+2(8)
+4(16)
+8(32)
+# of nodes (# of GPUs)
+0.0
+0.2
+0.4
+0.6
+0.8
+Runtime (s)
+0.43
+0.60
+0.50
+0.39
+484, A100
+(b) Strong scalability
+Figure 3: Time taken by smearing three color sources with 𝑛 = 50 QUDA two-link smearing on NVIDIA
+A100 GPUs in Big Red 200. The unshaded region represents the time taken by the two-link computation,
+and the shaded region represents the time taken by 3 × 𝑛 iterations of smearing. Lattices are divided by
+(𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 1, 1) for 4 GPUs, (2, 2, 2, 1) for 8 GPUs, (4, 2, 2, 1) for 16 GPUs, and (4, 4, 2, 1) for 32
+GPUs.
+243 × 64
+323 × 96
+403 × 96
+Lattice volume
+0.0
+0.5
+1.0
+1.5
+Runtime (s)
+0.19
+0.45
+0.71
+0.14
+0.36
+0.53
+8 × V100 (w/o NVSHMEM)
+8 × V100 (w/ NVSHMEM)
+643 × 96
+1.55
+1.36
+Figure 4: Time taken by smearing three color sources with 𝑛 = 50 QUDA two-link smearing on NVIDIA
+V100 GPUs in Summit with/without NVSHMEM support enabled in QUDA. The unshaded (shaded) region
+represents the time taken by the two-link computation (smearing iterations). Here, we use two Summit nodes,
+but only four V100 GPUs out of six per node, because six is not suitable for dividing the spatial dimension
+of some representative lattices. All lattices are divided by (𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 2, 1).
+is slower than the one-node run. Even the two-link computation taking three times longer. This
+implies the communication between off-nodes affects the performance significantly, and it is more
+prominent for the two-link computation. Excluding the one-node result, the performance improves
+as the number of nodes increases, but it scales very poorly. Figure 3a and 3b imply that this routine
+is a communication-intensive calculation.
+The observation above suggests that the QUDA two-link smearing may perform better with a
+faster communication environment. Summit at OLCF supports NVIDIA NVSHMEM technology.
+NVSHMEM improves strong scaling of GPU operations by enabling direct communication between
+GPUs with shared memory space [14]. QUDA supports NVSHMEM [15]. Figure 4 shows the
+performance improvement of the QUDA two-link smearing on Summit by enabling NVSHMEM.
+Here, the two-link computation takes more than half of the total runtime. We find that NVSHMEM
+reduces the two-link computation time by around 30–50%.
+6
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+244
+484
+964
+Lattice volume
+0
+2
+4
+Runtime (s)
+0.22
+0.41
+2.25
+8 × MI250X
+(16 GPUs)
+(a) Volume scalability
+1(8)
+2(16)
+4(32)
+8(64)
+# of nodes (# of GPUs)
+0.0
+0.2
+0.4
+0.6
+0.8
+Runtime (s)
+0.37
+0.41
+0.44
+0.38
+484, MI250X
+(b) Strong scalability
+Figure 5: Time taken by smearing three color sources with 𝑛 = 50 QUDA two-link smearing on AMD
+MI250X GPUs on Crusher. Note that each MI250X contains two GCDs, so the actual number of GPUs used
+is twice of the number of MI250Xs. The unshaded (shaded) region represents the time taken by the two-link
+computation (smearing iterations). Lattices are divided by (𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 2, 1) for 8 GPUs, (4, 2, 2, 1) for
+16 GPUs, (4, 4, 2, 1) for 32 GPUs, and (4, 4, 4, 1) for 64 GPUs.
+Big Red 200
+Crusher
+GPU
+4 × NVIDIA A100
+4 × AMD MI250X (=8 GPUs)
+(≈ 40 TFLOPS)
+(≈ 210 TFLOPS)
+GPU
+1555 GB/s
+3277 GB/s
+Mem. BW
+NIC
+2 HPE × Slingshot-10
+4 × HPE Slingshot-11
+(200 Gbps)
+(800 Gbps)
+Table 1: GPU and NIC specification of Big Red 200 and Crusher [17–20]. FLOPS numbers represent the
+double precision peak performance.
+5.
+Performance on AMD GPU
+QUDA also supports HIP on AMD ROCm platform [16], allowing it to run on AMD GPUs.
+In this section, we report the performance of the QUDA two-link smearing on Crusher at OLCF, an
+AMD GPU-based system containing hardware identical to that on Frontier. As in Sec. 4, we smear
+three different color wall sources in order, where the two-link product is computed only at the first
+smearing and reused for others.
+In Fig. 5, we measure the performance and scalability of the QUDA two-link smearing on
+the AMD MI250X GPUs in Crusher in the same manner as in Fig. 3.2 Figure 5a and 5b plot the
+volume and strong scalabilities, respectively. The y-axis ranges are the same as in Fig. 3a and 3b for
+easy comparison. The smearing iteration performs approximately twice faster here with MI250X
+compared to that with A100. This result agrees with our expectation that the bottleneck of this
+2These plots are updated from the ones we presented in the poster, where we observed an abnormal slowdown in
+the two-link computation on AMD GPUs. It turned out the culprit was a HIP API which was not directly related to the
+two-link computation. We found a way to avoid this problem and reran the benchmark. We are also working on resolving
+the issue.
+7
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+243 × 64
+323 × 96
+Lattice volume
+0
+500
+1000
+Runtime (s)
+301
+1126
+105
+420
+197
+717
+2
+6
+w/o QUDA smearing
+w/ QUDA smearing
+Figure 6: Total time taken by a baryon correlator measurement employing 72 source/sinks which are smeared
+by the 𝑛 = 30 QUDA two-link smearing. The measurement is carried out on two nodes of Big Red 200’s
+GPU partition using four CPU cores and GPUs per node. With or without the QUDA two-link smearing
+enabled, all other QUDA-supported calculations are performed on the GPU. The shaded region represents
+the total smearing time.
+calculation is the GPU memory bandwidth because its compute-to-communication ratio is similar
+to a typical dslash routine (see Eq. (5)), and a MI250X has twice faster memory bandwidth than
+A100 (see Table 1). On the other hand, the two-link computation performs similarly or slower on
+the MI250X compared to the A100. The only exception is the 964 lattice result, where we observe
+the expected twice faster performance. Regarding the scalability, we observe a poor scaling as we
+observed from the A100 GPU.
+6.
+Application: Baryon correlator measurement
+Our baryon correlator calculations include many sources and sinks. Figure 6 shows the total
+runtime taken by an example baryon correlator measurement. Most parts of the calculation were
+already implemented in QUDA. With the QUDA smearing disabled, the smearing is carried out
+on the CPU by the MILC code smearing routine while other major calculations are carried out on
+the GPU by QUDA. Note that here we use only the same number of CPUs as of GPUs. The result
+shows that the source/sink smearing takes up around 60–70% of the total measurement time when
+it is run on the CPU. However, enabling the QUDA smearing reduces the smearing time by a factor
+of 100–120, requiring only around 1–2% of the total measurement time. Thus, the total time for
+the job is reduced to about 30–40% of the original time.
+7.
+Conclusion
+We have implemented two-link staggered quark smearing in QUDA. When run on eight
+NVIDIA A100 GPUs, it performs up to six times faster than the MILC code run on total 256 cores
+of four AMD EPYC 7742 CPUs. Reusing the precomputed two-link product for another source on
+the same gauge field increases the difference up to 18 times.
+The scaling behavior of this routine on both NVIDIA A100 and AMD MI250X is rather poor,
+especially for the two-link computation part. NVSHMEM can improve the performance of the
+8
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+two-link computation by 30–50%. For a baryon correlator measurement, reusing the two-link
+product for all 72 sources/sinks reduces the total smearing time from 60–70% to 1–2% of the total
+measurement time. Thus, even without further optimization, this code can be useful for GPU jobs
+that require many smearings for sources or sinks.
+Acknowledgments
+This research was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative
+effort of the U.S. Department of Energy Office of Science and the National Nuclear Security
+Administration. We gratefully acknowledge support by the U.S. Department of Energy, Office of
+Science under award DE-SC0010120. This research was supported in part by Lilly Endowment, Inc.,
+through its support for the Indiana University Pervasive Technology Institute. This research used
+resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory,
+which is supported by the Office of Science of the U.S. Department of Energy under Contract No.
+DE-AC05-00OR22725. We thank the QUDA [8, 9] developers whose names can be found at their
+website[7] .
+References
+[1] S. Gusken, U. Low, K.H. Mutter, R. Sommer, A. Patel and K. Schilling, Phys. Lett. B 227
+(1989) 266.
+[2] S. Gusken, Nucl. Phys. B Proc. Suppl. 17 (1990) 361.
+[3] C. Alexandrou, F. Jegerlehner, S. Gusken, K. Schilling and R. Sommer, Phys. Lett. B 256
+(1991) 60.
+[4] UKQCD collaboration, Phys. Rev. D 47 (1993) 5128 [hep-lat/9303009].
+[5] G.M. von Hippel, B. Jäger, T.D. Rae and H. Wittig, JHEP 09 (2013) 014 [1306.1440].
+[6] Hadron Spectrum collaboration, Phys. Rev. D 80 (2009) 054506 [0905.2160].
+[7] https://github.com/lattice/quda.
+[8] M.A. Clark, R. Babich, K. Barros, R.C. Brower and C. Rebbi, Comput. Phys. Commun. 181
+(2010) 1517 [0911.3191].
+[9] R. Babich, M.A. Clark, B. Joo, G. Shi, R.C. Brower and S. Gottlieb, 9, 2011, DOI
+[1109.2935].
+[10] T.A. DeGrand and R.D. Loft, Comput. Phys. Commun. 65 (1991) 84.
+[11] S.N. Syritsyn et al., Phys. Rev. D 81 (2010) 034507 [0907.4194].
+[12] G.S. Bali, B. Lang, B.U. Musch and A. Schäfer, Phys. Rev. D 93 (2016) 094515
+[1602.05525].
+9
+
+Two-link Staggered Quark Smearing in QUDA
+Hwancheol Jeong
+[13] https://github.com/milc-qcd/milc_qcd/tree/develop.
+[14] https://developer.nvidia.com/nvshmem.
+[15] https://github.com/lattice/quda/wiki/Multi-GPU-with-NVSHMEM.
+[16] https://github.com/lattice/quda/wiki/Building-QUDA-With-HIP.
+[17] https://kb.iu.edu/d/brcc.
+[18] https://www.nvidia.com/content/dam/en-zz/Solutions/Data-Center/a100/
+pdf/nvidia-a100-datasheet-us-nvidia-1758950-r4-web.pdf.
+[19] https://docs.olcf.ornl.gov/systems/crusher_quick_start_guide.html.
+[20] https://www.amd.com/en/products/server-accelerators/instinct-mi250x.
+10
+

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+page_content='Two-link Staggered Quark Smearing in QUDA  Steven Gottlieb,𝑎 Hwancheol Jeong𝑎,∗ and Alexei Strelchenko𝑏  𝑎Department of Physics, Indiana University, Bloomington, Indiana 47405, USA  𝑏Scientific Computing Division, Fermi National Accelerator Laboratory, Batavia, Illinois, 60510, USA  E-mail: sg@indiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='edu, sonchac@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='com, astrel@fnal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='gov  Gauge covariant smearing based on the 3D lattice Laplacian can be used to create extended op-  erators that have better overlap with hadronic ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' For staggered quarks, we make use  of two-link parallel transport to preserve taste properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We have implemented the procedure  in QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We present the performance of this code on the NVIDIA A100 GPUs in Indiana Uni-  versity’s Big Red 200 supercomputer and on the AMD MI250X GPUs in Oak Ridge Leadership  Computer Facility’s (OLCF’s) Crusher and discuss its scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We also study the performance  improvement from using NVSHMEM on OLCF’s Summit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Reusing precomputed two-link prod-  ucts for all sources and sinks, it reduces the total smearing time for a baryon correlator measurement  by a factor of 100–120 as compared with the original MILC code and reduces the overall time by  60–70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory (Lattice2022),  8-13 August, 2022  Bonn, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='05518v1  [hep-lat]  13 Jan 2023  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Introduction  Lattice QCD calculations require operators that have a strong overlap with particular hadronic  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' For example, lattice QCD calculations that study the low energy hadron spectrum benefit  from extended operators that have better overlap with the ground state than local operators at a  single lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Decomposing a correlation function in terms of energy eigenstates, a two-point  correlation function 𝐶(𝑡) = �  x  �  O(𝑡, x) O†(0, 0)  �  can be expressed as  𝐶(𝑡) =  ∑︁  𝑛  | ⟨0|O|𝑛⟩|2 𝑒−𝐸𝑛𝑡 ,  (1)  where 𝐸𝑛 is the energy of the 𝑛-th energy eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We can extract some low energy properties  from it, including the mass from the ground state contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' If the configuration is gauge-  fixed, one can use extended operators without concern about parallel transport;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' however, by using  gauge-covariant smearing of the source one can avoid having to fix the gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  There are two popular kinds of gauge covariant smearing: Jacobi smearing and Gaussian  smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Jacobi smearing is an iterative version of Wuppertal smearing which takes the three-  dimensional scalar propagator as a smeared source [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The smeared source follows the ex-  ponential distribution on a free gauge configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Alternatively, Gaussian smearing applies a  hopping operator iteratively to the given source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The resulting smeared source follows the Gaussian  distribution on a free gauge configuration [2, 3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  In this paper, we are interested in a variant of Gaussian smearing, which replaces the hopping  operator with the three-dimensional lattice Laplacian operator for staggered quarks [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' To preserve  the taste symmetry of staggered quarks, the Laplacian should extend to the next-to-nearest-neighbor  sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  We define the two-link products joining the next-to-nearest-neighbor sites and call this  smearing method two-link staggered quark smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  The MILC code with which we started was doing two-site parallel transport by applying  single-site parallel transport twice in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This requires two communications and  two matrix-vector multiplies per direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We found that this smearing was taking an inordinate  amount of time when done on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The situation is even worse when other parts of the  calculation run on the GPU, because one allocates only one MPI rank per GPU, requiring multi-  threading using OpenMP to use more than one CPU core per rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Hence, we have implemented  the procedure in QUDA [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The exascale computers in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' all make use of GPUs, so more  and more of our projects will make use of this timely addition to our codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Section 2 briefly describes the two-link staggered quark smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Section 3 describes our GPU  implementation and algorithmic improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 4 and 5, we present benchmark results on  some (recent or latest) NVIDIA and AMD GPUs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We also apply our QUDA smearing  routine to a baryon correlator measurement in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 6 to show how our code performs in a production  job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We summarize our conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Two-link staggered quark smearing  Let us consider a hopping operator 𝐻 defined by  𝐻(𝑥, 𝑦) =  3  ∑︁  𝜇=1  𝑈𝜇(𝑥) 𝛿𝑥+ ˆ𝜇, 𝑦 + 𝑈†  𝜇(𝑥 − ˆ𝜇) 𝛿𝑥− ˆ𝜇, 𝑦 ,  (2)  2  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  x  0  5  10  15  20  y  0  5  10  15  20  |h(x, y)|z = 0, t = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='40  (a) Free gauge (𝑈 = 1)  x  0  5  10  15  20  y  0  5  10  15  20  |h(x, y)|z = 0, t = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='40  (b) HISQ gauge  Figure 1: Example distributions of a point source smeared by the two-link staggered quark smearing on a  free gauge configuration (left) and a HISQ gauge configuration (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Red data points represent norms of  smeared quark field at (𝑥, 𝑦, 𝑧, 𝑡) = (𝑥, 𝑦, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Contours are drawn by connecting these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  where 𝑥, 𝑦 are space-time coordinates and 𝑈𝜇(𝑥) is the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' An iterative Gaussian smearing  operation to a quark field 𝜓(𝑦) can be written as  �𝜓 = 𝐶(1 + 𝛼𝐻)𝑛𝜓 ,  (3)  where 𝐶, 𝛼 ∈ R and 𝑛 ∈ Z are tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' For the unit gauge configuration 𝑈𝜇(𝑥) = 1, the  smeared field �𝜓(𝑥) approaches the Gaussian distribution as the iteration count 𝑛 grows [2, 3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  The 3D lattice Laplacian ∇2 is defined by  ∇2(𝑥, 𝑦) =  3  ∑︁  𝜇=1  �  𝑈𝜇(𝑥) 𝛿𝑥+ ˆ𝜇, 𝑦 + 𝑈†  𝜇(𝑥 − ˆ𝜇) 𝛿𝑥− ˆ𝜇, 𝑦  �  − 6 𝛿𝑥,𝑦  = 𝐻(𝑥, 𝑦) − 6 𝛿𝑥,𝑦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  (4)  Replacing 𝑈𝜇(𝑥) with the two-link product 𝑉𝜇(𝑥) ≡ 𝑈𝜇(𝑥)𝑈𝜇(𝑥 + ˆ𝜇) and adjusting the coordinate,  we define (ignoring factors of 𝑎) the two-link 3D lattice Laplacian ∇2  two:  4∇2  two(𝑥, 𝑦) ≡  3  ∑︁  𝜇=1  �  𝑉𝜇(𝑥) 𝛿𝑥+2 ˆ𝜇, 𝑦 + 𝑉†  𝜇(𝑥 − 2 ˆ𝜇) 𝛿𝑥−2 ˆ𝜇, 𝑦  �  − 6 𝛿𝑥,𝑦  (5)  =  3  ∑︁  𝜇=1  �  𝑈𝜇(𝑥) 𝑈𝜇(𝑥 + ˆ𝜇) 𝛿𝑥+2 ˆ𝜇, 𝑦 + 𝑈†  𝜇(𝑥 − ˆ𝜇) 𝑈†  𝜇(𝑥 − 2 ˆ𝜇) 𝛿𝑥−2 ˆ𝜇, 𝑦  �  − 6 𝛿𝑥,𝑦 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  (6)  Note that ∇2  two preserves taste properties for staggered quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  If we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' (3) in terms of the Laplacian ∇2,  �𝜓 = 𝐶(1 + 𝛼(∇2 + 6))𝑛𝜓 ≡ 𝐶′(1 + 𝛼′∇2)𝑛𝜓 ,  (7)  3  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  where 𝐶′ ≡ 𝐶(1 + 6𝛼)𝑛 and 𝛼′ ≡  𝛼  1 + 6𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Now, we define the two-link staggered quark smearing  as  �𝜓 =  �  1 + 𝜎  𝑛 ∇2  two  �𝑛  𝜓 ,  (8)  where 𝜎 and 𝑛 are tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This is a taste-preserving gauge covariant smearing for  staggered quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Figure 1 shows the distributions of a point source after this smearing is applied  on the free gauge configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 1a) and a HISQ gauge configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Although the  latter is distorted by the existence of the gauge field, these results imply we can get a better overlap  with hadronic ground states with a suitable choice of tuning parameters 𝜎 and 𝑛 [2, 4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Using a  gauge-smeared link in the place of 𝑈𝜇 can relax the distortion as well as increase the overlap with  the ground state [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' There are also other approaches that enhance the overlap with some good  properties [5, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  GPU implementation  Before this project began, the MILC code library [13] provided a CPU based routine for the  two-link staggered quark smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' In need of the GPU equivalent, we have implemented it in  QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We have also added an interface for this QUDA smearing in the MILC code, which runs  all QUDA benchmarks in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='1  We also improved upon the CPU algorithm in the MILC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Instead of carrying out two  consecutive parallel transports (as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' (6)) at each smearing iteration, we precompute the two-link  product 𝑉𝜇 in advance of smearing, and every iteration just loads it from the memory and uses  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This two-link product can even be reused for different sources or sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' In the GPU  algorithm, we perform the smearing with significantly less memory traffic, fewer floating point  operations, and less communication between MPI ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 2, we compare the smearing time required by the MILC code to that of our QUDA  code using all CPUs and GPUs, respectively, on Big Red 200 at Indiana University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Big Red 200 is  similar to Perlmutter at NERSC in that it has a CPU partition in which each node contains two AMD  64-core EPYC 7742 processors and a GPU partition in which each node contains an AMD processor  and four NVIDIA A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Big Red 200 has a Slingshot 10 network, whereas Perlmutter was  upgraded from Slingshot 10 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We ran benchmarks for four different lattice volumes on either  two CPU or two GPU nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' For production jobs, we like to run with a local volume of at least 244  per GPU and 324 or larger is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Thus, these tests with a fixed number of nodes were not  designed for maximum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' In the case of QUDA runs, we also measure the second source  smearing that reuses the precomputed two-link product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The first source smearing includes the  computation of the two-link product and 𝑛 iterations of smearing, while the second source smearing  does only 𝑛 iterations of smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Typical computations make use of multiple sources and sinks on  the same gauge configuration, so the second source smearing result is quite pertinent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The results  show that the first source smearing by QUDA on the GPU takes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='79 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='15 times as long as  the MILC code smearing on the CPU, and the second source smearing by QUDA takes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='16  1We are working on merging the QUDA code and the MILC code interface into the main (develop) branches of  QUDA and the MILC code’s GitHub repositories, respectively[7, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  4  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  243 × 64  323 × 96  403 × 96  Lattice volume  0  2  4  6  8  Runtime (s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='15  MILC code  (2-node×2×64-core AMD EPYC 7742)  QUDA 1st source (2-node×4×A100)  QUDA 2nd source (2-node×4×A100)  643 × 96  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='42  Figure 2: Time taken by smearing a source quark field with 𝑛 = 50 two-link staggered quark smearing  by the MILC code and QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' They are measured on two nodes of Big Red 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The MILC code is run  with one MPI rank per CPU core (total peak double precision FLOPS ≈ 14 TFLOPS), and QUDA is run  with one MPI rank per GPU (total peak double precision FLOPS ≈ 80 TFLOPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Lattices are divided by  (𝑥, 𝑦, 𝑧, 𝑡) = (8, 8, 4, 1) for MILC code runs and (𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 2, 1) for QUDA runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Note that splitting  in 𝑡-dimension does not improve the performance of 3D Gaussian smearing applied to fixed time sources  such as point sources or wall sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='06 times as long as the MILC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The improvement increases for larger lattice volumes, as  the smaller cases are too small for the GPU code to run efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  In this section, we have compared the maximum — from the point of view of using all available  CPU/GPU resources — performances of the MILC code smearing and the QUDA smearing on  two nodes of a computer with both CPU and GPU nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' In practice, however, our primary goal  is to implement the QUDA smearing in measurements running on GPU nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' In this situation,  smearing running on the CPU can become a severe bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Section 6 discusses an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Performance on NVDIA GPU  In this section, we report the performance of the two-link staggered quark smearing in QUDA  (QUDA two-link smearing) on two NVIDIA GPU based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We smear three different color  wall sources in succession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Only the first color source smearing computes the two-link product,  while the others reuse it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 3, we measure the performance and scalability of the QUDA two-link smearing on  NVIDIA A100 GPUs in Big Red 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The total runtime includes one run of two-link computation  (unshaded) and three 𝑛 = 50 smearing iterations (shaded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  We find that the single two-link  computation takes around 10–40% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This indicates the advantage of reusing the two-link  product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Figure 3a presents the smearing time by varying the lattice volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' While the lattice volume  increases geometrically by a factor of 16, the total smearing time increases by factors of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='75, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='45,  and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='53, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This indicates the computation has not been saturated yet up to the largest  lattice volume, so its performance (FLOPS) would be better for a bigger lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Still, for small  lattices, it would be a fraction of time compared to typical lattice simulation scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Figure 3b  presents the smearing time by varying the number of nodes and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Note that the two-node run  5  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  124  244  484  964  Lattice volume  0  2  4  Runtime (s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='60  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='52  8 × A100  (a) Volume scalability  1(4)  2(8)  4(16)  8(32)  # of nodes (# of GPUs)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='8  Runtime (s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='39  484, A100  (b) Strong scalability  Figure 3: Time taken by smearing three color sources with 𝑛 = 50 QUDA two-link smearing on NVIDIA  A100 GPUs in Big Red 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The unshaded region represents the time taken by the two-link computation,  and the shaded region represents the time taken by 3 × 𝑛 iterations of smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Lattices are divided by  (𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 1, 1) for 4 GPUs, (2, 2, 2, 1) for 8 GPUs, (4, 2, 2, 1) for 16 GPUs, and (4, 4, 2, 1) for 32  GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  243 × 64  323 × 96  403 × 96  Lattice volume  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='5  Runtime (s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='53  8 × V100 (w/o NVSHMEM)  8 × V100 (w/ NVSHMEM)  643 × 96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='36  Figure 4: Time taken by smearing three color sources with 𝑛 = 50 QUDA two-link smearing on NVIDIA  V100 GPUs in Summit with/without NVSHMEM support enabled in QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The unshaded (shaded) region  represents the time taken by the two-link computation (smearing iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Here, we use two Summit nodes,  but only four V100 GPUs out of six per node, because six is not suitable for dividing the spatial dimension  of some representative lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' All lattices are divided by (𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  is slower than the one-node run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Even the two-link computation taking three times longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This  implies the communication between off-nodes affects the performance significantly, and it is more  prominent for the two-link computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Excluding the one-node result, the performance improves  as the number of nodes increases, but it scales very poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Figure 3a and 3b imply that this routine  is a communication-intensive calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  The observation above suggests that the QUDA two-link smearing may perform better with a  faster communication environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Summit at OLCF supports NVIDIA NVSHMEM technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  NVSHMEM improves strong scaling of GPU operations by enabling direct communication between  GPUs with shared memory space [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' QUDA supports NVSHMEM [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Figure 4 shows the  performance improvement of the QUDA two-link smearing on Summit by enabling NVSHMEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Here, the two-link computation takes more than half of the total runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We find that NVSHMEM  reduces the two-link computation time by around 30–50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  6  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  244  484  964  Lattice volume  0  2  4  Runtime (s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='41  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='25  8 × MI250X  (16 GPUs)  (a) Volume scalability  1(8)  2(16)  4(32)  8(64)  # of nodes (# of GPUs)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='8  Runtime (s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='38  484, MI250X  (b) Strong scalability  Figure 5: Time taken by smearing three color sources with 𝑛 = 50 QUDA two-link smearing on AMD  MI250X GPUs on Crusher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Note that each MI250X contains two GCDs, so the actual number of GPUs used  is twice of the number of MI250Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The unshaded (shaded) region represents the time taken by the two-link  computation (smearing iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Lattices are divided by (𝑥, 𝑦, 𝑧, 𝑡) = (2, 2, 2, 1) for 8 GPUs, (4, 2, 2, 1) for  16 GPUs, (4, 4, 2, 1) for 32 GPUs, and (4, 4, 4, 1) for 64 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Big Red 200  Crusher  GPU  4 × NVIDIA A100  4 × AMD MI250X (=8 GPUs)  (≈ 40 TFLOPS)  (≈ 210 TFLOPS)  GPU  1555 GB/s  3277 GB/s  Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' BW  NIC  2 HPE × Slingshot-10  4 × HPE Slingshot-11  (200 Gbps)  (800 Gbps)  Table 1: GPU and NIC specification of Big Red 200 and Crusher [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' FLOPS numbers represent the  double precision peak performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Performance on AMD GPU  QUDA also supports HIP on AMD ROCm platform [16], allowing it to run on AMD GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  In this section, we report the performance of the QUDA two-link smearing on Crusher at OLCF, an  AMD GPU-based system containing hardware identical to that on Frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' As in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 4, we smear  three different color wall sources in order, where the two-link product is computed only at the first  smearing and reused for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 5, we measure the performance and scalability of the QUDA two-link smearing on  the AMD MI250X GPUs in Crusher in the same manner as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='2 Figure 5a and 5b plot the  volume and strong scalabilities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The y-axis ranges are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 3a and 3b for  easy comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The smearing iteration performs approximately twice faster here with MI250X  compared to that with A100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This result agrees with our expectation that the bottleneck of this  2These plots are updated from the ones we presented in the poster, where we observed an abnormal slowdown in  the two-link computation on AMD GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' It turned out the culprit was a HIP API which was not directly related to the  two-link computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We found a way to avoid this problem and reran the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We are also working on resolving  the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  7  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  243 × 64  323 × 96  Lattice volume  0  500  1000  Runtime (s)  301  1126  105  420  197  717  2  6  w/o QUDA smearing  w/ QUDA smearing  Figure 6: Total time taken by a baryon correlator measurement employing 72 source/sinks which are smeared  by the 𝑛 = 30 QUDA two-link smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The measurement is carried out on two nodes of Big Red 200’s  GPU partition using four CPU cores and GPUs per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' With or without the QUDA two-link smearing  enabled, all other QUDA-supported calculations are performed on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The shaded region represents  the total smearing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  calculation is the GPU memory bandwidth because its compute-to-communication ratio is similar  to a typical dslash routine (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' (5)), and a MI250X has twice faster memory bandwidth than  A100 (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' On the other hand, the two-link computation performs similarly or slower on  the MI250X compared to the A100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The only exception is the 964 lattice result, where we observe  the expected twice faster performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Regarding the scalability, we observe a poor scaling as we  observed from the A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Application: Baryon correlator measurement  Our baryon correlator calculations include many sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Figure 6 shows the total  runtime taken by an example baryon correlator measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Most parts of the calculation were  already implemented in QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' With the QUDA smearing disabled, the smearing is carried out  on the CPU by the MILC code smearing routine while other major calculations are carried out on  the GPU by QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Note that here we use only the same number of CPUs as of GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' The result  shows that the source/sink smearing takes up around 60–70% of the total measurement time when  it is run on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' However, enabling the QUDA smearing reduces the smearing time by a factor  of 100–120, requiring only around 1–2% of the total measurement time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Thus, the total time for  the job is reduced to about 30–40% of the original time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Conclusion  We have implemented two-link staggered quark smearing in QUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' When run on eight  NVIDIA A100 GPUs, it performs up to six times faster than the MILC code run on total 256 cores  of four AMD EPYC 7742 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Reusing the precomputed two-link product for another source on  the same gauge field increases the difference up to 18 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  The scaling behavior of this routine on both NVIDIA A100 and AMD MI250X is rather poor,  especially for the two-link computation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' NVSHMEM can improve the performance of the  8  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  two-link computation by 30–50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' For a baryon correlator measurement, reusing the two-link  product for all 72 sources/sinks reduces the total smearing time from 60–70% to 1–2% of the total  measurement time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Thus, even without further optimization, this code can be useful for GPU jobs  that require many smearings for sources or sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  Acknowledgments  This research was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative  effort of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Department of Energy Office of Science and the National Nuclear Security  Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We gratefully acknowledge support by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Department of Energy, Office of  Science under award DE-SC0010120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This research was supported in part by Lilly Endowment, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=',  through its support for the Indiana University Pervasive Technology Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' This research used  resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory,  which is supported by the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  DE-AC05-00OR22725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' We thank the QUDA [8, 9] developers whose names can be found at their  website[7] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Gusken, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Low, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Mutter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Sommer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Patel and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Schilling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' B 227  (1989) 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Gusken, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 17 (1990) 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Alexandrou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Jegerlehner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Gusken, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Schilling and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Sommer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' B 256  (1991) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [4] UKQCD collaboration, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' D 47 (1993) 5128 [hep-lat/9303009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' von Hippel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Jäger, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Rae and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Wittig, JHEP 09 (2013) 014 [1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='1440].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [6] Hadron Spectrum collaboration, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' D 80 (2009) 054506 [0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='2160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [7] https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='com/lattice/quda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Clark, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Babich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Barros, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Brower and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Rebbi, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 181  (2010) 1517 [0911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='3191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Babich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Clark, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Joo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Shi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Brower and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Gottlieb, 9, 2011, DOI  [1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='2935].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' DeGrand and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Loft, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' 65 (1991) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Syritsyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' D 81 (2010) 034507 [0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='4194].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Bali, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Lang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Musch and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Schäfer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content=' D 93 (2016) 094515  [1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='05525].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  9  Two-link Staggered Quark Smearing in QUDA  Hwancheol Jeong  [13] https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='com/milc-qcd/milc_qcd/tree/develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
+page_content='  [14] https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
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+page_content='  [15] https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE5T4oBgHgl3EQfRA7S/content/2301.05518v1.pdf'}
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+Forecasting Natural Gas Prices with Spatio-Temporal
+Copula-based Time Series Models
+Sven Papperta,∗, Antonia Arsovaa,b
+aChair of Econometrics, Department of Statistics, TU Dortmund University, Germany
+bRWI – Leibniz Institute for Economic Research, Germany
+Abstract
+Commodity price time series possess interesting features, such as heavy-tailedness,
+skewness, heteroskedasticity, and non-linear dependence structures. These features
+pose challenges for modeling and forecasting. In this work, we explore how spatio-
+temporal copula-based time series models can be effectively employed for these purposes.
+We focus on price series for fossil fuels and carbon emissions. Further, we illustrate
+how the t-copula may be used in conditional heteroskedasticity modeling.
+The
+possible emergence of non-elliptical probabilistic forecasts in this context is examined
+and visualized. The problem of finding an appropriate point forecast given a non-
+elliptical probabilistic forecast is discussed. We propose a solution where the forecast
+is augmented with an artificial neural network (ANN). The ANN predicts the best (in
+MSE sense) quantile to use as point forecast. In a forecasting study, we find that the
+copula-based models are competitive.
+Keywords:
+Commoditiy Prices, Copula-based time series, Conditional Volatility,
+Forecasting, Vine Copula
+∗Corresponding author
+Email addresses: pappert@statistik.tu-dortmund.de (Sven Pappert),
+arsova@statistik.tu-dortmund.de (Antonia Arsova)
+Preprint submitted to Contributions to Statistics
+January 10, 2023
+arXiv:2301.03328v1  [stat.AP]  9 Jan 2023
+
+1. Introduction
+Modeling and forecasting commodity prices is important for trading, political
+decision making and economic adjustments. Especially in recent times, forecasting
+natural gas prices has gained importance following the russian invasion of Ukraine. In
+this work we focus on modeling and forecsting short-term natural gas future prices
+jointly with related commodity prices. We model the time series jointly to exploit the
+additional information carried by their mutual dependence. To this aim we employ
+spatio-temporal copula based time series models.
+Copulas are popular choices to model the cross-sectional dependence in time series
+with the copula-GARCH approach in financial markets [16, 18] as well as in energy
+markets [2, 7]. In the copula-GARCH approach the temporal dependence of each
+time series is modeled by typical time series models, such as ARMA-GARCH. The
+cross-sectional dependence structure of the time series can be captured by finding the
+copula of the standardized residuals of the univariate time series model. However, such
+models are only able to allow for flexible dependence structures in the cross-sectional
+dimension. The mean process is modeled linearly.
+On the other hand, temporal copula modeling (or ’copula-based time series modeling’)
+as well as spatio-temporal copula time series modeling offers an alternative to classical
+linear time series approaches. The models are able to flexibly model cross-sectional as
+well as temporal dependencies. There is emerging literature on the topic. Chen & Fan
+[8] investigate the estimation of copula-based semiparametric time series models. The
+authors provide conditions for β-mixing and prove consistency as well as asymptotic
+normality using the Delta method. Beare [4] further investigates mixing conditions.
+Smith et al. [23] decompose serial dependence of intraday electricity load using pair
+copula constructions. Simard and Rémillard [21] investigate the forecasting perfor-
+mance of the spatio-temporal t-copula dependent on the strength and structure of the
+dependence as well as the marginal distributions. Beare & Seo [5] as well as Nagler et
+al. [19] examine spatio-temporal vine copula models.
+Examples where flexible dependence modeling can be important are the following. The
+cross-sectional dependence between international stock markets can be asymmetric
+with dominant lower tail dependence, indicating the phenomenon of contagion in
+financial markets. This was investigated by Hu [16]. It was found that the asymmetric
+dependence dominates. With regard to energy markets it was found by Aloui et al.
+[2] that crude oil and gas markets rather comove during bullish periods. Thereby also
+2
+
+displaying an asymmetric cross-sectional dependence structure. A concise example
+for a possible emergence of non-linear dependence structures in the temporal domain
+and hence requiring sophisticated dependence modeling is given in the introduction of
+the work by Beare [4]. The continous growth of financial time series contrasted with
+their sudden and quick decrease represent an asymmetric temporal relation. Thus
+cross-sectional as well as temporal dependence modeling can be important in many
+fields.
+In this paper we explore the possibilities and performances of spatio-temporal copula
+models for modeling energy market time series. The basic idea underlying spatio-
+temporal time series modeling with copulas is a decomposition of the joint distribution.
+Using Sklars theorem, [22] the joint distribution of consecutive observations is decom-
+posed into dependence and marginal structure, FXt,Xt−1(a, b) = C
+�
+FXt(a), FXt−1(b)
+�
+.
+Various copula specifications can be employed. In this work, the t-copula [11] and the
+gaussian copula are considered as basic copula models for spatio-temporal time series
+modeling. Vine copula models [1, 9] are also considered. Spatio-temporal forecasting
+with the t-copula was examined in [21]. The spatio-temporal vine copula modeling of
+multivariate time series is, among others, explored in [5, 19, 23].
+Using the notion of conditional copulas, the models can be used for forecasting. The
+resulting probabilistic forecasts can be non-elliptical. It is not obvious what constitutes
+a sensible point forecast in this case. The expectation value is a sensible point forecast
+for elliptical or almost elliptical probabilistic forecasts. For non-elliptical forecasts,
+e.g. a bimodular probabilistic forecast, the expecation value predicts points that are
+unlikely. One possible solution to this problem would be to take the mode of the
+probabilistic forecast as the point forecast. Another possibility is to augment the
+forecasting procedure by an artificial neural network (ANN)1 The ANN predicts which
+quantile of the probabilistic forecast is optimal (in MSE sense) as point forecast. The
+inputs of the ANN are past values of the times series and the last optimal quantiles.
+One advantage of the ANN-augmented forecast is that the ANN can be estimated and
+used for prediction completely independent from the probabilistic time series model
+estimation and forecast. It is well known that ANNs are very powerful with regards
+to point forecasting. In this approach the ANN point forecasts are also equipped
+with an underlying probabilistic distribution, enabling the calculation of confidence
+1We refer to [14] for a concise introduction.
+3
+
+intervals and other distributional properties. The models performances are examined
+in a forecasting study. A model closely related to the model from [6], which was shown
+to outperform other popular models, is considered as benchmark. We find that the
+spatio-temporal copula time series modeling with ANN-augmented point forecasts are
+competitive for natural gas and related commoditiy prices forecasting.
+The main contributions of this paper are two-fold. The first contribution is the appli-
+cation oriented exploration of spatio-temporal (vine) copula time series models for the
+energy market. We evaluate the performance of the models in a forecasting study and
+find that they perform well. The second contribution is the methological exploration
+of point forecasting from non-elliptical probabilistic forecasts and the inclusion of
+ANN-augmented forecasts.
+In the next Section, Sect. 2, the data used in this work is introduced briefly. Sect. 3
+describes the statistical methods used in this work. The empirical results of the
+forecasting study are presented in Sect. 4, while Sect. 5 summarizes the results and
+outlines some avenues for future research.
+2. Data description
+The time series analyzed in this work are extracted from the web-platform
+investing.com using the Python package investpy [10]. We analyze month-ahead
+natural gas futures (NGas) from the Netherlands (TTF Hub). The related commodities
+used for modeling are short-term carbon emission futures (CEF), short term brent oil
+futures (oil) and short term coal futures (coal). The analyzed time series are comprised
+of daily observations. The obervation period spans from March 2010 to February 2021.
+In total, the time series comprises 2861 observations. Missing values, which occur
+especially during the holidays are trivially imputated as the last known value. The
+original time series are non-stationary. To obtain stationarity, which is necessary for
+the methods used in this report, the time series are differenced once. The differenced
+time series are displayed in Fig. 1. The hypothesis of non-stationarity in the first
+differences is rejected by the Dickey-Fuller test at the level α = 0.01 for all time series.
+3. Statistical Methods
+This section comprises the description of the methods used in this report. First
+copulas and related notions are introduced. The copula specifications used in the
+4
+
+analysis are presented and discussed. The application of copulas to time series modeling
+follows. The emergence of non-elliptical probabilistic forecasts is examined with regard
+to the t-copula. It is shown how the t-copula may be used to model conditional
+heteroskedasticity. The need for new point forecast methods is presented.
+3.1. Copulas
+Copulas are distribution functions on the unit cube with uniform marginals:
+C : [0, 1]d → [0, 1].
+(1)
+Copulas gain their relevance by Sklars Theorem [22]. It states that every multivari-
+ate distribution can be decomposed into a copula and marginal distributions. Let
+X1, . . . , Xd be real valued random variables with joint distribution FX1,...,Xd and
+marginal distributions FX1, . . . , FXd. Then it holds that there exists a copula C such
+that
+FX1,...,Xd(x1, . . . , xd) = CU1,...,Ud [FX1(x1), . . . , FXd(xd)] ,
+(2)
+where (U1, . . . , Ud) := (FX1(X1), . . . , FXd(Xd)). In the following the indices of the cop-
+ula will be dropped. If the random variables X1, . . . , Xd are continous then the decom-
+position is unique [17]. The pseudo-observation (u1, . . . , ud) := FX1(x1), . . . , FXd(xd)
+are, by virtue of the probability integral transformation, realizations from a uniform
+distribution, Ui = FXi(Xi) ∼ U[0, 1], i ∈ {1, . . . , d} [3]. This permits the copula to
+be interpreted as the dependence structure of the random variables X1, . . . , Xd. The
+−5.0
+−2.5
+0.0
+2.5
+5.0
+Natural Gas
+−5.0
+−2.5
+0.0
+2.5
+5.0
+Oil
+−5.0
+−2.5
+0.0
+2.5
+5.0
+Coal
+−5.0
+−2.5
+0.0
+2.5
+5.0
+2010
+2015
+2020
+Date
+Carbon
+Figure 1: First differences of the respective commodity price time series.
+5
+
+copula density, c which couples the joint density fX1,...,Xd and marginal densities
+fX1, . . . , fXd can be derived directly from Eq. 2 by taking derivatives,
+fX1,...,Xd(x1, . . . , xd) =
+c [FX1(x1), . . . , FXd(xd)] fX1(x1) . . . fXd(xd),
+(3)
+c[u1, . . . , ud] =
+∂dC[u1,...,ud]
+∂u1...∂ud
+.
+(4)
+The copula density is important for estimation via maximum likelihood as well as for
+the visualization of dependence structures. In this paper the copula density will also
+be used to introduce the notion of vine copula models. Another important notion
+for dependence modeling is the conditional copula of U1, . . . , Ui given Ui+1, . . . , Ud,
+respectively the conditional copula density. The conditional copula (density) can also
+be derived from Eq. 2. It is given by [21],
+C[u1, . . . , ui|ui+1, . . . , ud] =
+∂ui+1...∂udC[u1,...,ud]
+c[ui+1,...,ud]
+,
+(5)
+c[u1, . . . , ui|ui+1, . . . , ud] =
+c[u1,...,ud]
+c[ui+1,...,ud].
+(6)
+Conditional copulas are especially relevant for conditional time series models as
+presented in this paper. The relation between the conditional density and the copula
+is as follows,
+fX1,...,Xi|Xi+1...Xd(x1, . . . , xi|xi+1, . . . , xd) =
+c[FX1(x1),...,FXd(xd)]
+c[ui+1,...,ud]
+(7)
+×fX1(x1) . . . fXi(xi).
+The copula approach to multivariate modeling allows for separate modeling of marginal
+properties and dependence structure. This feature renders the approach far more
+flexible than standard multivaritate modeling. Joint distributions such as the multi-
+variate normal or students t-distribution restrict the choice of marginal distributions.
+In the copula approach, marginal distributions can be arbitrary. Also the dependence
+structure of random variables can have various features, that have to be accounted for
+by choosing an appropriate copula specification. In this work, the gaussian, Clayton,
+Gumbel and t-copula are utilized. In the following they will be introduced briefly as
+the joint distribution of random variables Ui ∼ U[0, 1], i ∈ {1, . . . , d}. The gaussian
+copula is a popular choice for the modeling of linear dependence structures. The gaus-
+sian copula is constructed by extracting the dependence structure of the multivariate
+6
+
+normal distribution and filtering the marginal influences,
+Cgaussian[u1, . . . , ud] = ΦΣ[φ−1(u1), . . . , φ−1(ud)],
+(8)
+where φ is the cumulative distribution function of the standard normal distribution
+and ΦΣ is the d-variate cumulative distribution function of the normal distribution
+with correlation matrix Σ.
+The correlation matrix Σ ∈ [0, 1]d×d contains
+d(d−1)
+2
+dependence parameters, ρ1, . . . , ρ d(d−1)
+2
+, governing the linear dependencies among
+the random variables U1, . . . , Ud. The density of a bivariate gaussian copula with
+dependence parameter ρ = 0.4 is displayed in the upper left panel of Fig. 2. The
+density only displays a linear relation between the variables. Similar to recovering
+linear dependence structures from the multivariate normal distribution, heavy-tailed
+dependence structures can be recovered from the multivariate students t-distribution
+using the t-copula
+Ct[u1, . . . , ud] = tΣ,ν[t−1
+ν (u1), . . . , t−1
+ν (ud)].
+(9)
+where tν is the cumulative distribution function of the students t-distribution with
+degree of freedom ν and tΣ,ν is the cumulative distribution function of the multivariate
+t-distribution with correlation matrix Σ and degree of freedom ν. Incorporating
+the degree of freedom ν ∈ (0, ∞) permits heavy tailed dependence structures. The
+heavy-tailedness can be interpreted as extreme events coinciding. A lower degree of
+freedom ν implies heavier tails. The density of a bivariate t-copula with dependence
+parameter ρ = 0.4 and degree of freedom ν = 4 is displayed in the upper right panel
+of Fig. 2. The density displays the linear relation between the variables as well as
+the coincidence of extreme events. Another class of dependence structures can be
+described as asymmetric dependence structures. Two relevant copulas are the Gumbel
+and Clayton copula. The Gumbel copula exhibits dominant upper tail dependence
+while the Clayton copula exhibits dominant lower tail dependence. Their bivariate
+densities are displayed in the lower left, respectively lower right panel of Fig. 2. Both
+copulas are part of the archimedean copula family. Hence they are constructed as
+C[u1, . . . , ud] = Ψ−1 (Ψ(u1) + . . . + Ψ(ud)) with a suitable generator function Ψ [12],
+7
+
+[15]. The generators for the Gumbel, respectively Clayton copula are given by
+ΨClayton(t) =
+(1 + t)− 1
+θ ,
+(10)
+ΨGumbel(t) =
+e−t
+1
+θ .
+(11)
+The dominant lower tail dependence of the Clayton copula can be interpreted as lower
+tail events coinciding more often than upper tail events and vice versa for the dominant
+upper tail dependence of the Gumbel copula. An even more flexible copula model is
+the vine copula model. Vine copula models will be explained next.
+3.2. Vine Copulas
+Vine copulas are special pair copula constructions.
+The idea of pair copula
+constructions amounts to decomposing a d-variate dependence structure into a product
+of bivariate copulas. The joint density of d random variables can, by virtue of the
+law of total probability, be decomposed into a product of conditional densities. Using
+the relation between conditional densities and copula densities (Eq. 8), one possible
+decomposition can be derived as [1, 9],
+c[u1, . . . , ud] =
+d−1
+�
+j=1
+d−j
+�
+i=1
+c[ui, ui+j|ui+1, . . . , uj−1].
+(12)
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+u1
+u2
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+u1
+u2
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+u1
+u2
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+u1
+u2
+Figure 2: Simulated density plots of four different two-dimensional copula specifications. The upper
+left plot shows the gaussian copula density with dependence parameter set to ρ = 0.4. Upper right
+shows the t-copula density with dependence parameter ρ = 0.4 and degree of freedom ν = 4. The
+lower left plot displays the Gumbel copula density with dependence parameter θ = 2. The lower right
+plot displays the Clayton copula density with dependence parameter θ = 2. All plots were created
+with simulations with 2000 samples.
+8
+
+The decomposition is not unique. The decomposition in Eq. 12 is called drawable
+vine (D-vine). The unconditional copulas in the product all capture the dependence
+structure of neighboring variables, e.g. c[ui, ui+1], c[ui+1, ui+2] and so forth. In a
+graphical representation, the connection between the variables resembles a straight
+line, hence the name D-vine.2 The vine copula approach allows for flexible dependence
+modeling. It is advantageous in contexts where the bivariate dependence structures
+between variables can take different shapes, e.g. the dependence structure between
+variable U1 and U2 is linear whereas the dependence structure between U2 and U3
+is heavy-tailed and so forth. Vine copula models can be estimated by maximum
+likelihood, we refer to [1] for details.
+3.3. Modeling Time Series with Spatio-Temporal Copulas
+In this subsection, the copula-based time series models will be reviewed and
+summarized. It will be explained how these models can be used for forecasting. First,
+the temporal copula modeling (see for example [8], [4]) will be introduced. Eventually
+the combination of cross-sectional and temporal copula modeling, the spatio-temporal
+copula modeling, will be introduced. The exposition is based on the spatio-temporal
+t-copula modeling from [21] and vine copula modeling from [5, 19, 23]. Further it
+will be examined how conditional temporal copula models offer a new approach to
+conditional heteroskedasticity modeling. The emergence of non-elliptical conditional
+distributions, respectively probabilistic forecasts, will be exemplified. The consequences
+for forecasting and the need for new point forecasting methods will be discussed.
+Let Xt be a univariate stationary Markov(1) time series. The temporal evolution of
+the time series is completely specified by the joint distribution of random variables
+from consecutive time points i.e. FXt,Xt−1. Using Sklars theorem (Eq. 2), the joint
+distribution can be decomposed into copula and marginal distributions,
+FXt,Xt−1(a, b) = C
+�
+FXt(a), FXt−1(b)
+�
+.
+(13)
+By the stationarity of Xt, FXt = FXt−1 =: FX. Hence the model can be determined
+by choosing an appropriate marginal distribution FX and an appropriate copula
+specification. Note that the marginal distribution FX is the unconditional distribution
+2Another special class of decompositions are canonical vines (C-vine). In this decomposition, the un-
+conditional dependence structures are all centered around one variable, e.g. c[ui, ui+1], c[ui, ui+2], . . ..
+In a graphical representation, the unconditional connection between variables resembles a star. In
+this work only D-vines are used.
+9
+
+of Xt. Conditional properties of the time series are completely determined by the
+conditional copula. The conditional density of Xt|Xt−1 is given by
+fXt|Xt−1(a|b) = c [FX(a), FX(b)] fX(a).
+(14)
+Hence, for forecasting, the conditional density of Xt|Xt−1 = xt−1 can be used as
+probabilistic forecast.
+This model can be understood as a generalization of the
+AR(1) model [23]3. The gaussian autoregressive model can be recovered by choosing
+C = Cgaussian and FX = Φ. When allowing other dependence structures, any temporal
+dependency representable by a copula can be reproduced. The concept of the copula
+based time series models can be further illustrated by its conditional model equation,
+Xt|(Xt−1 = xt−1) = F −1
+X (C−1 [ut|FX(xt−1)]),
+ut ∼ U[0, 1].
+(15)
+In this formulation, the non-linear connection between Xt and Xt−1 becomes obvious.
+The generalization of the temporal copula time series model to d-variate time series,
+hence spatio-temporal time series models, is straight forward. Let Xt = (X1,t, . . . , Xd,t)
+be stationary Markov(1) time series. The structure of the time series is completely
+captured by the joint distribution of Xt and Xt−1,
+FXt,Xt−1(a, b) = C [FX1(a1), . . . , FXd(ad), FX1(b1), . . . , FXd(bd)] .
+(16)
+The conditional density given observations from time point t − 1 is as follows,
+fXt|Xt−1(a|b) =
+c[FX1(a1),...,FXd(ad),FX1(b1),...,FXd(bd)]
+c[FX1(b1),...,FXd(bd)]
+(17)
+×fX1(a1) · . . . · fXd(ad).
+To sample from the conditional distribution, as necessary for Monte-Carlo approx-
+imations of conditional forecasts, the following procedure is employed [21]. First
+transform the observations at time t − 1 to pseudo-observations. This is done by
+applying the probability integral transform to the observations, (ut−1,1, . . . , ut−1,d) :=
+(FX1(xt−1,1), . . . , FXd(xt−1,d)). Then sample n d-dimensional realizations from the
+conditional copula, Eq. 5. (Details on how to sample from the t-copula can be found
+3The generalization to AR(p) models can be achieved by permiting the time series to be a
+Markov(p) process.
+10
+
+in [21]. Details to sampling from vine copulas can be found in [1]). At last, the n
+realizations have to be quantile transformed with their respective marginal distri-
+bution, yielding the n samples of the conditional distribution. Relevant models for
+this work are the following. First, the spatio-temporal time series model where the
+copula is specified as the gaussian copula (Eq. 8). The marginals are approximated
+non-parametrically by the empirical distribution.
+FXt,Xt−1(a, b) =
+ΦΣ[φ−1(F emp
+X1 (a1)), . . . φ−1(F emp
+Xd (ad)),
+(18)
+φ−1(F emp
+X1 (b1)), . . . , φ−1(F emp
+Xd (bd))].
+This model is sensible to use when the dependence structure between the variables as
+well as the temporal dependence is linear. When the dependence strucure exhibits
+heavy-tailedness, the spatio-temporal t-copula model with non-parametric marginals
+poses a viable option,
+FXt,Xt−1(a, b) =
+tν,Σ[t−1
+ν (F emp
+X1 (a1)), . . . t−1
+ν (F emp
+Xd (ad)),
+(19)
+t−1
+ν (F emp
+X1 (b1)), . . . , t−1
+ν (F emp
+Xd (bd))].
+For more flexible modeling, the spatio-temporal D-vine copula with non-parametric
+marginals can be utilized. For convenience, the model is presented in terms of its joint
+density and with variables (a, b) =: p
+fXt,Xt−1(p) =
+�2d−1
+j=1
+�2d−j
+i=1 c[FXi(pi), FXi+j(pi+j)|FXi+1(pi+1), . . . , FXj−1(pj−1)]
+×fX1(p1) · . . . · fXd(pd)fX1(pd+1) · . . . · fXd(p2d).
+(20)
+This model is sensible to use when the dependence between variables differs in its
+structure or when the temporal dependence differs from the cross-sectional dependence.
+As for solely temporal modeling, the temporal t-copula is employed,
+FXt,Xt−1(a, b) = tν,Σ[t−1
+ν (F emp
+X
+(a)), t−1
+ν (F emp
+X
+(b))].
+(21)
+The heavy-tailed temporal dependence that this model exhibits is suitable for condi-
+tional heteroskedasticity modeling as will be discussed next.
+The conditional distributions, respectively probabilistic forecasts from (spatio)-temporal
+copula time series models can be non-elliptical because of non-linear influences of the
+11
+
+conditioning variable. In the following the behavior of the conditional distributions
+will be examined with regard to the temporal t-copula with standard normal marginal
+distribution4. The emergence of non-elliptical conditional densities from the heavy-
+tailed t-copula is visualized in Fig. 3. When the conditioning variable takes moderate
+values around u1 = 0.5 the resulting conditional density is approximately elliptical.
+However, when the conditioning variable takes extreme values e.g. u1 = 0.03 and
+u1 = 0.97, the conditional density becomes bimodular. Thus, depending on the value
+of the conditioning variable, the resulting conditional density can have fundamentally
+different structures. This behavior offers a new approach to conditional heteroskedas-
+ticity modeling. Instead of widening the conditional density as in GARCH models,
+the density gets bimodular. This can be viewed as a sensible approach to volatility
+because the extreme behavior in volatile phases is mirrored in this model: When the
+time series takes a very low value at time point t − 1 it can be expected that the value
+at time point t will either be also very low or very high. The variance at time point t
+is increased nevertheless, but the mechanism for the increased variance is a new one.
+The temporal t-copula approach to conditional volatility, however, holds a problem.
+When the conditional density is non-elliptical it is not clear what constitutes a sensible
+point forecast. The expectation value may not be suitable in extreme cases where the
+conditional density is bimodular because the expectation value will take a value which
+is less probable than e.g. the modes. Taking the mode as point forecast could be a
+solution. Another possible solution to the problem of point forecasting is to augment
+the forecast by a artificial neural network (ANN). The ANN predicts which quantile
+of the conditional distribution is best (in terms of MSE) to use as point forecast. The
+inputs of the ANN are past values of the time series and the last optimal quantiles.
+The ANN architecture used in this work is the basic multi-layer perceptron (MLP)
+structure. We refer to [14] for an introduction to the topic.
+4. Results
+This section comprises the results of the expanding window forecasting study,
+investigating the performance of different models. The first 1000 observations (ranging
+4The choice of the standard normal distribution is just for convenience. The example would still
+be valid with other marginal distributions, e.g. students t-distribution.
+12
+
+from 2010-03-16 to 2013-12-08) are used as training data set. The following models
+are considered for evaluation.
+1) The temporal t-copula model with non-parametric marginals, Eq. 21, henceforth
+denoted by Tem-t,
+2) The spatio-temporal D-vine copula model Eq. 20, henceforth denoted by S-Tem
+D-vine,
+3) The spatio-temporal t-copula model, Eq. 20, henceforth denoted by S-Tem-t,
+4) The spatio-temporal gaussian copula model, Eq. 19, henceforth denoted by
+S-Tem-gaussian,
+5) The Autoregressive moving average model with external regressors and absolute
+value, threshhold generalized autoregressive conditional heteroskedasticity model,
+henceforth denoted by ARMAX-AVTGARCH (closely related to the model from
+[6]).
+The models distributional forecasting performance is examined by the continous ranked
+probability score (CRPS) [13]. Further, the ANN assisted point forecasts of the S-Tem
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+u1
+u2
+0.0
+0.1
+0.2
+0.3
+−2
+0
+2
+Φ−1(u2|u1=0.03)
+density
+0.0
+0.1
+0.2
+0.3
+−3
+−2
+−1
+0
+1
+2
+3
+Φ−1(u2|u1=0.97)
+density
+0.0
+0.2
+0.4
+0.6
+−2
+−1
+0
+1
+Φ−1(u2|u1=0.5)
+density
+Figure 3: Visualization of the conditional density structure depending on the value of the conditioning
+variable. The underlying model assumes the t-copula with dependence parameter ρ = 0.4 and degree
+of freedom ν = 2. The marginal distribution is assumed as the standard normal distribution. The
+upper left panel shows the copula density of 2000 realizations of the before mentioned t-copula. The
+three lines indicate the three cases where the conditinal density is examined. The conditional density
+is calculated by aggregating all values in the neighborhood (u1 ± 0.025) of the conditioning variable
+and quantile-transforming them. The density in the upper right panel displays the conditional density
+given u1 = 0.03. The lower panels display the conditional densities given u1 = 0.5, respectively
+u1 = 0.97.
+13
+
+Table 1: Aggregated CRPS values of the competing models for their one day-ahead probabilistic
+forecast for the four commodities. The CRPS is evaluated for the period 2013-12-19 – 2021-02-23,
+comprising 1861 obervations.
+Model/
+Commodity
+S-Tem
+D-Vine
+ARMAX-
+AVTGARCH
+Tem-t
+S-Tem-t
+S-Tem-
+gaussian
+Natural Gas
+0.236
+0.227
+0.230
+0.234
+0.234
+Oil
+0.564
+0.548
+0.551
+0.559
+0.558
+Coal
+0.400
+0.389
+0.392
+0.398
+0.397
+CEF
+0.236
+0.222
+0.229
+0.234
+0.234
+D-vine model and the Tem-t model are compared with the point forecasts from the
+ARMAX-AVTGARCH model. For each time series the ARMAX-AVTGARCH model
+is fitted individually. All models are estimated via Maximum Likelihood. However, the
+marginals of the copula models are estimated non-parametrically to avoid transmitting
+estimation errors [20]. The order of the variables in the S-Tem D-vine copula model is
+fixed as
+CEF – coal – oil – NGas – NGas lag – oil lag – coal lag – CEF lag.
+(22)
+This order is chosen to enable the lagged natural gas price to directly interact with
+the non-lagged natural gas price. The gaussian, Gumbel, Clayton and t-copula are
+allowed as bivariate copulas in the D-vine decomposition (Eq. 12). The probabilistic
+forecasts of all models are approximated by Monte-Carlo simulations with 1000
+samples for each forecast. Table 1 displays the models performances in terms of the
+CRPS. The ARMAX-AVTGARCH model performs best with regard to univariate
+distributional forecasting. However, the S-Tem D-vine model, the S-Tem-t and the
+Tem-t model are competitive. The performance of the copula models may be enhanced,
+when the marginal distributions are modeled parametrically. The empirical marginal
+distributions of the copula may not capture all marginal features of the time series.
+More versatile copula models could be used to enhance the forecast. The conditional
+dependence modeling may only be able to capture parts of the conditional effects. The
+probabilistic forecasts from the Tem-t model during a volatile period is displayed in
+Fig. 4. During volatile times the probabilistic forecasts are non-elliptical. During these
+times the ANN-augmented point forecasts can be valuable. The point forecasting
+performance of the models can be found in Table 2. The evaluation starts at the
+2001st observation, because the first 1000 probabilist forecasts are used to train the
+ANN. The hybrid, ANN-augmented S-Tem vine and the ANN-augmented Tem-t model
+14
+
+Table 2: Aggregated RMSE values of the competing point forecasting procedures for the four
+commodities.
+The RMSE is evaluated for the period 2017-10-23 – 2021-02-23, comprising 861
+observations.
+Model/
+Commodity
+S-Tem
+D-Vine
+ANN
+ARMAX-
+AVTGARCH
+S-Tem
+D-Vine
+Mean
+S-Tem
+D-Vine
+Mode
+Tem-t
+ANN
+Gas
+0.594
+0.600
+0.599
+0.597
+0.589
+oil
+1.222
+1.222
+1.236
+1.246
+1.220
+coal
+1.000
+0.999
+1.009
+1.002
+0.997
+CEF
+0.605
+0.608
+0.614
+0.604
+0.607
+generate the best point forecasts. The point forecasts of the ARMAX-AVTGARCH
+model are competitive though. Note that the ANN model used for forecasting is build
+according to the basic multi-layer perceptron architecture. It is not perfectly suitable
+for catching sequential patterns. Using recurrent neural networks, especially long
+short-term memory architectures could enhance the performance even more and could
+be subject to future research. Also incorporating a measure for the structure of the
+probabilistic forecast could enhance the performance. However, this would requiere
+more advanced architectures.
+5. Conclusion
+The application of copula-based time series models to natural gas and related
+commoditiy prices is explored in this work. An expanding window forecasting study
+is conducted. The time series used for analysis are extracted from investing.com
+Aug 15
+Sep 01
+Sep 15
+−2
+−1
+0
+1
+2
+density
+Figure 4: Probabilistic forecasts for natural gas futures generated from the temporal t-copula model.
+The forecast densities can be non-elliptical during volatile times (August 2019 – September 2019)
+15
+
+via the Python package investpy. The time series comprises short term future price
+series of natural gas, crude oil, coal and carbon emissions.
+After introducing the basic notions of dependence modeling with copulas and the
+D-Vine copula, the copula based time series models from the literature are reviewed.
+The emergence of non-elliptical probabilistic forecasts is exemplified using the tem-
+poral t-copula. It is visualized how the temporal t-copula offers a new approach to
+conditional heteroskedasticity modeling. It is not clear what constitutes a sensible
+point forecast when the probabilistic forecast is non-elliptic. To this end a artificial
+neural network is employed to predict what quantile of the probabilistic forecast is
+best to use as point forecast. The inputs of the artificial neural network are past values
+of the multivariate time series and the last best quantiles of the probabilistic forecast.
+In the forecasting study, the predictive performance of the temporal t-copula, the
+spatio-temporal t-copula and the spatio-temporal D-Vine copula is examined. The
+marginal distributions are estimated by the respective empirical distribution. The per-
+formance is compared with the performance of an autoregressive moving-average model
+with external regressors and absolute value, threshhold generalized autoregressive
+conditional heteroskedasticity modeling (ARMAX-AVTGARCH). A closely related
+model was recently shown to be the best model for natural gas forecasting. Hence
+it is understood as benchmark model. The distributional predicitive performance
+is examined by the continious ranked probability score (CRPS). We find that the
+copula-based time series models are competitive with the ARMAX-AVTGARCH
+model. The point forecasts are evaluated by the root mean squared error (RMSE).
+The ANN-augmented point forecasts perform best, although the forecasts from the
+ARMAX-AVTGARCH model are still competitive.
+The performance of the copula-based time series models could be enhanced by mod-
+eling the marginal distributions parametrically. The non-parametric modeling may
+not catch all marginal features of the time series. However, this procedure requieres
+the estimation to be conducted in one step to guarantee efficient estimation. Another
+possibility to enhance the performance is to consider more versatile copula models.
+The current modeling may not capture all conditional features of the time series.
+Another possibility, with regard to the vine copula model, is to consider other vine
+structures. In this work the D-vine structure was imposed. Other structures may
+be able to capture the dependencies better. As for the point forecasts, it was shown
+that the ANN-augmented forecasts perform well. Even though we choose to utilize
+16
+
+the standard multi-layer perceptron architecture, which can not model sequential
+information perfectly well, the precision was increased. Using more sophisticated
+architectures that are more suited to catch sequential information will be subject to
+future research. It would also be interesting to use other models to predict the best
+quantile for point forecasting.
+Acknowledgement:
+The authors gratefully acknowledge the computing time provided on the Linux HPC
+cluster at Technical University Dortmund (LiDO3), partially funded in the course of
+the Large-Scale Equipment Initiative by the German Research Foundation (DFG) as
+project 271512359.
+References
+[1] Aas, K., Czado, C., Frigessi, A. & Bakken, H. (2009). Pair-copula construtions
+of multiple dependence. Insurance: Mathematics and economics, 44(2):182-198,
+2009.
+[2] Aloui, R., Aïssa, M. S. B., Hammoudeh, S. & Nguyen D. K. (2014). Dependence
+and extreme dependence of crude oil and natural gas prices with applications to
+risk management. Energy Economics, 42:332-243, 2014.
+[3] Angus, J. E. (1994). The probability integral transform and related results. SIAM
+review, 36(4), 652-654.
+[4] Beare, B. K. (2010). Copulas and temporal dependence. Econometrica, 78(1),
+395-410.
+[5] Beare, B. K., & Seo, J. (2015). Vine copula specifications for stationary multi-
+variate Markov chains. Journal of Time Series Analysis, 36(2), 228-246.
+[6] Berrisch, J., & Ziel, F. (2022). Distributional modeling and forecasting of natural
+gas prices. Journal of Forecasting.
+[7] Berrisch, J., Pappert, S., Ziel, F., & Arsova, A. (2022). Modeling Volatil-
+ity and Dependence of European Carbon and Energy Prices. arXiv preprint
+arXiv:2208.14311.
+17
+
+[8] Chen, X., & Fan, Y. (2006). Estimation of copula-based semiparametric time
+series models. Journal of Econometrics, 130(2), 307-335.
+[9] Czado, C. (2010). Pair-copula constructions of multivariate copulas. In Copula
+theory and its applications (pp. 93-109). Springer, Berlin, Heidelberg.
+[10] Del Canto, A. (2021). Investpy–Financial Data Extraction from Investing. com
+with Python. GitHub Repository.
+[11] Demarta, S., & McNeil, A. J. (2005). The t copula and related copulas. Interna-
+tional statistical review, 73(1), 111-129.
+[12] Genest, C., & Rivest, L. P. (1993). Statistical inference procedures for bivariate
+Archimedean copulas. Journal of the American statistical Association, 88(423),
+1034-1043.
+[13] Gneiting, T., & Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and
+estimation. Journal of the American statistical Association, 102(477), 359-378.
+[14] Higham, C. F., & Higham, D. J. (2019). Deep learning: An introduction for
+applied mathematicians. Siam review, 61(4), 860-891.
+[15] Hofert, M. (2008). Sampling archimedean copulas. Computational Statistics &
+Data Analysis, 52(12), 5163-5174.
+[16] Hu, L. (2006). Dependence patterns across financial markets: a mixed copula
+approach. Applied financial economics, 16(10), 717-729.
+[17] Joe, H. (2014). Dependence modeling with copulas. CRC press.
+[18] Jondeau, E., & Rockinger, M. (2006). The copula-garch model of conditional
+dependencies: An international stock market application. Journal of international
+money and finance, 25(5), 827-853.
+[19] Nagler, T., Krüger, D., & Min, A. (2022). Stationary vine copula models for
+multivariate time series. Journal of Econometrics, 227(2), 305-324.
+[20] Patton, A. (2013). Copula methods for forecasting multivariate time series.
+Handbook of economic forecasting, 2, 899-960.
+[21] Simard, C., & Rémillard, B. (2015). Forecasting time series with multivariate
+copulas. Dependence modeling, 3(1).
+18
+
+[22] Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. Publ.
+inst. statist. univ. Paris, 8, 229-231.
+[23] Smith, M., Min, A., Almeida, C., & Czado, C. (2010). Modeling longitudinal data
+using a pair-copula decomposition of serial dependence. Journal of the American
+Statistical Association, 105(492), 1467-1479.
+19
+

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+arXiv:2301.04880v1  [gr-qc]  12 Jan 2023
+Relaxation-Time Model for the Post-Newtonian Boltzmann Equation
+Gilberto M. Kremer1, ∗
+1Departamento de F´ısica, Universidade Federal do Paran´a, Curitiba 81531-980, Brazil
+The non-equilibrium contributions to the post-Newtonian hydrodynamic equations are deter-
+mined from a relaxation-time model of the post-Newtonian Boltzmann equation. The Chapman-
+Enskog method is used to calculate the non-equilibrium distribution function. The components of
+the energy-momentum tensor are found from the knowledge of the non-equilibrium and the post-
+Newtonian equilibrium Maxwell-J¨uttner distribution functions. The linearized field equations for
+the mass, momentum and internal energy densities coupled with the three Poisson equations of the
+post-Newtonian approximation are investigated by considering a plane wave representation of the
+fields. The constitutive equations for the viscous stress and heat flux vector are obtained and it
+is shown that the transport coefficients of shear viscosity and heat conductivity do depend on the
+Newtonian gravitational potential.
+I.
+INTRODUCTION
+In the seminal work of Einstein, Infeld and Hoffman [1] it was proposed a method of successive
+approximations in powers of 1/c2 for the solution of Einstein’s field equations, which become the basis of
+the post-Newtonian approximation for the determination of the energy-momentum tensor components
+as well as the Eulerian hydrodynamic equations in the first [2, 3] and second [4] approximations.
+The post-Newtonian version of the Boltzmann equation in the first and in the second approximations
+were determined in [5, 6] and [7, 8], respectively. In [7, 8] the energy-momentum tensor components
+were obtained from the equilibrium Maxwell-J¨uttner distribution function [9] in the first and second
+post-Newtonian approximations and the Eulerian hydrodynamic equations from a collisionless post-
+Newtonian Boltzmann equation were determined.
+The inclusion of non-equilibrium terms in the post-Newtonian theory was investigated in [10, 11]
+within the framework of a phenomenological theory of a viscous, heat conducting and compressible
+fluid.
+On the other hand, the inclusion of non-equilibrium terms in the hydrodynamic equations
+which follow from the post-Newtonian Boltzmann equation was considered in [12]. In this work the
+hydrodynamic equations resulted from a post-Newtonian Maxwell-Enskog transfer equation together
+with a post-Newtonian Grad’s distribution function which takes into account the non-equilibrium fields
+of viscous stress and heat flux vector.
+One interesting subject to be investigate is the determination of the post-Newtonian hydrodynamic
+equations for a viscous and heat conducting fluid from the post-Newtonian Boltzmann equation where
+the particle collisions are taken into account through the collision operator of the Boltzmann equation.
+Here we shall adopt a relaxation-time model for the collision operator which is known in the non-
+relativistic framework as the Bhatnagar-Gross-Krook (BGK) model (see e.g.
+[13, 14]) and in the
+relativistic one as the Marle model [15, 16].
+We use the Chapman-Enskog method to determine the non-equilibrium distribution function from
+the post-Newtonian BGK (Marle) model of the Boltzmann equation and the post-Newtonian Maxwell-
+J¨uttner distribution function. From the knowledge of the non-equilibrium distribution function the
+non-equilibrium contributions to the energy-momentum tensor are calculated.
+The linearized field
+equations for the mass, momentum and internal energy densities are determined from the particle four-
+flow and energy-momentum tensor conservation laws. These linearized field equations are coupled with
+three Poisson equations from the post-Newtonian approximation and a solution of the coupled system of
+equations is found in terms of a plane wave representation of the fields. Furthermore, the constitutive
+equations for the viscous stress and heat flux vector – which correspond to the Navier-Stokes and
+Fourier laws, respectively – are obtained from the Eckart decomposition [17] of the energy-momentum
+tensor. It is shown that the transport coefficients of shear viscosity and heat conductivity do depend
+on the Newtonian gravitational potential.
+The paper is structured as follows: in Section II we introduce the relaxation-time model of the
+post-Newtonian Boltzmann equation and determine the non-equilibrium distribution function. The
+∗ kremer@fisica.ufpr.br
+
+2
+particle four-flow and the energy-momentum tensor components are calculated on the basis of the
+equilibrium Maxwell-J¨uttner and non-equilibrium distribution functions in Section III. The linearized
+field equations are determined in Section IV and a plane wave solution of the linearized field equations
+coupled with the three Poisson equations of the post-Newtonian approximation is analyzed. In Section
+V the constitutive equations for the viscous stress and heat flux vector are obtained and the transport
+coefficients of shear viscosity and thermal conductivity are identified. In the last section the conclusions
+of the work are stated.
+II.
+RELAXATION-TIME MODEL
+In the phase space spanned by the space coordinates x and velocity of the particles v a state of
+a monatomic gas is characterized by the one-particle distribution function f(x, v, t) and its space-
+time evolution is governed by Boltzmann equation. In the first post-Newtonian approximation the
+Boltzmann equation is given by [5, 7, 8]
+�∂f
+∂t + vi
+∂f
+∂xi + ∂f
+∂vi
+∂U
+∂xi
+��
+1 + 1
+c2
+�v2
+2 + U
+� �
++ 1
+c2
+∂f
+∂vi
+�
+vj
+�∂Πi
+∂xj − ∂Πj
+∂xi
+�
+−3vi
+∂U
+∂t + ∂Πi
+∂t + 2 ∂Φ
+∂xi − 4U ∂U
+∂xi − 4vivj
+∂U
+∂xj + v2 ∂U
+∂xi
+�
+= Q(f, f).
+(1)
+Here Q(f, f) denotes the collision operator of the Boltzmann equation which takes into account the
+binary collisions of the particles and refers to an integral of the product of two particle distribution
+functions at collision. Furthermore, the Newtonian gravitational potential U, the scalar gravitational
+potential Φ and the vector gravitational potential Πi satisfy Poisson equations, which are obtained
+from the first post-Newtonian approximation of Einstein’s field equations and read [2, 8]
+∇2U = −4πGρ,
+∇2Φ = −4πGρ
+�
+V 2 + U + ε
+2 + 3p
+2ρ
+�
+,
+(2)
+∇2Πi = −16πGρVi + ∂2U
+∂t∂xi .
+(3)
+Above V denotes the hydrodynamic three-velocity, G the universal gravitational constant and ε, p
+the specific internal energy and hydrostatic pressure of the gas, respectively. The gauge condition
+3∂U/∂t + ∂Πi/∂xi = 0 for the gravitational potentials U and Πi holds.
+In the BGK (Marle) model the collision operator is replaced by the difference between the one-
+particle distribution function and its equilibrium value multiplied by a frequency ν which is of order
+of the collision frequency.
+The one-particle distribution function at equilibrium is determined from the relativistic Boltzmann
+equation by considering that the collision operator vanishes at equilibrium. In the relativistic theory
+the equilibrium distribution function is the Maxwell-J¨uttner distribution function (see e.g [16]) and its
+first post-Newtonian approximation was determined in [9] and reads
+fMJ = f0
+�
+1 − 1
+c2
+�15kT
+8m + m(ViVi)2
+2kT
++ 2mUV2
+kT
++ 3mV4
+8kT
++ mV 2V2
+2kT
++ m(ViVi)V2
+kT
+��
+,
+(4)
+where f0 denotes the non-relativistic Maxwellian distribution function, namely
+f0 =
+ρ e− mV2
+2kT
+(2πm
+5
+3 kT )
+3
+2 .
+(5)
+In the above equation ρ is the mass density, T the absolute temperature, m the rest mass of a gas
+particle and k the Boltzmann constant. Furthermore, Vi = vi − Vi is the so-called peculiar velocity
+which is the difference of the particle velocity vi and the hydrodynamic velocity Vi.
+By considering that the relativistic equilibrium distribution function is the Maxwell-J¨uttner distri-
+bution fMJ, the collision operator is written as
+Q(f, f) = −ν(f − fMJ) = −νfNE,
+(6)
+where fNE is the non-equilibrium distribution function.
+
+3
+For the determination of the non-equilibrium distribution function we shall rely on the Chapman-
+Enskog method (see e.g. [13, 14] and insert the equilibrium Maxwell-J¨uttner distribution function (4)
+into the left-hand side of the Boltzmann equation (1) and compute the non-equilibrium distribution
+function by considering the BGK (Marle) model (6). Hence it follows
+�
+1 + 1
+c2
+�v2
+2 + U
+� ��∂fMJ
+∂ρ
+�dρ
+dt + Vi
+∂ρ
+∂xi
+�
++ ∂fMJ
+∂T
+�dT
+dt + Vi
+∂T
+∂xi
+�
++ ∂fMJ
+∂Vi
+�dVi
+dt + Vj
+∂Vi
+∂xj
+�
++∂fMJ
+∂U
+�dU
+dt + Vi
+∂U
+∂xi
+�
++ ∂fMJ
+∂vi
+∂U
+∂xi
+�
++ 1
+c2
+∂fMJ
+∂vi
+�
+vj
+�∂Πi
+∂xj − ∂Πj
+∂xi
+�
+− 3vi
+∂U
+∂t + ∂Πi
+∂t
++2 ∂Φ
+∂xi − 4U ∂U
+∂xi − 4vivj
+∂U
+∂xj + v2 ∂U
+∂xi
+�
+= −νfNE,
+(7)
+where d/dt = ∂/∂t + Vi∂/∂xi denotes the material time derivative and
+∂fMJ
+∂ρ
+= fMJ
+ρ ,
+∂fMJ
+∂U
+= −f0
+2mV2
+kT c2 ,
+(8)
+∂fMJ
+∂T
+= f0
+T
+�mV2
+2kT − 3
+2 + 1
+c2
+�15kT
+16m
+�
+1 − mV2
+kT
++ m2V4
+k2T 2
+�
++ 5m
+2kT
+�
+2UV2 + (ViVi)2
+2
++V 2V2
+2
++ (ViVi)V2
+�
+−
+m2
+2k2T 2
+�
+2UV4 + (ViVi)2V2
+2
++ V 2V4
+2
++ (ViVi)V4 + 3V6
+8
+���
+,
+(9)
+∂fMJ
+∂V i
+= mf0
+kT
+�
+Vi + 1
+c2
+�
+4UVi
+�
+1 − mV2
+2kT
+�
+− 15kT
+8m Vi + (VjVj)Vi + ViV 2
+�
+1 − mV2
+2kT
+�
++(VjVj)Vi
+�
+1 − mV2
+kT
+�
++ ViV2
+2
+�
+1 − 3mV2
+4kT
+�
+− m(VjVj)2
+2kT
+Vi
+��
+,
+(10)
+∂fMJ
+∂vi
+= −mf0
+kT
+�
+Vi + 1
+c2
+�
+4UVi
+�
+1 − mV2
+2kT
+�
++ Vi(V2 + VjVj) − 15kT
+8m Vi
++Vi
+�
+V 2 + 2VjVj + 3V2
+2
+�
+− mVi
+kT
+�(VjVj)2
+2
++ V 2V2
+2
++ (VjVj)V2 + 3V4
+8
+���
+.
+(11)
+As usual in the Chapman-Enskog method the material time derivatives are eliminated from the
+non-equilibrium distribution function by using the Eulerian balance equations for the mass density ρ,
+hydrodynamic velocity Vi and absolute temperature T .
+The Eulerian mass density and the momentum density balance equations in the first post-Newtonian
+approximation are [2, 8]
+dρ
+�
+1 + 1
+c2
+�
+V 2
+2 + 3U
+��
+dt
++ ρ
+�
+1 + 1
+c2
+�V 2
+2 + 3U
+�� ∂Vi
+∂xi = 0,
+(12)
+ρdVi
+dt + ∂p
+∂xi
+�
+1 − 1
+c2
+�
+V 2 + 4U + ε + p
+ρ
+��
+− ρ ∂U
+∂xi
+�
+1 + 1
+c2 (V 2 − 4U)
+�
++ ρ
+c2
+��1
+ρ
+∂p
+∂t − ∂U
+∂t + 4dU
+dt
+�
+Vi − 2 ∂Φ
+∂xi − dΠi
+dt + Vj
+∂Πj
+∂xi
+�
+= 0.
+(13)
+For the determination of the Eulerian internal energy density balance equation ρε in the first post-
+Newtonian approximation one has to go to the second post-Newtonian approximation, since within
+the framework of the first post-Newtonian approximation one recover only its Newtonian expression.
+The Eulerian internal energy density balance equation reads1
+dε
+dt + p
+ρ
+∂Vi
+∂xi + 3p
+ρc2
+dU
+dt + pVi
+ρc2
+� ∂U
+∂xi − 1
+ρ
+∂p
+∂xi
+�
+= 0.
+(14)
+From the above equation follows the expression for the material time derivative of the absolute temper-
+ature, if we take into account the relationship for the specific internal energy in the first post-Newtonian
+approximation which comes from the relativistic kinetic theory of gases (see e.g. [16])
+ε = 3kT
+2m
+�
+1 + 5kT
+4mc2
+�
+.
+(15)
+1 This equation corrects some misprints in [7, 8]
+
+4
+III.
+PARTICLE FOUR-FLOW AND ENERGY-MOMENTUM TENSOR COMPONENTS
+In the relativistic kinetic theory of gases the particle four-flow N µ and the energy-momentum tensor
+T µν are given in terms of the one-particle distribution function f(x, v, t) by [8, 16]
+N µ = m4c
+�
+uµf
+√−g d3u
+u0
+,
+T µν = m4c
+�
+uµuνf
+√−g d3u
+u0
+.
+(16)
+Here uµ = pµ/m (with uµuµ = c2) denotes the gas particle four-velocity whose components in the first
+post-Newtonian approximation read [2, 3, 8]
+u0 = c
+�
+1 + 1
+c2
+�v2
+2 + U
+��
+,
+ui = vi
+u0
+c ,
+(17)
+where v is the particle three-velocity. Furthermore, √−g d3u/u0 is an invariant integration element
+whose first post-Newtonian approximation was determined in [9] and is given by
+√−g d3u
+u0
+=
+�
+1 + 1
+c2
+�
+2v2 + 6U
+�� d3v
+c .
+(18)
+Once the one-particle distribution function f = fMJ + fNE and the invariant integration element
+are known, one can determine the components of the particle four-flow N µ and energy-momentum
+tensor T µν. Indeed, if we insert (4), (7), (17) and (18) into (16) and integrate the resulting equations
+we get
+N 0 = ρc
+m
+�
+1 + 1
+c2
+�V 2
+2 + U
+��
+,
+N i = N 0 Vi
+c ,
+(19)
+T 00 = ρc2
+�
+1 + 1
+c2
+�
+V 2 + 3kT
+2m + 2U
+�
++ O(c−4)
+�
+,
+(20)
+T i0 = cρVi
+�
+1 + 1
+c2
+�
+V 2 + 2U + 5kT
+2m
+��
++ T i0
+NE,
+(21)
+T ij = ρViVj
+�
+1 + 1
+c2
+�
+V 2 + 2U + 5kT
+2m
+��
++ p
+�
+1 − 2U
+c2
+�
+δij + T ij
+NE.
+(22)
+Note that there are no non-equilibrium contributions to the components of the particle four-flow (19.
+The non-equilibrium contribution to T 00 is of order O(c−4) (the order of the nth inverse power of light
+speed is denoted by O(c−n)) while the non-equilibrium contributions to the energy-momentum tensor
+components T 0i
+NE and T ij
+NE are associate with terms related with the collision frequency ν and read
+T i0
+NE = −p
+νc
+� 5k
+2m
+∂T
+∂xi + ∆ijklVj
+∂Vk
+∂xl
+�
+,
+(23)
+T ij
+NE = − p
+ν
+��
+1 + 1
+c2
+�5kT
+2m − U + V 2
+2
+��
+∆ijklVj
+∂Vk
+∂xl + 1
+c2 ∆ijklVk
+� ∂U
+∂xl − 1
+ρ
+∂p
+∂xl
+�
+− 2
+3c2 ViVj
+∂Vk
+∂xk + 1
+c2
+�
+Vj
+∂
+∂xi + Vi
+∂
+∂xj
+��5kT
+2m + V 2
+2
+��
+.
+(24)
+Here we have introduced the fourth-order tensor
+∆ijkl = δikδjl + δilδjk − 2
+3δijδkl.
+(25)
+IV.
+LINEARIZED FIELD EQUATIONS
+The thermodynamic theory of a single relativistic fluid is described by the fields of particle four-flow
+N µ and energy-momentum tensor T µν where their hydrodynamic equations follow from the conserva-
+tion laws
+N µ;µ = ∂N µ
+∂xµ + ΓµµσN σ = 0,
+T µν;ν = ∂T µν
+∂xν + ΓµνσT σν + ΓννσT µσ = 0.
+(26)
+
+5
+Above the semicolon refers to the covariant derivative and Γµνσ to the Christoffel symbols.
+From the knowledge of the expressions of the particle four-flow and energy momentum tensor com-
+ponents (19) – (24) and the conservation laws (26) one can obtain the field equations for the particle
+number density, momentum density and specific internal energy for a viscous and heat conducting
+fluid in the first post-Newtonian approximation.
+Here we are interested in determining the linearized field equations and for that end we consider
+a background state of constant values for the mass density, absolute temperature and Newtonian
+gravitational potential denoted by ρ0, T0 and U0, respectively, which are superposed by linear perturbed
+fields denoted by ρ1, T1, U1, V 1
+i , Φ1, Π1
+i , namely
+ρ(x, t) = ρ0 + ρ1(x, t),
+T (x, t) = T0 + T1(x, t),
+U(x, t) = U0 + U1(x, t),
+(27)
+Vi(x, t) = V 1
+i (x, t),
+Φ(x, t) = Φ1(x, t),
+Πi(x, t) = Π1
+i (x, t).
+(28)
+From the insertion of (19) into (26)1 follows the linearized field equation for the mass density, by tak-
+ing into account the expressions of the Christoffel symbols in the first post-Newtonian approximation
+– which can be found in [2, 7, 8] – and of the representations (27), yielding
+∂ρ1
+∂t + ρ0
+∂V 1
+i
+∂xi + 3ρ0
+c2
+∂U1
+∂t = 0,
+(29)
+The linearized field equations for the mass-energy and momentum densities are obtained from the
+time and spatial components of (26)2, respectively, by considering the expressions (19) – (24), the
+representations (27), (28) and the Christoffel symbols in the first post-Newtonian approximation.
+Hence it follows
+∂ρ1
+∂t + ρ0
+�
+1 + kT0
+mc2
+� ∂V 1
+i
+∂xi + ρ0
+c2
+�3kT0
+2m
+∂T1
+∂t + 3∂U1
+∂t
+�
+− 5k2ρ0T0
+2m2c2ν0
+∂2T1
+∂xi∂xi = 0,
+(30)
+ρ0
+∂V 1
+i
+∂t + k
+m
+�
+1 − 1
+c2
+�5kT0
+2m + 4U0
+�� �
+T0
+∂ρ1
+∂xi + ρ0
+∂T1
+∂xi
+�
+− ρ0
+�
+1 − 4U0
+c2
+� ∂U1
+∂xi
+− 5k2ρ0T0
+2m2c2ν0
+∂2T1
+∂t∂xi − kρ0T0
+mν0
+�
+1 − 3U0
+c2
+� �
+∂2V 1
+i
+∂xj∂xj + 1
+3
+∂2V 1
+j
+∂xj∂xi
+�
+− ρ0
+c2
+�
+2∂Φ1
+∂xi + ∂Π1
+i
+∂t
+�
+= 0.
+(31)
+Since the constant values of the background state does not satisfy the Poisson equations (2) and (3)
+it is usual to take into account the ”Jeans swindle” (see e.g. [18–20]) which requires that the Poisson
+equations are valid only for the perturbed fields. Hence, by considering that ε = 3kT/2m = 3p/2ρ,
+the linearized Poisson equations become
+∇2U1 = −4πGρ1,
+∇2Φ1 = −4πGρ1
+�
+U0 + 9k
+4mT0
+�
+− 4πGρ0
+�
+U1 + 9k
+4mT1
+�
+,
+(32)
+∇2Π1
+i = −16πGρ0V 1
+i + ∂2U1
+∂t∂xi .
+(33)
+Let us find a solution of the coupled system of partial differential equations (29) – (33) in terms of
+a plane wave representation of the perturbed fields, namely
+ρ1(x, t) = ρe[i(κixi−ωt)],
+T1(x, t) = Te[i(κixi−ωt)],
+U1(x, t) = Ue[i(κixi−ωt)],
+(34)
+V 1
+i (x, t) = Vie[i(κixi−ωt)],
+Φ1(x, t) = Φe[i(κixi−ωt)],
+Π1
+i (x, t) = Πie[i(κixi−ωt)],
+(35)
+where κi denotes the wavenumber vector, ω the angular frequency and the overlined quantities the
+small amplitudes of the wave.
+We insert the plane wave representations (34) and (35) into the coupled system of partial differential
+
+6
+equations (29) – (33) and get a linearized system of algebraic equations for the amplitudes which reads
+ω∗ρ∗ − V∗ + 3U0
+c2 U∗ = 0,
+(36)
+ω∗ρ∗ −
+�
+1 + 3c2
+s
+5c2
+�
+V∗ + 3U0
+c2 U∗ + 9c2
+s
+10c2
+�
+ω∗ + iκ∗
+ν∗
+�
+T∗ = 0,
+(37)
+�
+ω∗ +
+4
+5ν∗
+�
+1 − 3U0
+c2
+�
+iκ2
+∗
+�
+V∗ − 3
+5κ2
+∗
+�
+1 − c2
+s
+c2
+�3
+2 + 4U0
+c2s
+��
+[ρ∗ + T∗]
++κ2
+∗
+U0
+c2s
+�
+1 − 4U0
+c2
+�
+U∗ −
+3c2
+s
+2c2ν∗
+iω∗κ2
+∗T∗ + c2
+s
+c2
+�
+2κ2
+∗Φ∗ − ω∗Π∗
+�
+= 0,
+(38)
+κ2
+∗
+U0
+c2s
+U∗ = ρ∗,
+(39)
+κ2
+∗Φ∗ =
+�U0
+c2s
++ 27
+20
+�
+ρ∗ +
+�U0
+c2s
+U∗ + 27
+20T∗
+�
+,
+(40)
+κ2
+∗Π∗ = 4V∗ − ω∗κ2
+∗
+U0
+c2s
+U∗.
+(41)
+Equations (38) and (41) result from the scalar product with κi. Furthermore, the above equations
+were written in terms of the dimensionless quantities
+κ∗
+i = κi
+κJ
+,
+ω∗ =
+ω
+√4πGρ0
+,
+ν∗ =
+ν0
+√4πGρ0
+,
+(42)
+ρ∗ = ρ
+ρ0
+,
+T∗ = T
+T0
+,
+V∗ = V iκi
+csκJ
+,
+U∗ = U
+U0
+,
+Φ∗ = Φ
+c4s
+,
+Π∗ = Πiκi
+c3sκJ
+,
+(43)
+where κJ = √4πGρ0/cs denotes the Jeans wavelength, cs =
+�
+5kT0/3m the sound speed and κ∗ =
+�κ∗
+i κ∗
+i .
+The system of algebraic equations for the amplitudes (36) – (41) admits a non-trivial solution
+if the determinant of the coefficients which correspond to the amplitudes vanish. Hence it follows
+the dispersion relation which connect the dimensionless angular frequency ω∗ with the dimensionless
+wavenumber κ∗, namely
+ω3
+∗ + 9i
+5ν∗
+�
+κ2
+∗ + 4
+3
+�
+1 − κ2
+∗
+� 5
+12 + U0
+c2s
+� c2
+s
+c2
+��
+ω2
+∗ +
+�
+1 − κ2
+∗ − 4κ4
+∗
+5ν∗
++
+�33
+10 + 2
+κ2∗
++3κ2
+∗
+2
+− 2U0
+c2s
+(1 − 2κ2
+∗) − 12κ2
+∗
+5ν2∗
+�
+1 − U0κ2
+∗
+c2s
+��c2
+s
+c2
+�
+ω∗ + i
+ν∗
+�
+κ2
+∗
+�
+1 − 3κ2
+∗
+5
+�
++
+�
+2 + 27κ2
+∗
+10
+�
+1 + κ2
+∗
+3
+�
+− 2κ2
+∗U0
+c2s
+�
+1 − 6κ2
+∗
+5
+��c2
+s
+c2
+�
+= 0.
+(44)
+Here terms up to the order O(c−2) were taken into account.
+In the case of a non relativistic and collisionless Boltzmann equation we have that cs/c → 0 and
+ν∗ → ∞ and we obtain from (44) Jeans solution [18]
+ω∗ = ±
+�
+λ2
+J
+λ2 − 1.
+(45)
+Above we have introduced the wavelengths λ and λJ (Jeans wavelength) through the relationship
+κ∗ = κ/κJ = λJ/λ. In the case of small wavelengths with respect to Jeans wavelength λJ/λ > 1 the
+dimensionless angular frequency is a real quantity and the perturbations propagate as harmonic waves
+in time. On the other hand, for big wavelengths λJ/λ < 1 the angular frequency becomes a pure
+imaginary quantity and the perturbations will grow or decay in time, which will depend on the sign
+of the solution (45). The perturbations which grow in time are referred as Jeans instability, which is
+associated with the gravitational collapse of self-gravitating gas clouds.
+The analysis of Jeans instability within the first and second post-Newtonian approximation by
+considering the Eulerian hydrodynamic equations were investigated in [21–23] and [24], respectively.
+Here if we consider a collisionless Boltzmann equation where ν∗ → ∞ (44) reduces to
+ω3
+∗ +
+�
+1 − κ2
+∗ +
+�33
+10 + 2
+κ2∗
++ 3κ2
+∗
+2
+− 2U0
+c2s
+(1 − 2κ2
+∗)
+�c2
+s
+c2
+�
+ω∗ = 0,
+(46)
+
+7
+which is the dispersion relation in the first post-Newtonian approximation where dissipative effects
+are not considered. There is a difference of this expression with the one in [8], since here the constant
+value is 33/10 while there is 9/2. The reason of this difference is that here we have considered the
+mass, mass-energy and momentum densities hydrodynamic equations while in the former work only
+the mass and momentum densities hydrodynamic equations were taken into account.
+For big wavelengths with respect to Jeans wavelength λJ/λ < 1 three different values associated with
+the dimensionless angular frequencies can be obtained from (44) which correspond to the growth/decay
+of the perturbations:
+ω∗ = − i
+ν∗
+λ2
+J
+λ2
+�
+1 − 7c2
+s
+5c2
+�
++ . . . ,
+(47)
+ω∗ = i
+�
+1 − 1
+2
+λ2
+J
+λ2
+�
+1 +
+4
+5ν∗
+�
++
+�43
+20 − U0
+c2s
++ λ2
+λ2
+J
+−
+6
+5ν∗
+� c2
+s
+c2
+�
++ . . . ,
+(48)
+ω∗ = −i
+�
+1 − 1
+2
+λ2
+J
+λ2
+�
+1 −
+4
+5ν∗
+�
++
+�43
+20 − U0
+c2s
++ λ2
+λ2
+J
++
+6
+5ν∗
+� c2
+s
+c2
+�
++ . . . .
+(49)
+On the other hand, if we expand the dimensionless wavenumber in power series of the reduced
+angular frequency κ∗ = a0 + a1ω∗ + . . . we get from the dispersion relation (44) the solution where
+the perturbations propagate as harmonic waves
+κ∗ =
+�
+5
+3
+�
+1 +
+�27
+10 + U
+� c2
+s
+c2
+�
++ 2i
+3ν∗
+�
+5
+3
+�
+1 + 3ν2
+∗
+10 +
+�24
+5 + 2U − ν2
+∗
+�3U
+10 + 36
+25
+�� c2
+s
+c2
+�
+ω∗+. . . . (50)
+V.
+CONSTITUTIVE EQUATIONS
+As was previously said the thermodynamic theory of a single relativistic fluid is characterized by
+the fields of particle four-flow N µ and energy-momentum tensor T µν whose hydrodynamic equations
+are the conservation laws (26).
+The representation of the particle four-flow and energy-momentum tensor in terms of non-relativistic
+quantities makes use of the four-velocity U µ –where U µUµ = c2 – and of the projector ∆µν =
+gµν − U µU ν/c2 – where gµν denotes the metric tensor. The projector has the properties ∆µνUν = 0,
+∆µν∆νσ = ∆µσ and in a local Minkowski rest frame where U µ = (c, 0) it reduces to ∆µν =
+diag(0, −1, −1, −1).
+Two representations for the particle four-flow and energy-momentum tensor in terms of non-
+relativistic quantities are the Eckart [17] and the Landau-Lifshitz [25] decompositions. Here we shall
+use the Eckart decomposition where the particle four-flow and energy-momentum tensor are written
+as
+N µ = nU µ,
+(51)
+T µν = p⟨µν⟩ − (p + ̟) ∆µν + ǫ
+c2 U µU ν + 1
+c2
+�
+U µq(ν) + U νq(µ)
+�
+.
+(52)
+Above n is the particle number density, p the hydrostatic pressure, ̟ the non-equilibrium pressure,
+p⟨µν⟩ the pressure deviator, q(µ) the heat flux and ǫ the energy density. The energy density is a sum
+of two terms one related with the internal energy density ρε while the other with the mass density ρ,
+namely ǫ = ρc2(1 + ε/c2). The following projections of the particle four-flow and energy-momentum
+tensor define the non-relativistic quantities (see e.g [16]):
+n = 1
+c2 N µUµ,
+ǫ = 1
+c2 UµT µνUν,
+(p + ̟) = −1
+3∆µνT µν
+(53)
+p⟨µν⟩ =
+�
+∆µ
+σ∆ν
+τ − 1
+3∆µν∆στ
+�
+T στ,
+q(µ) = ∆µ
+νUσT νσ,
+(54)
+In the first post-Newtonian approximation the components of the four-velocity read [2, 3, 8]
+U 0 = c
+�
+1 + 1
+c2
+�V 2
+2 + U
+��
+,
+U i = ViU 0
+c
+,
+(55)
+where V denotes the hydrodynamic three velocity.
+
+8
+From the knowledge of the components of the metric tensor in the first post-Newtonian approxima-
+tion
+g00 = 1 − 2U
+c2 + 2
+c4
+�
+U 2 − 2Φ
+�
+,
+g0i = Πi
+c3 ,
+gij = −
+�
+1 + 2U
+c2
+�
+δij,
+(56)
+and of the four-velocity components (55) we can determine the components of the projector, which
+read
+∆00 = −V 2
+c2 − 1
+c4
+�
+6UV 2 + V 4 − 2ΠiVi
+�
+,
+∆0i = −Vi
+c − 1
+c3
+�
+2UVi + V 2Vi − Πi
+�
+,
+(57)
+∆ij = −
+�
+1 − 2U
+c2
+�
+δij − ViVj
+c2 .
+(58)
+Now we introduce the non-relativistic pressure deviator
+pij = pij − pkkδij/3
+whit
+δijpij = 0,
+(59)
+so that the components of the pressure deviator p⟨µν⟩ become [12]
+p⟨ij⟩ = pij + 1
+2c2 (pikVkVj + pjkVkVi) ,
+(60)
+p⟨00⟩ = pij
+ViVj
+c2 ,
+p⟨0i⟩ = pij
+Vj
+c .
+(61)
+In terms of the non-relativistic heat flux vector qi the components of the heat flux q(µ) are
+q(i) = qi,
+q(0) = qi
+Vi
+c .
+(62)
+In the five field thermodynamic theory – where the basic fields are the mass density, momentum
+density and internal energy density – the pressure deviator, the dynamic pressure and the heat flux
+vector are given by constitutive equations.
+Here we can obtain the desired constitutive equations
+from the components of the energy-momentum tensor (19) – (24) combined with the decomposition
+expressions (53) and (54) and the components of the projection (57) and (58). Hence it follows the
+constitutive equations for the non-relativistic heat flux vector and pressure deviator
+qi = − 5kp
+2mν
+�
+1 − c2
+s
+c2
+U
+c2s
+� ∂T
+∂xi +
+p
+νc2 ∆ijkl
+∂Vk
+∂xl
+��5kT
+2m + 3U + V 2
+2
+�
+Vj − Πj
+�
++ p
+νc2 (V 2δij − ViVj)
+�
+Vk
+∂Vk
+∂xj − ∂T
+∂xj
+�
++
+p
+νc2
+�
+V 2δij + ViVj
+3
+�� ∂U
+∂xj − 1
+ρ
+∂p
+∂xj
+�
+,
+(63)
+pij = − p
+ν
+�
+1 + c2
+s
+c2
+�3
+2 − U
+c2s
+��
+∆ijkl
+∂Vk
+∂xl +
+2p
+3νc2
+∂Vk
+∂xk
+�
+ViVj − 1
+3V 2δij
+�
+− p
+νc2 ∆ijkl
+�1
+2
+∂V 2Vk
+∂xl
++ Vk
+� ∂U
+∂xl − 1
+ρ
+∂p
+∂xl
+��
+.
+(64)
+The constitutive equation for the dynamic pressure ̟ does not show up in the first post-Newtonian
+approximation and it is known that in the kinetic theory of relativistic gases the coefficient of bulk
+viscosity – which relates the dynamic pressure with the velocity divergent – is of order O(c−4) (see
+e.g. [16]).
+Let us fix our attention in the underlined linearized terms in (63) and (64). Without the relativistic
+corrections they reduce to the non-relativistic constitutive equations of a viscous and heat conducting
+gas, namely
+qi = − 5kp
+2mν
+∂T
+∂xi ,
+pij = − p
+ν ∆ijkl
+∂Vk
+∂xl ,
+(65)
+where the thermal conductivity λ and the shear viscosity µ coefficients are those of the non-relativistic
+BGK model
+λ = 5kp
+2mν ,
+µ = p
+ν .
+(66)
+
+9
+With the first post-Newtonian correction these coefficients read
+λ = 5kp
+2mν
+�
+1 − c2
+s
+c2
+U
+c2s
+�
+,
+µ = p
+ν
+�
+1 + c2
+s
+c2
+�3
+2 − U
+c2s
+��
+.
+(67)
+We note that the coefficients of shear viscosity and thermal conductivity do depend on the Newtonian
+gravitational potential. On the basis of a non-relativistic kinetic theory the influence the gravity on the
+thermal coefficient was first reported in [26, 27]. Within the framework of a relativistic kinetic theory
+the transport coefficients of shear viscosity, thermal conductivity and bulk viscosity were obtained by
+considering a Schwarzschild metric in [28] and the diffusion coefficient in [29].
+VI.
+CONCLUSIONS
+In this work we have examined a relaxation-time model for the post-Newtonian Boltzmann equation
+and determined the non-equilibrium distribution function by using the Chapman-Enskog method and
+the equilibrium post-Newtonian Maxwell-J¨uttner distribution function. The components of the energy-
+momentum tensor were calculated by using the equilibrium and non-equilibrium distribution functions.
+From the conservation laws of the particle four-flow and energy-momentum tensor the linearized field
+equations for the mass, momentum and internal energy densities were determined.
+A plane wave
+solution of these linearized field equations coupled with the three post-Newtonian Poisson equations was
+found. By using the Eckart decomposition of the energy-momentum tensor the constitutive equations
+for the viscous stress and heat flux vector were obtained and it was shown that the transport coefficients
+of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.
+ACKNOWLEDGMENTS
+This work was supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq),
+grant No. 304054/2019-4.
+[1] A. Einstein, L. Infeld and B. Hoffmann, The gravitational equations and the problem of motion, Ann. of
+Math. 39, 65 (1938).
+[2] S. Chandrasekhar, The post-Newtonian equations of hydrodynamics in general relativity, Ap. J. 142, 1488
+(1965).
+[3] S. Weinberg, Gravitation and cosmology. Principles and applications of the theory of relativity (Wiley, New
+York, 1972).
+[4] S. Chandrasekhar and Y. Nutku, The second post-Newtonian equations of hydrodynamics in general
+relativity. Ap. J. 158, 55 (1969).
+[5] C. A. Ag´on, J. F. Pedraza and J. Ramos-Caro, Kinetic theory of collisionless self-gravitating gases: Post-
+Newtonian polytropes, Phys. Rev. D 83, 123007 (2011).
+[6] V. Rezania and Y. Sobouti, Liouville’s equation in post Newtonian approximation I. Static solutions,
+Astron. Astrophys. 354, 1110 (2000).
+[7] G.M. Kremer, Post-Newtonian kinetic theory, Ann. Phys. 426, 168400 (2021).
+[8] G. M. Kremer, Post-Newtonian hydrodynamics: theory and applications, (Cambridge Scholars Publishing,
+Newcastle upon Tyne, 2022).
+[9] G. M. Kremer, M. G. Richarte and K. Weber, Self-gravitating systems of ideal gases in the 1PN approxi-
+mation, Phys. Rev. D 93, 064073 (2016).
+[10] P. J. Greenberg, The post-Newtonian equations of hydrodynamics for a thermally conducting, viscous,
+compressible fluid in general relativity, Ap. J. 164, 569 (1971).
+[11] J.-C. Hwang and H. Noh, Special relativistic hydrodynamics with gravitation, Ap. J. 833, 180 (2016).
+[12] G.M. Kremer, Post-Newtonian non-equilibrium kinetic theory, Ann. Phys. 441, 168865 (2022).
+[13] S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases 3rd. (Cambridge University
+Press, Cambridge, 1970).
+[14] G. M. Kremer, An introduction to the Boltzmann equation and transport processes in gases (Springer,
+Berlin, 2010).
+[15] C. Marle, Mod`ele cin´etique pour l’´etablissement des lois de la conduction de la chaleur et de la viscosit´e
+en th´eorie de la relativit´e, C. R. Acad. Sc. Paris 260, 6539 (1965).
+
+10
+[16] C. Cercignani and G. M. Kremer, The relativistic Boltzmann equation:
+theory and applications
+(Birkh¨auser, Basel, 2002)
+[17] C. Eckart, The thermodynamics of irreversible processes, III. Relativistic theory of a simple fluid, Phys.
+Rev. 58, 919 (1940).
+[18] J. H. Jeans, The stability of a spherical nebula. Philos. Trans. R. Soc. A, 199, 1 (1902).
+[19] P. Coles and F. Lucchin, Cosmology. The origin and evolution of cosmic structures, 2nd, edn. (John Wiley,
+Chichester, 2002).
+[20] J. Binney and S. Tremaine, Galactic Dynamics, 2nd. edn. (Princeton University Press, Princeton, 2008).
+[21] E. Nazari, A. Kazemi, M. Roshan and S. Abbassi, Post-Newtonian Jeans analysis. Ap. J. 839, 75 (2017).
+[22] H. Noh and J.-C. Hwang, Gravitomagnetic instabilities of relativistic magnetohydrodynamics. Ap. J. 906,
+22 (2021).
+[23] G. M. Kremer, Jeans instability from post-Newtonian Boltzmann equation. Eur. Phys. J. C 81, 927
+(2021).
+[24] G. M. Kremer, Plane wave analysis of the second post-Newtonian hydrodynamic equations, Int. J. Geom.
+Methods Mod. Phys. 2350039 (2023).
+[25] L. D. Landau and E. M. Lifshitz, Fluid mechanics, 2nd ed. (Pergamon Press, Oxford, 1987).
+[26] T. Doi T, A. Santos and M. Tij M, Numerical study of the influence of gravity on the heat conductivity on
+the basis of kinetic theory Phys. Fluids 11, 3553 (1999).
+[27] M. Tij, V. Garz´o and A. Santos, On the influence of gravity on the thermal conductivity, in Rarefied Gas
+Dynamics, R. Brun , R. Campargue, R. Gatignol and J.-C. Lengrand , eds. 1999 (Toulouse: C´epadu`es) p.
+239
+[28] G. M. Kremer, Relativistic gas in a Schwarzschild metric, J. Stat. Mech. P04016 (2013).
+[29] G. M. Kremer, Diffusion of relativistic gas mixtures in gravitational field, Physica A 393 76 (2014).
+

AtE4T4oBgHgl3EQfEwyK/content/tmp_files/load_file.txt

@@ -0,0 +1,359 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf,len=358
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='04880v1  [gr-qc]  12 Jan 2023  Relaxation-Time Model for the Post-Newtonian Boltzmann Equation  Gilberto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer1, ∗  1Departamento de F´ısica, Universidade Federal do Paran´a, Curitiba 81531-980, Brazil  The non-equilibrium contributions to the post-Newtonian hydrodynamic equations are deter-  mined from a relaxation-time model of the post-Newtonian Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The Chapman-  Enskog method is used to calculate the non-equilibrium distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The components of  the energy-momentum tensor are found from the knowledge of the non-equilibrium and the post-  Newtonian equilibrium Maxwell-J¨uttner distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The linearized field equations for  the mass, momentum and internal energy densities coupled with the three Poisson equations of the  post-Newtonian approximation are investigated by considering a plane wave representation of the  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The constitutive equations for the viscous stress and heat flux vector are obtained and it  is shown that the transport coefficients of shear viscosity and heat conductivity do depend on the  Newtonian gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  INTRODUCTION  In the seminal work of Einstein, Infeld and Hoffman [1] it was proposed a method of successive  approximations in powers of 1/c2 for the solution of Einstein’s field equations, which become the basis of  the post-Newtonian approximation for the determination of the energy-momentum tensor components  as well as the Eulerian hydrodynamic equations in the first [2, 3] and second [4] approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The post-Newtonian version of the Boltzmann equation in the first and in the second approximations  were determined in [5, 6] and [7, 8], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' In [7, 8] the energy-momentum tensor components  were obtained from the equilibrium Maxwell-J¨uttner distribution function [9] in the first and second  post-Newtonian approximations and the Eulerian hydrodynamic equations from a collisionless post-  Newtonian Boltzmann equation were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The inclusion of non-equilibrium terms in the post-Newtonian theory was investigated in [10, 11]  within the framework of a phenomenological theory of a viscous, heat conducting and compressible  fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  On the other hand, the inclusion of non-equilibrium terms in the hydrodynamic equations  which follow from the post-Newtonian Boltzmann equation was considered in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' In this work the  hydrodynamic equations resulted from a post-Newtonian Maxwell-Enskog transfer equation together  with a post-Newtonian Grad’s distribution function which takes into account the non-equilibrium fields  of viscous stress and heat flux vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  One interesting subject to be investigate is the determination of the post-Newtonian hydrodynamic  equations for a viscous and heat conducting fluid from the post-Newtonian Boltzmann equation where  the particle collisions are taken into account through the collision operator of the Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Here we shall adopt a relaxation-time model for the collision operator which is known in the non-  relativistic framework as the Bhatnagar-Gross-Krook (BGK) model (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [13, 14]) and in the  relativistic one as the Marle model [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  We use the Chapman-Enskog method to determine the non-equilibrium distribution function from  the post-Newtonian BGK (Marle) model of the Boltzmann equation and the post-Newtonian Maxwell-  J¨uttner distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' From the knowledge of the non-equilibrium distribution function the  non-equilibrium contributions to the energy-momentum tensor are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The linearized field  equations for the mass, momentum and internal energy densities are determined from the particle four-  flow and energy-momentum tensor conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' These linearized field equations are coupled with  three Poisson equations from the post-Newtonian approximation and a solution of the coupled system of  equations is found in terms of a plane wave representation of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Furthermore, the constitutive  equations for the viscous stress and heat flux vector – which correspond to the Navier-Stokes and  Fourier laws, respectively – are obtained from the Eckart decomposition [17] of the energy-momentum  tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' It is shown that the transport coefficients of shear viscosity and heat conductivity do depend  on the Newtonian gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The paper is structured as follows: in Section II we introduce the relaxation-time model of the  post-Newtonian Boltzmann equation and determine the non-equilibrium distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The  ∗ kremer@fisica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='br  2  particle four-flow and the energy-momentum tensor components are calculated on the basis of the  equilibrium Maxwell-J¨uttner and non-equilibrium distribution functions in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The linearized  field equations are determined in Section IV and a plane wave solution of the linearized field equations  coupled with the three Poisson equations of the post-Newtonian approximation is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' In Section  V the constitutive equations for the viscous stress and heat flux vector are obtained and the transport  coefficients of shear viscosity and thermal conductivity are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' In the last section the conclusions  of the work are stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  RELAXATION-TIME MODEL  In the phase space spanned by the space coordinates x and velocity of the particles v a state of  a monatomic gas is characterized by the one-particle distribution function f(x, v, t) and its space-  time evolution is governed by Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' In the first post-Newtonian approximation the  Boltzmann equation is given by [5, 7, 8]  �∂f  ∂t + vi  ∂f  ∂xi + ∂f  ∂vi  ∂U  ∂xi  ��  1 + 1  c2  �v2  2 + U  � �  + 1  c2  ∂f  ∂vi  �  vj  �∂Πi  ∂xj − ∂Πj  ∂xi  �  −3vi  ∂U  ∂t + ∂Πi  ∂t + 2 ∂Φ  ∂xi − 4U ∂U  ∂xi − 4vivj  ∂U  ∂xj + v2 ∂U  ∂xi  �  = Q(f, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (1)  Here Q(f, f) denotes the collision operator of the Boltzmann equation which takes into account the  binary collisions of the particles and refers to an integral of the product of two particle distribution  functions at collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Furthermore, the Newtonian gravitational potential U, the scalar gravitational  potential Φ and the vector gravitational potential Πi satisfy Poisson equations, which are obtained  from the first post-Newtonian approximation of Einstein’s field equations and read [2, 8]  ∇2U = −4πGρ,  ∇2Φ = −4πGρ  �  V 2 + U + ε  2 + 3p  2ρ  �  ,  (2)  ∇2Πi = −16πGρVi + ∂2U  ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (3)  Above V denotes the hydrodynamic three-velocity, G the universal gravitational constant and ε, p  the specific internal energy and hydrostatic pressure of the gas, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The gauge condition  3∂U/∂t + ∂Πi/∂xi = 0 for the gravitational potentials U and Πi holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  In the BGK (Marle) model the collision operator is replaced by the difference between the one-  particle distribution function and its equilibrium value multiplied by a frequency ν which is of order  of the collision frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The one-particle distribution function at equilibrium is determined from the relativistic Boltzmann  equation by considering that the collision operator vanishes at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' In the relativistic theory  the equilibrium distribution function is the Maxwell-J¨uttner distribution function (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='g [16]) and its  first post-Newtonian approximation was determined in [9] and reads  fMJ = f0  �  1 − 1  c2  �15kT  8m + m(ViVi)2  2kT  + 2mUV2  kT  + 3mV4  8kT  + mV 2V2  2kT  + m(ViVi)V2  kT  ��  ,  (4)  where f0 denotes the non-relativistic Maxwellian distribution function, namely  f0 =  ρ e− mV2  2kT  (2πm  5  3 kT )  3  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (5)  In the above equation ρ is the mass density, T the absolute temperature, m the rest mass of a gas  particle and k the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Furthermore, Vi = vi − Vi is the so-called peculiar velocity  which is the difference of the particle velocity vi and the hydrodynamic velocity Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  By considering that the relativistic equilibrium distribution function is the Maxwell-J¨uttner distri-  bution fMJ, the collision operator is written as  Q(f, f) = −ν(f − fMJ) = −νfNE,  (6)  where fNE is the non-equilibrium distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  3  For the determination of the non-equilibrium distribution function we shall rely on the Chapman-  Enskog method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' [13, 14] and insert the equilibrium Maxwell-J¨uttner distribution function (4)  into the left-hand side of the Boltzmann equation (1) and compute the non-equilibrium distribution  function by considering the BGK (Marle) model (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Hence it follows  �  1 + 1  c2  �v2  2 + U  � ��∂fMJ  ∂ρ  �dρ  dt + Vi  ∂ρ  ∂xi  �  + ∂fMJ  ∂T  �dT  dt + Vi  ∂T  ∂xi  �  + ∂fMJ  ∂Vi  �dVi  dt + Vj  ∂Vi  ∂xj  �  +∂fMJ  ∂U  �dU  dt + Vi  ∂U  ∂xi  �  + ∂fMJ  ∂vi  ∂U  ∂xi  �  + 1  c2  ∂fMJ  ∂vi  �  vj  �∂Πi  ∂xj − ∂Πj  ∂xi  �  − 3vi  ∂U  ∂t + ∂Πi  ∂t  +2 ∂Φ  ∂xi − 4U ∂U  ∂xi − 4vivj  ∂U  ∂xj + v2 ∂U  ∂xi  �  = −νfNE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (7)  where d/dt = ∂/∂t + Vi∂/∂xi denotes the material time derivative and  ∂fMJ  ∂ρ  = fMJ  ρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  ∂fMJ  ∂U  = −f0  2mV2  kT c2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (8)  ∂fMJ  ∂T  = f0  T  �mV2  2kT − 3  2 + 1  c2  �15kT  16m  �  1 − mV2  kT  + m2V4  k2T 2  �  + 5m  2kT  �  2UV2 + (ViVi)2  2  +V 2V2  2  + (ViVi)V2  �  −  m2  2k2T 2  �  2UV4 + (ViVi)2V2  2  + V 2V4  2  + (ViVi)V4 + 3V6  8  ���  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (9)  ∂fMJ  ∂V i  = mf0  kT  �  Vi + 1  c2  �  4UVi  �  1 − mV2  2kT  �  − 15kT  8m Vi + (VjVj)Vi + ViV 2  �  1 − mV2  2kT  �  +(VjVj)Vi  �  1 − mV2  kT  �  + ViV2  2  �  1 − 3mV2  4kT  �  − m(VjVj)2  2kT  Vi  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (10)  ∂fMJ  ∂vi  = −mf0  kT  �  Vi + 1  c2  �  4UVi  �  1 − mV2  2kT  �  + Vi(V2 + VjVj) − 15kT  8m Vi  +Vi  �  V 2 + 2VjVj + 3V2  2  �  − mVi  kT  �(VjVj)2  2  + V 2V2  2  + (VjVj)V2 + 3V4  8  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (11)  As usual in the Chapman-Enskog method the material time derivatives are eliminated from the  non-equilibrium distribution function by using the Eulerian balance equations for the mass density ρ,  hydrodynamic velocity Vi and absolute temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The Eulerian mass density and the momentum density balance equations in the first post-Newtonian  approximation are [2, 8]  dρ  �  1 + 1  c2  �  V 2  2 + 3U  ��  dt  + ρ  �  1 + 1  c2  �V 2  2 + 3U  �� ∂Vi  ∂xi = 0,  (12)  ρdVi  dt + ∂p  ∂xi  �  1 − 1  c2  �  V 2 + 4U + ε + p  ρ  ��  − ρ ∂U  ∂xi  �  1 + 1  c2 (V 2 − 4U)  �  + ρ  c2  ��1  ρ  ∂p  ∂t − ∂U  ∂t + 4dU  dt  �  Vi − 2 ∂Φ  ∂xi − dΠi  dt + Vj  ∂Πj  ∂xi  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (13)  For the determination of the Eulerian internal energy density balance equation ρε in the first post-  Newtonian approximation one has to go to the second post-Newtonian approximation, since within  the framework of the first post-Newtonian approximation one recover only its Newtonian expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The Eulerian internal energy density balance equation reads1  dε  dt + p  ρ  ∂Vi  ∂xi + 3p  ρc2  dU  dt + pVi  ρc2  � ∂U  ∂xi − 1  ρ  ∂p  ∂xi  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (14)  From the above equation follows the expression for the material time derivative of the absolute temper-  ature, if we take into account the relationship for the specific internal energy in the first post-Newtonian  approximation which comes from the relativistic kinetic theory of gases (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' [16])  ε = 3kT  2m  �  1 + 5kT  4mc2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (15)  1 This equation corrects some misprints in [7, 8]  4  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  PARTICLE FOUR-FLOW AND ENERGY-MOMENTUM TENSOR COMPONENTS  In the relativistic kinetic theory of gases the particle four-flow N µ and the energy-momentum tensor  T µν are given in terms of the one-particle distribution function f(x, v, t) by [8, 16]  N µ = m4c  �  uµf  √−g d3u  u0  ,  T µν = m4c  �  uµuνf  √−g d3u  u0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (16)  Here uµ = pµ/m (with uµuµ = c2) denotes the gas particle four-velocity whose components in the first  post-Newtonian approximation read [2, 3, 8]  u0 = c  �  1 + 1  c2  �v2  2 + U  ��  ,  ui = vi  u0  c ,  (17)  where v is the particle three-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Furthermore, √−g d3u/u0 is an invariant integration element  whose first post-Newtonian approximation was determined in [9] and is given by  √−g d3u  u0  =  �  1 + 1  c2  �  2v2 + 6U  �� d3v  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (18)  Once the one-particle distribution function f = fMJ + fNE and the invariant integration element  are known, one can determine the components of the particle four-flow N µ and energy-momentum  tensor T µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Indeed, if we insert (4), (7), (17) and (18) into (16) and integrate the resulting equations  we get  N 0 = ρc  m  �  1 + 1  c2  �V 2  2 + U  ��  ,  N i = N 0 Vi  c ,  (19)  T 00 = ρc2  �  1 + 1  c2  �  V 2 + 3kT  2m + 2U  �  + O(c−4)  �  ,  (20)  T i0 = cρVi  �  1 + 1  c2  �  V 2 + 2U + 5kT  2m  ��  + T i0  NE,  (21)  T ij = ρViVj  �  1 + 1  c2  �  V 2 + 2U + 5kT  2m  ��  + p  �  1 − 2U  c2  �  δij + T ij  NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (22)  Note that there are no non-equilibrium contributions to the components of the particle four-flow (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The non-equilibrium contribution to T 00 is of order O(c−4) (the order of the nth inverse power of light  speed is denoted by O(c−n)) while the non-equilibrium contributions to the energy-momentum tensor  components T 0i  NE and T ij  NE are associate with terms related with the collision frequency ν and read  T i0  NE = −p  νc  � 5k  2m  ∂T  ∂xi + ∆ijklVj  ∂Vk  ∂xl  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (23)  T ij  NE = − p  ν  ��  1 + 1  c2  �5kT  2m − U + V 2  2  ��  ∆ijklVj  ∂Vk  ∂xl + 1  c2 ∆ijklVk  � ∂U  ∂xl − 1  ρ  ∂p  ∂xl  �  − 2  3c2 ViVj  ∂Vk  ∂xk + 1  c2  �  Vj  ∂  ∂xi + Vi  ∂  ∂xj  ��5kT  2m + V 2  2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (24)  Here we have introduced the fourth-order tensor  ∆ijkl = δikδjl + δilδjk − 2  3δijδkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (25)  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  LINEARIZED FIELD EQUATIONS  The thermodynamic theory of a single relativistic fluid is described by the fields of particle four-flow  N µ and energy-momentum tensor T µν where their hydrodynamic equations follow from the conserva-  tion laws  N µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='µ = ∂N µ  ∂xµ + ΓµµσN σ = 0,  T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='ν = ∂T µν  ∂xν + ΓµνσT σν + ΓννσT µσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (26)  5  Above the semicolon refers to the covariant derivative and Γµνσ to the Christoffel symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  From the knowledge of the expressions of the particle four-flow and energy momentum tensor com-  ponents (19) – (24) and the conservation laws (26) one can obtain the field equations for the particle  number density, momentum density and specific internal energy for a viscous and heat conducting  fluid in the first post-Newtonian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Here we are interested in determining the linearized field equations and for that end we consider  a background state of constant values for the mass density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' absolute temperature and Newtonian  gravitational potential denoted by ρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' T0 and U0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' which are superposed by linear perturbed  fields denoted by ρ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' U1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' V 1  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Π1  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' namely  ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t) = ρ0 + ρ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  T (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t) = T0 + T1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t) = U0 + U1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (27)  Vi(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t) = V 1  i (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t) = Φ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Πi(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t) = Π1  i (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (28)  From the insertion of (19) into (26)1 follows the linearized field equation for the mass density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' by tak-  ing into account the expressions of the Christoffel symbols in the first post-Newtonian approximation  – which can be found in [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 8] – and of the representations (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' yielding  ∂ρ1  ∂t + ρ0  ∂V 1  i  ∂xi + 3ρ0  c2  ∂U1  ∂t = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (29)  The linearized field equations for the mass-energy and momentum densities are obtained from the  time and spatial components of (26)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' by considering the expressions (19) – (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' the  representations (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' (28) and the Christoffel symbols in the first post-Newtonian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Hence it follows  ∂ρ1  ∂t + ρ0  �  1 + kT0  mc2  � ∂V 1  i  ∂xi + ρ0  c2  �3kT0  2m  ∂T1  ∂t + 3∂U1  ∂t  �  − 5k2ρ0T0  2m2c2ν0  ∂2T1  ∂xi∂xi = 0,  (30)  ρ0  ∂V 1  i  ∂t + k  m  �  1 − 1  c2  �5kT0  2m + 4U0  �� �  T0  ∂ρ1  ∂xi + ρ0  ∂T1  ∂xi  �  − ρ0  �  1 − 4U0  c2  � ∂U1  ∂xi  − 5k2ρ0T0  2m2c2ν0  ∂2T1  ∂t∂xi − kρ0T0  mν0  �  1 − 3U0  c2  � �  ∂2V 1  i  ∂xj∂xj + 1  3  ∂2V 1  j  ∂xj∂xi  �  − ρ0  c2  �  2∂Φ1  ∂xi + ∂Π1  i  ∂t  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (31)  Since the constant values of the background state does not satisfy the Poisson equations (2) and (3)  it is usual to take into account the ”Jeans swindle” (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' [18–20]) which requires that the Poisson  equations are valid only for the perturbed fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Hence, by considering that ε = 3kT/2m = 3p/2ρ,  the linearized Poisson equations become  ∇2U1 = −4πGρ1,  ∇2Φ1 = −4πGρ1  �  U0 + 9k  4mT0  �  − 4πGρ0  �  U1 + 9k  4mT1  �  ,  (32)  ∇2Π1  i = −16πGρ0V 1  i + ∂2U1  ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (33)  Let us find a solution of the coupled system of partial differential equations (29) – (33) in terms of  a plane wave representation of the perturbed fields, namely  ρ1(x, t) = ρe[i(κixi−ωt)],  T1(x, t) = Te[i(κixi−ωt)],  U1(x, t) = Ue[i(κixi−ωt)],  (34)  V 1  i (x, t) = Vie[i(κixi−ωt)],  Φ1(x, t) = Φe[i(κixi−ωt)],  Π1  i (x, t) = Πie[i(κixi−ωt)],  (35)  where κi denotes the wavenumber vector, ω the angular frequency and the overlined quantities the  small amplitudes of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  We insert the plane wave representations (34) and (35) into the coupled system of partial differential  6  equations (29) – (33) and get a linearized system of algebraic equations for the amplitudes which reads  ω∗ρ∗ − V∗ + 3U0  c2 U∗ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (36)  ω∗ρ∗ −  �  1 + 3c2  s  5c2  �  V∗ + 3U0  c2 U∗ + 9c2  s  10c2  �  ω∗ + iκ∗  ν∗  �  T∗ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (37)  �  ω∗ +  4  5ν∗  �  1 − 3U0  c2  �  iκ2  ∗  �  V∗ − 3  5κ2  ∗  �  1 − c2  s  c2  �3  2 + 4U0  c2s  ��  [ρ∗ + T∗]  +κ2  ∗  U0  c2s  �  1 − 4U0  c2  �  U∗ −  3c2  s  2c2ν∗  iω∗κ2  ∗T∗ + c2  s  c2  �  2κ2  ∗Φ∗ − ω∗Π∗  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (38)  κ2  ∗  U0  c2s  U∗ = ρ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (39)  κ2  ∗Φ∗ =  �U0  c2s  + 27  20  �  ρ∗ +  �U0  c2s  U∗ + 27  20T∗  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (40)  κ2  ∗Π∗ = 4V∗ − ω∗κ2  ∗  U0  c2s  U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (41)  Equations (38) and (41) result from the scalar product with κi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Furthermore, the above equations  were written in terms of the dimensionless quantities  κ∗  i = κi  κJ  ,  ω∗ =  ω  √4πGρ0  ,  ν∗ =  ν0  √4πGρ0  ,  (42)  ρ∗ = ρ  ρ0  ,  T∗ = T  T0  ,  V∗ = V iκi  csκJ  ,  U∗ = U  U0  ,  Φ∗ = Φ  c4s  ,  Π∗ = Πiκi  c3sκJ  ,  (43)  where κJ = √4πGρ0/cs denotes the Jeans wavelength, cs =  �  5kT0/3m the sound speed and κ∗ =  �κ∗  i κ∗  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The system of algebraic equations for the amplitudes (36) – (41) admits a non-trivial solution  if the determinant of the coefficients which correspond to the amplitudes vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Hence it follows  the dispersion relation which connect the dimensionless angular frequency ω∗ with the dimensionless  wavenumber κ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' namely  ω3  ∗ + 9i  5ν∗  �  κ2  ∗ + 4  3  �  1 − κ2  ∗  � 5  12 + U0  c2s  � c2  s  c2  ��  ω2  ∗ +  �  1 − κ2  ∗ − 4κ4  ∗  5ν∗  +  �33  10 + 2  κ2∗  +3κ2  ∗  2  − 2U0  c2s  (1 − 2κ2  ∗) − 12κ2  ∗  5ν2∗  �  1 − U0κ2  ∗  c2s  ��c2  s  c2  �  ω∗ + i  ν∗  �  κ2  ∗  �  1 − 3κ2  ∗  5  �  +  �  2 + 27κ2  ∗  10  �  1 + κ2  ∗  3  �  − 2κ2  ∗U0  c2s  �  1 − 6κ2  ∗  5  ��c2  s  c2  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (44)  Here terms up to the order O(c−2) were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  In the case of a non relativistic and collisionless Boltzmann equation we have that cs/c → 0 and  ν∗ → ∞ and we obtain from (44) Jeans solution [18]  ω∗ = ±  �  λ2  J  λ2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (45)  Above we have introduced the wavelengths λ and λJ (Jeans wavelength) through the relationship  κ∗ = κ/κJ = λJ/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' In the case of small wavelengths with respect to Jeans wavelength λJ/λ > 1 the  dimensionless angular frequency is a real quantity and the perturbations propagate as harmonic waves  in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' On the other hand, for big wavelengths λJ/λ < 1 the angular frequency becomes a pure  imaginary quantity and the perturbations will grow or decay in time, which will depend on the sign  of the solution (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The perturbations which grow in time are referred as Jeans instability, which is  associated with the gravitational collapse of self-gravitating gas clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The analysis of Jeans instability within the first and second post-Newtonian approximation by  considering the Eulerian hydrodynamic equations were investigated in [21–23] and [24], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Here if we consider a collisionless Boltzmann equation where ν∗ → ∞ (44) reduces to  ω3  ∗ +  �  1 − κ2  ∗ +  �33  10 + 2  κ2∗  + 3κ2  ∗  2  − 2U0  c2s  (1 − 2κ2  ∗)  �c2  s  c2  �  ω∗ = 0,  (46)  7  which is the dispersion relation in the first post-Newtonian approximation where dissipative effects  are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' There is a difference of this expression with the one in [8], since here the constant  value is 33/10 while there is 9/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The reason of this difference is that here we have considered the  mass, mass-energy and momentum densities hydrodynamic equations while in the former work only  the mass and momentum densities hydrodynamic equations were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  For big wavelengths with respect to Jeans wavelength λJ/λ < 1 three different values associated with  the dimensionless angular frequencies can be obtained from (44) which correspond to the growth/decay  of the perturbations:  ω∗ = − i  ν∗  λ2  J  λ2  �  1 − 7c2  s  5c2  �  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' ,  (47)  ω∗ = i  �  1 − 1  2  λ2  J  λ2  �  1 +  4  5ν∗  �  +  �43  20 − U0  c2s  + λ2  λ2  J  −  6  5ν∗  � c2  s  c2  �  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' ,  (48)  ω∗ = −i  �  1 − 1  2  λ2  J  λ2  �  1 −  4  5ν∗  �  +  �43  20 − U0  c2s  + λ2  λ2  J  +  6  5ν∗  � c2  s  c2  �  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (49)  On the other hand, if we expand the dimensionless wavenumber in power series of the reduced  angular frequency κ∗ = a0 + a1ω∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' we get from the dispersion relation (44) the solution where  the perturbations propagate as harmonic waves  κ∗ =  �  5  3  �  1 +  �27  10 + U  � c2  s  c2  �  + 2i  3ν∗  �  5  3  �  1 + 3ν2  ∗  10 +  �24  5 + 2U − ν2  ∗  �3U  10 + 36  25  �� c2  s  c2  �  ω∗+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' (50)  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  CONSTITUTIVE EQUATIONS  As was previously said the thermodynamic theory of a single relativistic fluid is characterized by  the fields of particle four-flow N µ and energy-momentum tensor T µν whose hydrodynamic equations  are the conservation laws (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  The representation of the particle four-flow and energy-momentum tensor in terms of non-relativistic  quantities makes use of the four-velocity U µ –where U µUµ = c2 – and of the projector ∆µν =  gµν − U µU ν/c2 – where gµν denotes the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The projector has the properties ∆µνUν = 0,  ∆µν∆νσ = ∆µσ and in a local Minkowski rest frame where U µ = (c, 0) it reduces to ∆µν =  diag(0, −1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Two representations for the particle four-flow and energy-momentum tensor in terms of non-  relativistic quantities are the Eckart [17] and the Landau-Lifshitz [25] decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Here we shall  use the Eckart decomposition where the particle four-flow and energy-momentum tensor are written  as  N µ = nU µ,  (51)  T µν = p⟨µν⟩ − (p + ̟) ∆µν + ǫ  c2 U µU ν + 1  c2  �  U µq(ν) + U νq(µ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (52)  Above n is the particle number density, p the hydrostatic pressure, ̟ the non-equilibrium pressure,  p⟨µν⟩ the pressure deviator, q(µ) the heat flux and ǫ the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The energy density is a sum  of two terms one related with the internal energy density ρε while the other with the mass density ρ,  namely ǫ = ρc2(1 + ε/c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The following projections of the particle four-flow and energy-momentum  tensor define the non-relativistic quantities (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='g [16]):  n = 1  c2 N µUµ,  ǫ = 1  c2 UµT µνUν,  (p + ̟) = −1  3∆µνT µν  (53)  p⟨µν⟩ =  �  ∆µ  σ∆ν  τ − 1  3∆µν∆στ  �  T στ,  q(µ) = ∆µ  νUσT νσ,  (54)  In the first post-Newtonian approximation the components of the four-velocity read [2, 3, 8]  U 0 = c  �  1 + 1  c2  �V 2  2 + U  ��  ,  U i = ViU 0  c  ,  (55)  where V denotes the hydrodynamic three velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  8  From the knowledge of the components of the metric tensor in the first post-Newtonian approxima-  tion  g00 = 1 − 2U  c2 + 2  c4  �  U 2 − 2Φ  �  ,  g0i = Πi  c3 ,  gij = −  �  1 + 2U  c2  �  δij,  (56)  and of the four-velocity components (55) we can determine the components of the projector, which  read  ∆00 = −V 2  c2 − 1  c4  �  6UV 2 + V 4 − 2ΠiVi  �  ,  ∆0i = −Vi  c − 1  c3  �  2UVi + V 2Vi − Πi  �  ,  (57)  ∆ij = −  �  1 − 2U  c2  �  δij − ViVj  c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (58)  Now we introduce the non-relativistic pressure deviator  pij = pij − pkkδij/3  whit  δijpij = 0,  (59)  so that the components of the pressure deviator p⟨µν⟩ become [12]  p⟨ij⟩ = pij + 1  2c2 (pikVkVj + pjkVkVi) ,  (60)  p⟨00⟩ = pij  ViVj  c2 ,  p⟨0i⟩ = pij  Vj  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (61)  In terms of the non-relativistic heat flux vector qi the components of the heat flux q(µ) are  q(i) = qi,  q(0) = qi  Vi  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (62)  In the five field thermodynamic theory – where the basic fields are the mass density, momentum  density and internal energy density – the pressure deviator, the dynamic pressure and the heat flux  vector are given by constitutive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Here we can obtain the desired constitutive equations  from the components of the energy-momentum tensor (19) – (24) combined with the decomposition  expressions (53) and (54) and the components of the projection (57) and (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Hence it follows the  constitutive equations for the non-relativistic heat flux vector and pressure deviator  qi = − 5kp  2mν  �  1 − c2  s  c2  U  c2s  � ∂T  ∂xi +  p  νc2 ∆ijkl  ∂Vk  ∂xl  ��5kT  2m + 3U + V 2  2  �  Vj − Πj  �  + p  νc2 (V 2δij − ViVj)  �  Vk  ∂Vk  ∂xj − ∂T  ∂xj  �  +  p  νc2  �  V 2δij + ViVj  3  �� ∂U  ∂xj − 1  ρ  ∂p  ∂xj  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (63)  pij = − p  ν  �  1 + c2  s  c2  �3  2 − U  c2s  ��  ∆ijkl  ∂Vk  ∂xl +  2p  3νc2  ∂Vk  ∂xk  �  ViVj − 1  3V 2δij  �  − p  νc2 ∆ijkl  �1  2  ∂V 2Vk  ∂xl  + Vk  � ∂U  ∂xl − 1  ρ  ∂p  ∂xl  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (64)  The constitutive equation for the dynamic pressure ̟ does not show up in the first post-Newtonian  approximation and it is known that in the kinetic theory of relativistic gases the coefficient of bulk  viscosity – which relates the dynamic pressure with the velocity divergent – is of order O(c−4) (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Let us fix our attention in the underlined linearized terms in (63) and (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Without the relativistic  corrections they reduce to the non-relativistic constitutive equations of a viscous and heat conducting  gas, namely  qi = − 5kp  2mν  ∂T  ∂xi ,  pij = − p  ν ∆ijkl  ∂Vk  ∂xl ,  (65)  where the thermal conductivity λ and the shear viscosity µ coefficients are those of the non-relativistic  BGK model  λ = 5kp  2mν ,  µ = p  ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (66)  9  With the first post-Newtonian correction these coefficients read  λ = 5kp  2mν  �  1 − c2  s  c2  U  c2s  �  ,  µ = p  ν  �  1 + c2  s  c2  �3  2 − U  c2s  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  (67)  We note that the coefficients of shear viscosity and thermal conductivity do depend on the Newtonian  gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' On the basis of a non-relativistic kinetic theory the influence the gravity on the  thermal coefficient was first reported in [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Within the framework of a relativistic kinetic theory  the transport coefficients of shear viscosity, thermal conductivity and bulk viscosity were obtained by  considering a Schwarzschild metric in [28] and the diffusion coefficient in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  CONCLUSIONS  In this work we have examined a relaxation-time model for the post-Newtonian Boltzmann equation  and determined the non-equilibrium distribution function by using the Chapman-Enskog method and  the equilibrium post-Newtonian Maxwell-J¨uttner distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The components of the energy-  momentum tensor were calculated by using the equilibrium and non-equilibrium distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  From the conservation laws of the particle four-flow and energy-momentum tensor the linearized field  equations for the mass, momentum and internal energy densities were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  A plane wave  solution of these linearized field equations coupled with the three post-Newtonian Poisson equations was  found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' By using the Eckart decomposition of the energy-momentum tensor the constitutive equations  for the viscous stress and heat flux vector were obtained and it was shown that the transport coefficients  of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq),  grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 304054/2019-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Einstein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Infeld and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Hoffmann, The gravitational equations and the problem of motion, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' of  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 39, 65 (1938).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Chandrasekhar, The post-Newtonian equations of hydrodynamics in general relativity, Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 142, 1488  (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Weinberg, Gravitation and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Principles and applications of the theory of relativity (Wiley, New  York, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Chandrasekhar and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Nutku, The second post-Newtonian equations of hydrodynamics in general  relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 158, 55 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Ag´on, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Pedraza and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Ramos-Caro, Kinetic theory of collisionless self-gravitating gases: Post-  Newtonian polytropes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' D 83, 123007 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Rezania and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Sobouti, Liouville’s equation in post Newtonian approximation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Static solutions,  Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 354, 1110 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, Post-Newtonian kinetic theory, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 426, 168400 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, Post-Newtonian hydrodynamics: theory and applications, (Cambridge Scholars Publishing,  Newcastle upon Tyne, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Richarte and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Weber, Self-gravitating systems of ideal gases in the 1PN approxi-  mation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' D 93, 064073 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Greenberg, The post-Newtonian equations of hydrodynamics for a thermally conducting, viscous,  compressible fluid in general relativity, Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 164, 569 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Hwang and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Noh, Special relativistic hydrodynamics with gravitation, Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 833, 180 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, Post-Newtonian non-equilibrium kinetic theory, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 441, 168865 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Chapman and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Cowling, The mathematical theory of non-uniform gases 3rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' (Cambridge University  Press, Cambridge, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, An introduction to the Boltzmann equation and transport processes in gases (Springer,  Berlin, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Marle, Mod`ele cin´etique pour l’´etablissement des lois de la conduction de la chaleur et de la viscosit´e  en th´eorie de la relativit´e, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Paris 260, 6539 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  10  [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Cercignani and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, The relativistic Boltzmann equation:  theory and applications  (Birkh¨auser, Basel, 2002)  [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Eckart, The thermodynamics of irreversible processes, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Relativistic theory of a simple fluid, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 58, 919 (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Jeans, The stability of a spherical nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' A, 199, 1 (1902).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Coles and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Lucchin, Cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' The origin and evolution of cosmic structures, 2nd, edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' (John Wiley,  Chichester, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Binney and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Tremaine, Galactic Dynamics, 2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' (Princeton University Press, Princeton, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Nazari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kazemi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Roshan and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Abbassi, Post-Newtonian Jeans analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 839, 75 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Noh and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Hwang, Gravitomagnetic instabilities of relativistic magnetohydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 906,  22 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, Jeans instability from post-Newtonian Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' C 81, 927  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [24] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, Plane wave analysis of the second post-Newtonian hydrodynamic equations, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  Methods Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 2350039 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Lifshitz, Fluid mechanics, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' (Pergamon Press, Oxford, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Doi T, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Santos and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Tij M, Numerical study of the influence of gravity on the heat conductivity on  the basis of kinetic theory Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Fluids 11, 3553 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Tij, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Garz´o and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Santos, On the influence of gravity on the thermal conductivity, in Rarefied Gas  Dynamics, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Brun , R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Campargue, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Gatignol and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Lengrand , eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' 1999 (Toulouse: C´epadu`es) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  239  [28] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, Relativistic gas in a Schwarzschild metric, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' P04016 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content='  [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}
+page_content=' Kremer, Diffusion of relativistic gas mixtures in gravitational field, Physica A 393 76 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQfEwyK/content/2301.04880v1.pdf'}

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+Collective Vortical Motion and Vorticity Reversals
+of Self-Propelled Particles on Circularly Patterned Substrates∗
+Haosheng Wen,1, 2 Yu Zhu,1 Chenhui Peng,1, 3 P.B. Sunil Kumar,4, 5 and Mohamed Laradji1, †
+1Department of Physics and Materials Science, The University of Memphis, Memphis, TN 38152, USA
+2Biophysics Graduate Program, The Ohio State University, Columbus, OH 43210, USA
+3Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
+4Department of Physics, Indian Institute of Technology Palakkad, Palakkad 668557, Kerala, India
+5Department of Physics, Indian Institute of Technology Madras, Chennai 600036, Tamil Nadu, India
+The collective behavior of self-propelled particles (SPPs) under the combined effects of a circu-
+larly patterned substrate and circular confinement is investigated through coarse-grained molecular
+dynamics simulations of polarized and disjoint ring polymers.
+The study is performed over a wide
+range of values of the SPPs packing fraction ¯φ, motility force FD, and area fraction of the patterned
+region. At low packing fractions, the SPPs are excluded from the system’s center and exhibit a
+vortical motion that is dominated by the substrate at intermediate values of FD.
+This exclusion
+zone is due to the coupling between the driving force and torque induced by the substrate, which
+induces an outward spiral motion of the SPPs.
+For high values of FD, the SPPs exclusion from the
+center is dominated by the confining boundary. At high values of ¯φ, the substrate pattern leads to
+reversals in the vorticity, which become quasi-periodic with increasing ¯φ.
+We also found that the
+substrate pattern is able to separate SPPs based on their motilities.
+I.
+INTRODUCTION
+Active matter systems, which are collections of individ-
+ual self-driven units that consume energy from the envi-
+ronment to move, have been the subject of a significant
+amount of research over the last few decades [1–3]. Active
+matter systems range widely from macroscopic systems,
+including schools of fish [4], flocks of birds [5], and granu-
+lar media [6–8], to microscopic systems including colonies
+of bacteria, eukaryotic cells [9–12], actin filaments and
+microtubules that are propelled by their respective mo-
+tor proteins [13], and active colloids [14]. Active matter
+systems often exhibit intriguing collective behavior char-
+acterized by clustering of the units and large-scale col-
+lective motion [15]. This collective behavior is used, for
+example, in bacteria colonies to reduce competition for
+nutrients, accelerate growth of the colony, or to increase
+resilience in hostile environments [16]. Likewise, the col-
+lective behavior of assemblies of eukaryotic cells, such as
+epithelial monolayers and cancer cells, has physiological
+and pathological implications. These include embryoge-
+nesis, wound healing and tumor metastasis [17–19].
+Many studies have shown that clustering and collec-
+tive motion of self-propelled particles (SPPs) are influ-
+enced by various physical factors, including the packing
+fraction of the SPPs, nature of the coupling between
+neighboring SPPs, and the type of motion of a single
+SPP [20, 21]. Other physical factors include environmen-
+tal constraints [3] such as anisotropy of the embedding
+fluid [22, 23], geometric confinement [24–36] and obsta-
+cles [37, 38]. An interesting effect of circular confinement,
+for example, is an induced vortical motion of the SPPs
+∗ Physical Review E (in press)
+† Corresponding author. mlaradji@memphis.edu
+that is concentric with the boundary [26–28].
+While in most studies of SPPs’ collective behavior, the
+substrate is non-patterned, the effect of patterned sub-
+strates on SPPs collective behavior has recently been in-
+vestigated in a few studies. For example, it was shown
+that patterning the substrate, into periodic linear fur-
+rows, aligns Pseudomonas aeruginosa along the furrows
+while greatly supresses their migration across them [39].
+Likewise, collective migration of epithelial cells is sub-
+stantially promoted by linear grooves of patterned sub-
+strates [40, 41]. However, computational investigations
+of the effect of patterned substrates on SPPs collective
+behavior are lacking. In this article, we address the effect
+of substrates, which are partially circularly patterned, on
+the collective behavior of soft SPPs with the ability to
+switch their polarity. In particular, we investigate how
+a circular confinement that is concentric with the sub-
+strate’s pattern further influences their collective motion.
+II.
+MODEL AND METHOD
+We consider a total number of P SPPs in two dimen-
+sions, each modeled as a semi-flexible ring polymer com-
+posed of N beads in a good solvent.
+This model was
+recently introduced by us to investigate SPPs collective
+behavior on a non-patterned substrate [42] and is a gen-
+eralization of an earlier model for strongly adsorbed dis-
+joint ring polymers [43].
+The potential energy of the
+arXiv:2301.11239v1  [cond-mat.soft]  26 Jan 2023
+
+2
+SPPs is given by
+Unet=
+P
+�
+l=1
+� N
+�
+i=1
+Ubond
+�
+r(l)
+i,i+1
+�
++
+N
+�
+i=1
+Ubend(r(l)
+i−1, r(l)
+i , r(l)
+i+1)
++
+N
+�
+i=1
+Uwall
+�
+r(l)
+i
+�
++ Uarea
+�
+{r(l)
+i }
+�
++ Usub
+�
+r(l)
+p1 , r(l)
+p2
+� �
++
+�
+l1,l2
+�
+i,j
+Urep
+�
+|r(l1)
+i
+− r(l2)
+j
+|
+�
+,
+(1)
+where r(l)
+i
+is the coordinate of bead i belonging to SPP
+l and r(l)
+i
+= |r(l)
+i |. The lth SPP has two symmetrically
+positioned poles with indices
+p1 = 1 and p2 = N/2 +
+1. Ubond is a harmonic potential ensuring connectivity
+between consecutive beads within an SPP and is given
+by
+Ubond
+�
+r(l)
+i,i+1
+�
+= 1
+2k
+�
+r(l)
+i,i+1 − rb
+�2
+,
+(2)
+where k is the spring constant, r(l)
+i,i+1 = |r(l)
+i+1−r(l)
+i | and rb
+is the preferred bond length.
+In Eq. (2), r(l)
+N,N+1 = r(l)
+N,1.
+The semi-flexibility of an SPP’s boundary is maintained
+through a three-body interaction
+Ubend(r(l)
+i−1, r(l)
+i , r(l)
+i+1) = κ
+�
+1 − cos θ(l)
+i
+�
+,
+(3)
+where κ is the bending stiffness of the polymers and
+cos θ(l)
+i
+= r(l)
+i−1,i·r(l)
+i+1,i/r(l)
+i−1,ir(l)
+i+1,i.
+In Eq. (3), r(l)
+0
+= r(l)
+N
+and r(l)
+N+1 = r(l)
+1 . Eq. (3) implies that the preferred bend-
+ing angle of a triplet is 180◦. To account for the polariza-
+tion of the SPPs, triplets of beads centered at the pole
+beads with indices p1 and p2 have a preferred bending
+angle θp ≤ 180◦. Since Eq. (3) does not allow for pre-
+ferred angles different from 180◦, beads p1 and p2 are
+assigned the following slightly different three-body inter-
+action, which allows for any arbitrary splay angle θs,
+Ubend(r(l)
+p−1, r(l)
+p , r(l)
+p+1) = 1
+2κ′ �
+cos θ(l)
+p − cos θs
+�2
+,
+(4)
+where κ′ is the bending stiffness at the poles. Due to
+the softness of the potential given by Eq. (4), we found
+that achieving the same persistence length of the polymer
+with this potential as with that given by Eq. (3) requires
+κ′ ≈ 10κ.
+The disjointness of the ring polymers is maintained
+by the following fully repulsive two-body interaction be-
+tween any two non-bonded beads
+Urep (r) =
+�
+1
+2ζ
+�
+1 − r
+rc
+�2
+if r ≤ rc,
+0
+if r > rc,
+(5)
+where ζ and rc are the strength and range of the repulsive
+interaction, respectively. Finally, the area constraint of
+each SPP is maintained by the effective potential energy
+Uarea
+�
+{r(l)
+i }
+�
+= 1
+2χA0
+�
+�1 −
+A
+�
+{r(l)
+i }
+�
+A0
+�
+�
+2
+,
+(6)
+Pl
+!"
+!#
+$%&%'
+(%& + (%' /2
+,-
+,
+Non-patterned 
+substrate
+Patterned 
+substrate
+Confining 
+wall
+FIG. 1. Schematic of the system. Solid black circle of radius R
+corresponds the confining wall of the system. The yellow disk
+of radius Rp corresponds the region of the substrate that is
+patterned. The green annulus corresponds to the region of the
+substrate that is non-patterned. The mid-point of the polarity
+vector Pl of an arbitrary SPP l is at a distance (rl1 + rl2)/2,
+where rl1 and rl2 are the coordinates of the two poles (The
+origin of the coordinate system is at the center of the system).
+The effect of the patterned substrate is to reorient the SPP’s
+polarity through a torque, whose forces are indicated by the
+green vectors). In this schematic, the size of the SPP is not
+to scale with the system size and the size of the patterned
+region.
+where χ is the area-stretch modulus, A0 is the SPP’s
+preferred area, and
+A
+�
+{r(l)
+i }
+�
+is the area enclosed by
+the SPP’s boundary and depends on the coordinates of
+the beads belonging to the SPP through the shoelace
+formula,
+A
+�
+{r(l)
+i }
+�
+= 1
+2
+����
+N
+�
+i=1
+�
+x(l)
+i y(l)
+i+1 − x(l)
+i+1y(l)
+i
+� ����,
+(7)
+with x(l)
+N+1 = x(l)
+1
+and y(l)
+N+1 = y(l)
+i .
+Finally, the SPPs are confined within a circle of radius
+R by the interaction potential
+Uwall (r) =
+�
+εwall (r − R + a)n /an if R − a ≤ r < R,
+0
+if
+r < R − a,
+(8)
+where εwall and a are the strength and range of this
+interaction, respectively.
+We choose n = 4 since this
+value is large enough to prevent the SPPs from crossing
+the circular confining wall. The main difference between
+this model and prior models for the collective behavior
+of elongated self-propelled particles is that the present
+model accounts for the elongation of the self-propelled
+particles and their flexibility.
+This is in contrast with
+previous studies wherein particles are either rigid [44–46]
+or deformable with high aspect ratio and with practically
+no account for the enclosed volume of the particles [47].
+We consider the case where a region of the substrate
+is circularly patterned. Experimentally, this would cor-
+
+3
+respond, for example, to a substrate that is circularly
+grooved [40, 41].
+The effect of the substrate’s pattern
+on an SPP is to align it along the local direction of the
+pattern. This is achieved by a simple effective potential
+energy between the SPP’s poles that produces a torque
+on the SPP,
+Usub
+�
+r(l)
+p1 , r(l)
+p2
+�
+= ks
+2 sin2 ϕl,
+(9)
+where ks is the strength of the interaction and ϕl is the
+angle between the polarity Pl = r(l)
+p2 − r(l)
+p1 and the local
+tangent to a circle of radius (r(l)
+p1 +r(l)
+p2 )/2 centered at the
+origin. This torque tends to align an SPP’s polarity with
+the local tangent of a circle centered at the origin and
+passing by the mid-point of the two poles, as schemati-
+cally shown by Fig. 1. We focus on the case where the
+substrate is patterned only within the region (r ≤ Rp).
+Otherwise, the substrate is uniform (non-patterned) for
+Rp < r ≤ R.
+Each SPP is propelled by a motility force of magnitude
+FD, along its polarity, that is given by
+fl(t) = FD (Pl(t)/Pl(t)) g (¯vl(t), Pl(t)) ,
+(10)
+where g(A, B) = +1 or -1 if A·B > 0 or < 0, respectively,
+and where ¯vl(t) is the SPP’s average velocity over the
+time interval [t − τm, t], i.e.
+¯vl(t) = 1
+τm
+� t
+t−τm
+vl(t′)dt′,
+(11)
+with vl(t) = (1/N) �N
+i=1 v(l)
+i (t).
+In Eq. (11), we take
+τm = τ where τ = rb
+�
+µ/ε, rb is the preferred bond
+length, ε is the energy scale and µ is the bead’s mass.
+Beads are moved according to a molecular dynamics
+scheme,
+˙r(l)
+i (t) = v(l)
+i (t), and
+µ ˙v(l)
+i (t) = −∇(l)
+i Unet + fl(t)
+N
+− Γv(l)
+i (t)
++Γ
+√
+2DΞ(l)
+i (t),
+(12)
+where ∇(l)
+i
+= (∂x(l)
+i , ∂y(l)
+i , ∂z(l)
+i ) and v(l)
+i
+is the instanta-
+neous velocity of bead i belonging to SPP l. In Eq. (12),
+Γ is the friction coefficient, D is the diffusion coefficient
+of the beads in the ideal limit (i.e. in the absence of inter-
+actions and beads connectivity), and Ξ(l)
+i (t) is a random
+vector that has zero-mean and is δ-correlated for the same
+particle and same component, i.e. Ξ(l)
+i (t) satisfies
+⟨Ξ(l)
+i (t)⟩ = 0,
+⟨Ξ(l1)
+i,α (t) Ξ(l2)
+j,β (t′)⟩ = δl1l2δijδαβδ (t − t′) ,
+(13)
+where α, β = x or y, δnm is the Kronecker delta, and
+δ(t) is the Dirac delta-function.
+The equations of motion are integrated using the
+velocity-Verlet algorithm with a time step ∆t = 0.01τ.
+The numerical value of a component of the random force
+is given by
+Ξ(l)
+i,α =
+� 3
+∆t
+�1/2
+λ(l)
+i,α,
+(14)
+where λ(l)
+i,α is a random number generated from a uni-
+form distribution in the interval [−1, 1].
+Each SPP is
+composed of N = 40 beads. The values of the param-
+eters of the model SPPs,
+which are kept fixed in the
+present study, are given by
+k = 100ε/r2
+b, κ = 100ε, κ′ = 1000ε, θs = 120◦, ζ = 50ε,
+rc = rb, χ = 1ε/r2
+b, A0 = 100r2
+b, τm = τ,
+D = 1.0r2
+b/τ, and Γ = 1.0µ/τ.
+(15)
+III.
+RESULTS
+A.
+Effects of Patterned Substrate and Motility
+Force on SPPs’ Collective Behavior
+We first focus on the combined effect of the patterned
+substrate and circular confining wall on the SPPs col-
+lective behavior at an average packing fraction ¯φ =
+PA0/πR2 = 0.398 with R = 200rb. This corresponds
+to P = 500.
+Steady-state snapshot (a) in Fig. 2(A)
+and Movie 1, at FD = 20ε/rb and non-patterned sub-
+strate (ks = 0), indicate a small amount of clustering
+and a weak collective motion, in agreement with prior
+results [42]. Fig. 2(C) shows that at these conditions, the
+radial distribution of the SPPs packing fraction, φ(r), is
+almost uniform. As FD is increased to 24ε/rb at ks = 0,
+the motility force drives many SPPs to the boundary
+leading to their accumulation as shown by snapshot (b)
+in Fig. 2(A) and collective unidirectional vortical motion
+(see Movie 2). This is also demonstrated by the time de-
+pendence of the average tangential velocity of the SPPs
+in an annulus of thickness 10rb near the boundary (red
+graph in Fig. 2(B) at ks = 0 and FD = 24ε/rb). In con-
+trast, the SPPs motion in an annulus close to the center
+is fairly turbulent (blue graph in Fig. 2(B) at ks = 0 and
+FD = 24ε/rb). SPPs accumulation at the boundary is
+due to the asymmetry between the effect of the motil-
+ity force, which drives the SPPs toward the boundary,
+and thermal effects, which drive the SPPs away from the
+boundary, and has been observed in earlier studies [3].
+In contrast, although the SPPs that are away from the
+boundary move collectively in clusters, they do not ex-
+hibit a net vortical motion, as demonstrated by the fluc-
+tuations around 0 of the average tangential velocity of
+the SPPs in the annulus close to the center (blue graph
+in Fig. 2(B) at ks = 0 and FD = 24ε/rb).
+Interaction between the SPPs and the patterned sub-
+strate leads to a much richer dynamical behavior. Snap-
+shots (c) and (d) in Fig. 2(A) and their corresponding
+tangential velocities vs.
+time in Fig. 2(B) show that,
+at ks = 100ε and FD = 18 or 20ε/rb, the patterned
+
+4
+0
+50
+100
+150
+200
+0.00
+0.25
+0.50
+0.75
+1.00
+(A)
+!" = 0, &'= 20)/+,
+!" = 100), &'= 20)/+,
+!" = 100), &'= 24)/+,
++ [+,]
+!" = 0, &'= 24)/+,
+!" = 100), &'= 22)/+,
+(C)
+!" = 100), &'= 18)/+,
+_.
+2[3]
+45 +,/3
+b
+!" = 0, &' = 24)/+,
+c
+!"= 100), &' = 18 )/+,
+d
+!" = 100), &' = 20 )/+,
+(B)
+e
+!" = 100), &' = 22)/+,
+a
+!" = 0, &' = 20)/+,
+; +
+&' = 24)/+,
+f !" = 100)
+(a)
+(b)
+(f)
+(d)
+(e)
+(c)
+FIG. 2.
+Panel (A): Steady-state snapshots at (a) FD = 20ε/rb and ks = 0, (b) FD = 24ε/rb and ks = 0, (c) FD = 18ε/rb and
+ks = 100ε, (d) FD = 20ε/rb and ks = 100ε, (e) FD = 22ε/rb and ks = 100ε, and (f) FD = 24ε/rb and ks = 100ε. Panel B: Time
+dependence of the average tangential velocity for different values of ks and FD corresponding to those in Panel (A). The blue
+(red) graphs correspond to SPPs in the blue (red) annulus, shown in snapshot (A). Shaded yellow (green) region corresponds to
+the regime where the vortices in the patterned and non-patterned regions are in same (opposite) directions. Panel (C): Radial
+profiles of the packing fraction, ¯φ at values of FD and ks corresponding to those in Panel (A). All data shown in this figure are
+at ¯φ = 0.398, Rp = 100rb and R = 200rb.
+substrate and the driving force collectively lead to (1)
+a tangential alignment of the SPPs in the patterned re-
+gion, (2) their accumulation at the periphery of the pat-
+terned region, and (3) their exclusion from the center. At
+ks = 100ε and FD = 20ε/rb, Fig. 2(B) and Movie 3 show
+that the SPPs move as a vortex, in the patterned region of
+the substrate, with very few reversals in its direction. In
+contrast, the SPPs outside the patterned region exhibit
+a weak collective behavior, as demonstrated by the fact
+that the SPPs’ average tangential velocity in this region
+fluctuates around 0 (red graph in Fig. 2(B) at ks = 100ε
+and FD = 20ε/rb).
+As FD is further increased to FD = 22 or 24ε/rb, at
+ks = 100ε, the corresponding snapshots (d) or (e), re-
+spectively, shown in Fig. 2(A), show that more SPPs are
+driven to the confining wall. This is also demonstrated by
+increased packing fraction next to the boundary at these
+values of FD in Fig. 2(C). Fig. 2(B) shows that, at these
+values of FD, the SPPs exhibit collective vortical motion
+in both patterned and non-patterned regions. These vor-
+tices can move either in the same direction (shaded yellow
+regions in Fig. 2(B) and Movie 4) or opposite directions
+(shaded green regions and Movie 5) with frequent rever-
+sals. Inspection of the vorticity reversals indicates that
+they are due to collectively moving clusters in the non-
+patterned region, which collide with the vortices in the
+patterned region or in the boundary layer.
+The SPPs collectivity is quantified through the vortical
+order parameter defined as
+Sv = ⟨|
+P
+�
+l=1
+σl|⟩/P,
+(16)
+where σl = +1 (-1) if the direction of the tangential ve-
+locity of SPP l is clockwise (counter-clockwise). Fig. 3,
+which depicts Sv vs. FD at ¯φ = 0.398, shows that the
+substrate pattern shifts the onset of vortical collective
+motion to smaller values of FD. Four distinct regimes
+in the case of ks = 100ε are identified.
+In regime I
+(FD ≲ 16ε/rb), there is no collective motion. In regime
+II (16ε/rb ≲ FD ≲ 21ε/rb), the collective behavior is
+dominated by the patterned region, and is characterized
+by an almost unidirectional vortical motion. Fig. 2(C)
+shows that regime II is also characterized by an in-
+crease in the maximum of the SPPs packing fraction
+in the patterned region with increasing FD. In regime
+III (21ε/rb ≲ FD ≲ 25ε/rb), both patterned substrate
+and confining wall independently promote SPPs collec-
+10
+15
+20
+25
+30
+35
+0
+0.2
+0.4
+0.6
+0.8
+1
+!"
+#$ = 0
+#$ = 100( (opposite vorticities)
+#$ = 100( (same vorticities)
+I
+II
+III
+IV
+67 (/9:
+0
+50
+100 150
+0
+0.1
+0.2
+0.3
+0.4
+!"
+#$ (
+67 = 20(/9:
+FIG. 3.
+SV vs. FD at ¯φ = 0.398, Rp = 100rb and R = 200rb
+for ks = 0 (red circles) and ks = 100ε (blue circles).
+Full
+(open) blue circles correspond to Sv at ks = 100ε where the
+vortices in the patterned and non-patterned regions have same
+(opposite) directions.(Inset) Sv vs. ks at FD = 20ε/rb. The
+solid lines are simply guides to the eye.
+
+0.4
+0
+-0.4
+0.4
+0
+-0.4
+0.4
+0
+-0.4
+0.4
+0
+-0.4
+0.4
+0
+-0.4
+0.4
+0
+-0.4
+20000
+25000
+30000
+35000
+40000800
+Q05
+0
+50
+100
+150
+200
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+50
+100
+150
+200
+0.00
+0.25
+0.50
+0.75
+1.00
+10
+15
+20
+25
+30
+35
+0
+0.2
+0.4
+0.6
+0.8
+1
+ 
+ 
+ 
+ 
+  
+1 pt 
+                                
+                                    20 
+                                    32 
+ 
+ 
+ 
+(A)
+Circular boundary
+Periodic boundary 
+conditions
+! "
+" ["$]
+(C)
+0
+50
+100
+150
+200
+0.00
+0.02
+0.04
+0.06
+0.08
+" ["$]
+(B)
+&' "
+" ["$]
+(D)
+125"$
+150"$
+175"$
+25"$
+50"$
+75"$
+100"$
+FIG. 4. (A) trajectories of a single SPP starting from a po-
+sition near the center, for differemt values of FD and ks. (B)
+Radial profile of the radial velocity of the SPPs for the case
+of a circular confining wall. (B) Radial profile of the packing
+fraction for the case of a circular confining wall (solid line)
+and PBC (dashed line). Data shown in (B) and (C) are in
+the case of FD = 24ε/rb, ks = 100ε, ¯φ = 0.398, Rp = 100rb
+and R = 200rb. (D) Radial profiles of the packing fraction for
+different values of the radius of the patterned region, Rp, indi-
+cated in the legend. These data correspond to FD = 22ε/rb,
+ks = 100ε, R = 200rb and ¯φ = 0.398. The vertical dashed
+lines in (B-D) indicate the location of the boundary between
+the patterned (left) and non-patterned (right) regions of the
+substrate.
+tive motion, and lead to vortical motion in both regions
+with same or opposite directions. This results in a bi-
+furcation of Sv into two branches: one branch with high
+values of Sv (solid blue circles in Fig. 3) where the two
+vortices have same direction, and a second branch with
+low values of Sv (open blue circles in Fig. 3) where the
+two vortices have opposite directions. Regime III marks
+the beginning of the decrease in the value of the maxi-
+mum of the SPPs’ packing fraction in the patterned re-
+gions. Finally, in regime IV (FD ≳ 24ε/rb), the major-
+ity of the SPPs are accumulated near the confining wall,
+where they move as a unidirectional vortex.
+Interestingly, snapshots (c) to (f) of Fig. 2(A) and
+Fig. 2(C) demonstrate that the patterned substrate in-
+duces an exclusion zone in the center with a diameter
+that increases with FD. This is contrasted with the case
+of a non-patterned substrate, in which the radial profile
+of the packing fraction is almost uniform, except at the
+boundary. The source of this exclusion zone, is inferred
+from simulations of a single SPP (dilute regime) at finite
+values of ks and FD, starting from a location near the
+center. Fig. 4(A) (see Movie 6 as well) shows that the
+SPP’s trajectory is an outward spiral, with a number of
+turns that increases with increasing ks or decreasing FD.
+Hence, the motility force and the substrate’s pattern co-
+operatively drive the SPPs away from the patterned re-
+gion with a rate that increases with FD and decreases
+with ks, leading to an exclusion zone in the center.
+In addition to the exclusion zone in the center,
+Fig. 2(C) shows that the radial profile of the packing
+fraction exhibits a broad peak within the patterned re-
+gion, and close to the boundary between the patterned
+and non-patterned regions. The emergence of this peak
+is understood as follows. The motion of the SPPs within
+the patterned region is mainly tangential, while in the
+non-patterned region (but away from the confining wall),
+the motion is more turbulent.
+As a result vp
+⊥ < vn
+⊥,
+where vp
+⊥ and vn
+⊥ are the averages of the magnitudes of
+the radial components of the SPPs velocities in the pat-
+terned and non-patterned regions, respectively, as shown
+in Fig. 4(B). Steady state requires that the outflux of
+the SPPs from the patterned must be equal to the influx
+of the SPPs from the non-patterned regions to the pat-
+terned region, i.e. φpvp
+⊥,out = φnvn
+⊥,in, where φp (φn) is
+the packing fractions of the SPPs in the patterned (non-
+patterned) region, close the boundary between the pat-
+terned and non-patterned regions.
+vp
+⊥,out is the average
+of the radial component of the velocity of the SPPs out-
+going from the patterned region at the boundary between
+the patterned and non-patterned regions. Likewise, vn
+⊥,in
+is the average of the radial component of the velocity of
+the SPPs incoming from the non-patterned region at the
+boundary between the patterned and non-patterned re-
+gions. Therefore, mass balance between the outflux and
+influx of the SPPs across this boundary, at steady state,
+imposes φp > φn. Combined with the fact that the inter-
+play between the motility force and the torque induced
+by the patterned substrate, which leads to SPPs exclu-
+sion from the center, the argument above implies that the
+radial packing fraction profile must exhibit a peak within
+the patterned region, and close to the boundary between
+the patterned and non-patterned regions, as shown by
+Fig. 2(C). FD enhances the SPPs outflux from the pat-
+terned region, i.e. it increases vp
+⊥, while it decreases the
+influx from the non-patterned region, due to increased
+accumulation of the SPPs near the confining wall. As
+a result, the size of the exclusion zone increases with
+FD (see Fig. 2(C)). Elimination of SPPs accumulation
+at the boundary, through imposing periodic boundary
+conditions (PBC), enhances SPPs influx from the non-
+patterned region to the patterned region. This leads to
+a decrease in the size of the exclusion zone, as demon-
+strated by Fig. 4(C).
+The results thus far presented correspond to the case
+of a radius of the patterned region of the substrate,
+Rp = 100rb. To infer the effect of the size of the patterned
+region, we performed a series of simulations in the case
+of ¯φ = 0.398, FD = 22ε/rb, ks = 100ε, and R = 200rb.
+Fig. 4(D) shows the radial profile of the packing fraction
+of these systems with Rp varying between 25rb and 175rb.
+This figure demonstrates that the diameter of the deple-
+
+/r;ks = 100
+/rb; ks = 160c
+ε/rb; ks = 100c32206
+tion zone increases with Rp, which implies that the size
+depletion of the SPPs from the middle is also affected
+by the behavior of the SPPs in the non-patterned region
+of the substrate, in line with the arguments presented in
+the previous paragraph.
+B.
+Effect of SPPs’ Packing Fraction on their
+Collective Behavior on a Patterned Substrate
+We now turn to the effect of SPPs packing fraction
+on their collective motion. We consider the case where
+FD = 24ε/rb and ks = 100ε. The packing fraction is var-
+ied by changing the number of SPPs from P = 59 to 540,
+while the radius of the system is kept fixed at R = 138rb.
+Corresponding Sv vs. ¯φ, shown in Fig. 5, reveals three
+main regimes. For ¯φ ≲ 0.3, most SPPs accumulate at
+the boundary where they move as a unidirectional vor-
+tex (see Movie 7). For 0.3 ≲ ¯φ ≲ 0.8, the amount of SPPs
+is increased in the patterned region, where they move as
+a vortex with same direction as that in the boundary
+layer (see Movie 8). Fig. 5 shows that for ¯φ ≲ 0.8, Sv in-
+creases monotonically with ¯φ. Surprisingly, however, Sv
+decreases with ¯φ for ¯φ ≳ 0.8. This decrease is interest-
+ingly correlated with the disappearance of the exclusion
+zone in the center as demonstrated by the profiles of the
+packing fraction in the inset of Fig. 5. In fact, the inset
+of Fig. 5 shows that an excess of SPPs at the center is
+induced at ¯φ ≳ 0.8.
+Inspection of movies at ¯φ ≳ 0.8 reveals an emergence
+of reversals in the vorticity (demonstrated by SPPs ve-
+locities snapshots in Fig. 6(A) and by Movie 9). These
+reversals are quantified by the time dependence of vT (t),
+defined as the average of the tangential velocity of the
+SPPs in an annulus of thickness 10rb near the system’s
+boundary.
+Fig. 6(B) shows that vT is essentially con-
+stant in the case of a non-patterned substrate (ks = 0)
+at FD = 24ε/rb, indicating a unidirectional vortical mo-
+tion. At ks = 40ε and same FD, Fig. 6(B) shows that
+vT exhibits a single reversal during the time interval
+[20 000τ, 40 000τ].
+In stark contrast, however, vT ex-
+hibits many reversals at ks = 100ε and same FD dur-
+ing the same time interval. Therefore, at high packing
+fractions, the rate of vorticity reversals (i.e., number of
+reversals per unit of time), κ, increases with increasing ks
+beyond some threshold value. Likewise, Fig. 6(C) shows
+that κ increases with ¯φ for ¯φ ≳ 0.8. The decrease in Sv
+at ¯φ ≳ 0.8, shown in Fig. 5(B), is simply due to coexis-
+tence of two vortices with opposite directions during the
+reversal events, as demonstrated by a series of snapshots
+in Fig. S1 in Supplemental Information [48].
+Correlations between reversal events are inferred from
+the power spectrum F(ν), defined as the Fourier trans-
+form of the velocity autocorrelation f(t) = ⟨vT (t0 +
+t)vT (t0)⟩, where ν is frequency. Fig. 6(D) shows that,
+at ¯φ = 0.836, F(ν) is peaked at ν ≈ 0.
+This indi-
+cates that reversal events are weakly correlated at pack-
+ing fractions around this value of ¯φ.
+Fig. 6(D) shows
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+!"
+#$
+0
+25
+50
+75
+100 125
+0.6
+0.7
+0.8
+0.9
+" %
+% [%']
+" = 0.887
+" = 0.861
+" = 0.836
+" = 0.803
+" = 0.769
+" = 0.736
+FIG. 5.
+Vortical order parameter vs. packing fraction at
+ks = 100ε, FD = 24ε/rb, Rp = 100rb and R = 138rb. Vor-
+tical motion is dominated by the circular confining wall at
+low ¯φ (green region). Both circular confining wall and pat-
+terned substrate contribute to vortical motion at intermediate
+¯φ (blue region). At high ¯φ, vortical motion exhibits reversals
+(red region). Inset shows radial packing fraction profiles at
+different values of ¯φ. Steady state snapshots at different pack-
+ing fractions are shown at the top of the figure. The dashed
+circles in these snapshots indicate the boundary of the pat-
+terned region of the substrate.
+that F(ν) exhibits a well-defined peak at ¯φ = 0.887.
+Therefore, reversal events of the vorticity become inter-
+estingly quasi-periodic with increasing ¯φ. The emergence
+of quasi-periodic reversals at high densities is also demon-
+strated by the time dependence of the tangential velocity
+in Fig. 7.
+Inspection of Movie 9 shows that vorticity reversals
+always originate from the center of the system.
+This
+concurs with the fact that vorticity reversals are absent
+at low packing fractions, i.e. when the exclusion zone is
+present. To demonstrate that the geometry of the confin-
+ing wall has a weak effect on vorticity reversals, we per-
+formed a simulation on a system with a square boundary,
+of linear size Lx = 400rb, and same circular pattern with
+ks = 100ε, FD = 24ϵ/rb, ¯φ = 0.887 and Rp = 100rb,
+and found reversals in the vorticity similar to the case
+with circular boundary and with about same value of κ,
+as demonstrated by Fig. S2 [48]. Likewise, Fig. S3 [48]
+shows that systems with periodic boundary conditions,
+at same values of FD, ks, ¯φ, Rp and Lx, also exhibit vor-
+ticity reversals, albeit not as correlated as in the case of
+circular or square boundary. This is due to the fact the
+periodic boundary conditions induce more turbulent flow
+of the SPPs in the non-patterned region.
+As stated above, reversals in the vorticity are associ-
+ated with an increase in SPPs packing fraction at the
+
+7
+0
+0.01
+0.02
+0.03
+0.04
+0
+2
+4
+6
+8
+0.8
+0.85
+0.9
+0
+1
+2
+3
+4
+5
+6
+20000
+25000
+30000
+35000
+40000
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+ ks=0
+ ks=40
+ ks=100
+v (t)[rb/ ]
+t [ ]
+! = 37000&
+38000&
+39000&
+40000&
+ B
+ B
+ B
+ B
+(A)
+(B)
+(D)
+(C)
+*+
+, &-. (×10-2)
+4(5)
+5 &-.
++ = 0.836
++ = 0.887
+FIG. 6. (A) Time-sequence of velocity snapshots showing vorticity reversals at FD = 24ε/rb, ¯φ = 0.836, Rp = 100rb, R = 138rb
+and ks = 100ε. (B) Tangential velocity vT (t) vs. time at FD = 24ε/rb and ¯φ = 0.836. (C) Rate of vorticity reversals vs. ¯φ at
+ks = 100ε. (G) The Fourier transform, F(ν), of the velocity autocorrelation function f(t) = ⟨vT (t0 + t)vT (t0)⟩, vs. frequency
+at ks = 100ε at two high values of the packing fraction.
+center. This is found to also be associated with an in-
+crease in the misalignment between the SPPs polarities
+and velocities, as shown by Fig. S4 (A) [48]. This re-
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+35000
+37500
+40000
+42500
+45000
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+̅"# $ [&'/)]
+$[)]
+FIG. 7.
+Time dependence of the tangential velocity of an
+annulus of thickness 10rb near the system’s boundary for the
+case of FD = 24ε/rb, Rp = 100rb, R = 200rb and ks = 100ε.
+Top and bottom graphs correspond to ¯φ = 0.836 and 0.887,
+respectively.
+sults in a high degree of fluctuations in the average of
+the tangential velocity of the SPPs in the center as op-
+posed to those away from the center, as shown by Fig. S4
+(B) [48]. These increased fluctuations at the center leads
+some SPPs to move in a direction opposite to that of
+the vortex, and in some cases these SPPs force neighbor-
+ing SPPs to follow, leading to the observed intermittent
+vorticity reversals.
+C.
+Patterned-substrates induced segregation
+between fast and slow SPPs
+Our simulations show that at low and intermediate val-
+ues of the packing fraction, the SPPs spatial distribution
+depends on their motility force. One would therefore ex-
+pect that patterning the substrate may be used as a tool
+to spatially separate SPPs, based on their motility force.
+To verify this hypothesis, we performed a simulation of a
+binary system, at an average packing fraction ¯φ = 0.6, in
+which half of the SPPs are slow (with Fd = 20ε/rb) and
+the other half are fast (with FD = 24ε/rb). The two types
+of SPPs are otherwise identical.
+The packing fraction
+profiles of the two components and a steady-state snap-
+shot, depicted in Figs. 8(A) and (B), respectively, show
+that the fast and slow SPPs mostly segregate such that
+the fast SPPs are highly concentrated in the patterned
+region and the slow SPPs are more concentrated in the
+non-patterned region. In comparison, the two types of
+SPPs are mixed in the case where the substrate is fully
+
+8
+0
+25
+50
+75
+100 125 150
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+25
+50
+75
+100 125 150
+0.00
+0.25
+0.50
+0.75
+1.00
+! [!#]
+(A)
+(B)
+% !
+&' = 24+/!#
+&' = 20+/!#
+Overall packing 
+fraction
+% !
+&' = 24+/!#
+&' = 20+/!#
+Overall packing 
+fraction
+(C)
+(D)
+FIG. 8. (A) Radial profile of the packing fraction in the case
+of a binary system of fast SPPs, with FD = 24ε/rb (blue)
+and slow SPPs, with FD = 20ε/rb (red), in the case where
+the average packing fraction is 0.6, ks = 100ε, R = 162rb
+and Rp = 100rb.
+(B) A snapshot of the binary system at
+steady state. Blue and red SPPs correspond to fast and slow
+SPPs, respectively. The dashed vertical line and circle in (A)
+and (B), respectively, indicate the boundary of the patterned
+region. (C) and (D) same as in (A) and (B), respectively, but
+in the case of a non-patterned substrate (ks = 0).
+uniform, as shown by Figs. 8(C) and (D), except that the
+fast SPPs are more concentrated at the confining wall
+than the slow SPPs.
+The separation between the fast and slow SPPs shown
+in Figs. 8 (A) and (B) is counterintuitive in that the
+coupling between the pattern of the substrate and the
+motility force tend to expel the SPPs from the patterned
+region. Therefore, one would expect that the fast SPPs
+are more concentrated in the non-patterned region and
+that the slow SPPs are more present in the patterned
+region, as discussed earlier in Section III.A, which is op-
+posite to what is observed from Figs. 8 (A) and (B).
+The fact that the patterned substrate is able to segre-
+gate the SPPs based on their motilities is very interesting
+and potentially very useful. However, an explanation of
+this phenomenon is lacking at the moment and requires
+further systematic simulations. This segregation could
+be understood from a balance of the normal stresses ex-
+erted by the SPPs at the interface between the patterned
+and non-patterned regions, using for example the Irving-
+Kirkwood formalism [49]. This study is planned to be
+performed by the authors in the near future. Separation
+between SPPs may also be induced through differences in
+their interaction strength with the substrate and possibly
+the degree of their flexibility.
+IV.
+SUMMARY AND CONCLUSIONS
+We showed in this article that a complex collective be-
+havior is exhibited by SPPs that are confined in a circular
+geometry and that interact with a circularly patterned
+substrate, which tends to orient the SPPs polarities with
+the local tangent of the pattern. This collective behav-
+ior is characterized by SPPs vortical motion, accumula-
+tion in the outer portion of the patterned region and/or
+the system boundary, and SPPs exclusion from the cen-
+ter. This collective behavior is enhanced with increasing
+SPPs driving force. The size of the exclusion zone is de-
+termined by an interplay between, on one hand, the com-
+bined effects of the driving force and the patterned sub-
+strate, which tends to drive the SPPs outward, and, on
+the other hand, motion of the SPPs in the non-patterned
+region of the substrate which drives the SPPs into the
+patterned region. Interestingly, the vortices in the pat-
+terned and non-patterned regions, at intermediate values
+of the SPPs packing fraction, may have same or opposite
+directions.
+Another interesting feature of this system is that at
+intermediate packing fractions and intermediate values
+of the motility force, the radial profile of the packing
+fraction is non-monotonic, with a peak in the patterned
+region close to its boundary with the non-patterned re-
+gion. A simulation of a binary system, composed of slow
+and fast SPPs (i.e., SPPs with a low and motility forces,
+respectively) show that they can be segregated such that
+the fast SPPs are mostly trapped in the patterned region,
+while the fast SPPs are mainly in the non-patterned re-
+gion. This implies that SPPs can be segregated based on
+their motility.
+With increasing packing fraction, the exclusion zone in
+the center disappears. High misalignment between the
+SPPs polarities and tangential velocities, in the center of
+the system, leads to an increased degree of fluctuations
+in their tangential velocities and reversals in the vorticity
+that originate from the center. Interestingly, these rever-
+sals become quasi-periodic at high packing fractions. It
+is worth noting that while the system exhibits vorticity
+reversal at both intermediate and high packing fractions,
+the mechanisms leading to the two types of reversals are
+different. The results of the present work implies that
+circular patterning of the substrate can be used as a tool
+to guide the motion of SPPs into a collective vortical mo-
+tion, and that at high packing fractions, can be used to
+create quasi periodic reversals in their vortical motion.
+We also showed that the patterned substrate is able to
+segregate a binary mixture of slow and fast SPPs. We
+expect that SPPs can likewise be segregated based on
+their degrees of adhesion to the substrate. This segrega-
+tion can be enhanced by further increasing the adhesion
+strength of the fast SPPs to the substrate.
+We note that the present model of SPPs accounts for
+details often not accounted for in other models. These
+include elongation of the self-propelled particles, their
+flexibility, and enclosed area of the SPPs. It would of
+
+9
+course be very desirable to determine the effects of each
+of these ingredients on the details of the results. There is
+of course a close connection between the SPP dynamics
+described here with that of swimming bacteria.
+How-
+ever, it is important to note that the estimated value of
+the Reynolds number based on the parameters used in
+this study (Eq. (15)) is about 1, which is much larger
+than that of swimming bacteria. Using the present ap-
+proach to investigate the collective motion of cells such as
+bacteria requires a much smaller Reynolds number which
+can be achieved by increasing the value of the drag coef-
+ficient Γ in our model. We plan to investigate the effects
+of these parameters on the observed phenomena in the
+present study in the near future.
+V.
+ACKNOWLEDGEMENTS
+All simulations were performed on computers of the
+High Performance Computing Facility of the University
+of Memphis. This work was funded by the University of
+Memphis.
+[1] S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1,
+323 (2010).
+[2] S. Ramaswamy, J. Stat. Mech.: Theory Exp. 5, 054002
+(2017).
+[3] C. Bechinger, R. Di Leonardo, H. L¨owen, C. Reichhardt,
+G. Volpe,
+and G. Volpe, Rev. Mod. Phys. 88, 045006
+(2016).
+[4] S. Camazine, J. Deneubourg, N. R. Franks, J. Sneyd,
+G. Theraula, and E. Bonabeau, Self-Organization in Bi-
+ological Systems (Princeton University Press, Princeton,
+NJ, USA, 2001).
+[5] C. K. Hemelrijk and H. Hildenbrandt, Int. Focus 2, 726
+(2012).
+[6] D. L. Blair, T. Neicu, and A. Kudrolli, Phys. Rev. E 67,
+031303 (2003).
+[7] A. Kudrolli, G. Lumay, D. Volfson, and L. S. Tsimring,
+Phys. Rev. Lett. 100, 058001 (2008).
+[8] K.-D. N. T. Lam, M. Schindler,
+and O. Dauchot, New
+J. Phys. 17, 113056 (2015).
+[9] T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen,
+and
+O. Shochet, Phys. Rev. Lett. 75, 1226 (1995).
+[10] B. Szabo, G. Sz¨oll¨osi, B. G¨onci, Z. Jur´anyi, D. Selmeczi,
+and T. Vicsek, Phys. Rev. E 74, 061908 (2006).
+[11] S. Wang and P. G. Wolynes, Proc. Natl. Acad. Sci. U.S.A.
+108, 15184 (2011).
+[12] E. M´ehes and T. Vicsek, Integr. Biol. 6, 831 (2014).
+[13] D. Needleman and Z. Dogic, Nat. Rev. Mat. 2, 17048
+(2017).
+[14] I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert,
+and L. Bocquet, Phys. Rev. Lett. 108, 268303 (2012).
+[15] F. Schweitzer, Brownian Agents and Active Particles:
+Collective Dynamics in the Natural and Social Sciences
+(Springer-Verlag, Heidelberg, Germany, 2007).
+[16] D. Kaiser, Curr. Biol. R561, 2007.
+[17] N. S. Gov, Proc. Natl. Acad. Sci. U.S.A. 104, 15970
+(2007).
+[18] D. S. Li, J. Zimmermann, and H. Levine, Phys. Biol. 13,
+016006 (2016).
+[19] M. Lintz, A. Mu˜noz, and C. A. Reinhart-King, J.
+Biomech. Eng. 139, 0210051 (2017).
+[20] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B.
+Liverpool, J. Prost, M. Rao,
+and R. A. Simha, Rev.
+Mod. Phys. 85, 1143 (2013).
+[21] M. E. Cates and J. Tailleur, Annu. Rev. Condens. Matter
+Phys. 6, 219 (2015).
+[22] C. Peng, T. Turiv, Y. Guo, Q.-H. Wei, and O. D. Lavren-
+tovich, Science 354, 882 (2016).
+[23] S. Liu, S. Shankar, M. C. Marchetti, and Y. Wu, Nature
+590, 80 (2021).
+[24] H. H. Wensink and H. L¨owen, Phys. Rev. E 78, 031409
+(2008).
+[25] S. R. K. Vedula, M. C. Leong, T. L. Lai, P. Hersen, A. J.
+Kabla, C. T. Lim, and B. Ladoux, Proc. Natl. Acad. Sci.
+U.S.A. 109, 12974 (2012).
+[26] H. Wioland, F. G. Woodhouse, J. Dunkel, J. O. Kessler,
+and R. E. Goldstein, Phys. Rev. Lett. 110, 268102
+(2013).
+[27] E. Lushi, H. Wioland, and R. E. Goldstein, Proc. Natl.
+Acad. Sci. U.S.A. 111, 9733 (2014).
+[28] B. C. van Zuiden, J. Paulose, W. T. M. Irvine, and V.
+Vitelli, Proc. Natl. Acad. Sci. U.S.A. 113, 12919 (2016).
+[29] J. Elgeti and G. Gompper, Europhys. Lett. 109, 58003
+(2015).
+[30] C. A. Velasco, S. D. Ghahnaviyeh, H. J. Pishkenari, T.
+Auth, and G. Gompper, Soft Matter 13, 5865 (2017).
+[31] H. Wioland, E. Lushi, and R. E. Goldstein, New J. Phys.
+18, 075002 (2016).
+[32] G. Duclos, C. Blanch-Mercader, V. Yashunsky, G. Sal-
+breux, J. F. Joanny, J. Prost,
+and P. Silberzan, Nat.
+Phys. 14, 728 (2018).
+[33] F. Kempf, R. Mueller, E. Frey, J. M. Yeomans,
+and
+A. Doostmohammadi, Soft Matter 15, 7538 (2019).
+[34] S. Jain, V. M. L. Cachoux, G. H. N. S. Narayana,
+S. de Beco, J. D’Alessandro, V. Cellerin, T. Chen, M. L.
+Heuz´e, P. Marcq, R.-M. M`ege, A. J. Kabla, C. T. Lim,
+and B. Ladoux, Nat. Phys. 16, 802 (2020).
+[35] M. M. Norton, A. Baskaran, A. Opathalage, B. Langes-
+lay, S. Fraden, A. Baskaran,
+and M. F. Hagan, Phys.
+Rev. E 97, 012702 (2018).
+[36] J.-B. Gorce, H. Xia, N. Francois, and M. Shats, Proc.
+Natl. Acad. Sci. U.S.A. 116, 25424 (2019).
+[37] A. Kaiser, H. H. Wensink,
+and H. L¨owen, Phys. Rev.
+Lett. 108, 268307 (2012).
+[38] I. S. Aronson and A. Pikovsky, Phys. Rev. Lett. 128,
+108001 (2022).
+[39] E.
+S.
+Gloag,
+C.
+Elbadawi,
+C.
+J.
+Zachreson,
+I. Aharonovich,
+M. Toth,
+I. G. Charles,
+L. Turn-
+bull,
+and C. B. Whitchurch, Front. Microbiol. 7, 2157
+(2017).
+[40] K.-H. Nam, P. Kim, D. K. Wood, S. Kwon, P. P. Proven-
+zano, and D.-H. Kim, Sci. Rep. 6, 29749 (2016).
+
+10
+[41] G. Lee, L. Atia, B. Lan, Y. Sharma, L. Nissim, M.-R.
+Wu, E. Pery, T. K. Lu, C. Y. Park, J. P. Butler, and J.
+J. Fredberg, Connect. Tissue Res. 59, 309 (2018).
+[42] H. Wen, Y. Zhu, C. Peng, P. B. S. Kumar,
+and
+M. Laradji, Soft Matter 18, 1228 (2022).
+[43] Y. Zhu, P.B. Sunil Kumar, and M. Laradji, Soft Matter
+17, 5427 (2021).
+[44] F. Peruani, A. Deutsch, and M. B¨ar, Phys. Rev. E 74,
+030904 (2006).
+[45] Y. Yang, V. Marceau, and G. Gompper, Phys. Rev. E
+82, 031904 (2010).
+[46] M. Theers, E. Westphal, K. Qi, R.G. Winkler, and G.
+Gompper, Soft Matter 14, 8590 (2018).
+[47] ¨O. Duman, R. E. Isele-Holder, J. Elgeti, and G. Gomp-
+per, Soft Matter 14, 4483 (2018).
+[48] See Supplemental Material at [URL will be inserted by
+publisher] for further results.
+[49] J.H. Irving and J.G. Kirkwood, J. Chem. Phys. 18, 817
+(1950).
+

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