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+ANALYSIS OF THE SMOOTHLY AMNESIA-REINFORCED
+MULTIDIMENSIONAL ELEPHANT RANDOM WALK
+JIAMING CHEN AND LUCILE LAULIN
+Abstract. In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant
+random walk (MARW). The scaling limit of the MARW is shown to exist in the diffusive, critical
+and superdiffusive regimes. We also establish the almost sure convergence in all of the three
+regimes. The quadratic strong law is displayed in the diffusive regime as well as in the critical
+regime. The mean square convergence towards a non-Gaussian random variable is established
+in the superdiffusive regime. Similar results for the barycenter process are also derived. Finally,
+the last two Sections are devoted to a discussion of the convergence velocity of the mean square
+displacement and the Cram´er moderate deviations.
+Contents
+1.
+Introduction
+1
+2.
+The amnesia-reinforced elephant random walk
+4
+3.
+A correlated martingale approach
+6
+4.
+Scaling limit and convergence
+7
+5.
+Scaling limit of the barycenter process
+13
+6.
+Velocity of quadratic mean displacement
+21
+7.
+Cram´er moderate deviations
+23
+Appendix A.
+Technical Lemmas
+24
+References
+35
+1. Introduction
+The study of reinforced processes and reinforced random walks has known a growing interest
+over the last decades. In particular, random walks on graphs, or more precisely edge [37] or vertex
+[39] reinforced random walks, have been the subject of a great number of contributions, see also
+[1, 12, 27] and the references therein. The insight of introducing reinforcement mechanisms to
+stochastic processes has also shed light on more applied models. In [30], the adaptive strategy of
+an agent who plays a two-armed bandit machine was described as a self-reinforced random walk.
+The philosophy of stochastic reinforcement has also been discussed in the topics of evolutionary
+ecology [4] and machine learning theory [17]. Another manifestation of reinforced P´olya urn models
+on financial economics can be found in [35]. We also refer the readers to [38] for a comprehensive
+and extensive survey on the subject.
+The Elephant Random Walk (ERW) is a discrete-time random walk, introduced by Sch¨utz and
+Trimper [40] in 2004. It was referred to as the ERW in allusion to the traditional saying that
+elephants can always remember anywhere they have been. As it was pointed out [12] by Bertoin
+2010 Mathematics Subject Classification. 60G50, 60G42, 62M09.
+Key words and phrases. Reinforced random walk, scaling limit, Cram´er moderate deviation, martingale.
+1
+arXiv:2301.08644v1  [math.PR]  20 Jan 2023
+
+2
+JIAMING CHEN AND LUCILE LAULIN
+(a) Diffusive regime
+(b) Critical regime
+(c) Superdiffusive regime
+Figure 1. The two-dimensional ERW with amnesia (in blue) and its barycenter
+(in red).
+who relied on K¨ursten’s work [29], the ERW is a special case of step-reinforced random walk. In
+fact, the ERW is reinforced because its behavior is influenced by its past : the ERW may have
+a tendency to do the same thing over and over, or on the contrary, it may try to compensate its
+previous steps. This different types of behavior, here-called regimes, are ruled by the memory
+parameter p and it is well-known that the ERW shows three regimes of behavior and that the
+critical value is p = 3/4.
+The ERW in dimension d = 1 has received a lot of attention from mathematicians and physicists
+over the last two decades.
+The almost sure convergence and the asymptotic normality of the
+position of the ERW were established in the diffusive regime p < 3/4 and the critical regime
+p = 3/4, see [3, 9, 16] and the references therein. In the superdiffusive regime p > 3/4, Bercu
+[5] proved that the limit of the position of the ERW is not Gaussian and Kubota and Takei [28]
+showed that the fluctuation of the ERW around this limit is Gaussian. To obtain those asymptotics,
+various approaches have been followed : Baur and Bertoin [3] went with the connection to P´olya-
+type urns while martingales were used by Bercu [5] and Coletti et al. [16] and the construction of
+random trees with Bernoulli percolation have been explicited by K¨ursten [29] and Businger [13].
+Other quantities of interest regarding the ERW have been studied. For example, Fan et al.
+[20] provided the Cramer moderate deviations associated with the ERW in dimension 1 and, more
+recently, Hayashi et al. [26] studied the rate of quadratic mean displacement.
+Bercu and Laulin [9] introduced the multidimensional ERW (MERW), where d ≥ 1, and estab-
+lished the natural extensions of the results [5] in dimension d = 1. Then, they investigated the
+center of mass of the MERW [8]. In both papers, they extensively used a martingale approach.
+Bertenghi [10] made use of the connection to P´olya-type urns in order to establish functional
+results for the MERW.
+Finally, the ERW with changing memory has also been introduced. The ERW with linearly
+reinforced memory has been studied by Baur [1] via the urn approach, and Laulin [31] using
+martingales. Gut and Stadm¨uller [25] proposed an amnesic ERW where the elephant could stop
+and only remember the first (and second) step it tooks. They also investigated the case where
+the elephant only remembered a fixed or time-evolving portion of its past (recent or distant)
+[24]. In the recent work [32], Laulin introduced smooth amnesia to the memory of the ERW and
+established the asymptotic behavior of this new process.
+The idea of our paper is to generalise the work [32] in dimension 1 to the dimension d ≥ 1.
+In other words, we introduce smooth amnesia to the memory of the multidimensional elephant
+random walk.
+
+40
+19
+30
+20
+10
+0
+-10
+-20
+0
+10
+20
+30
+40
+50
+60400
+300
+200
+100
+0
+0
+50
+100
+150
+200
+250
+35060
+50
+40
+30
+20
+10
+0
+0
+200
+400
+600
+800MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+3
+Our paper is organized as follows. In Section 2, we introduce the basic setting of the elephant
+random walk (Sn)n∈N placed under an amnesia reinforcement mechanism, which is controlled
+by the memory sequence (βn)n∈N. This type of multidimensional reinforced random walked is
+named as the multidimensional amnesia-reinforced elephant random walk (MARW). Similar to
+the ERW with the amnesia reinforcement, the MARW also admits a martingale structure, which
+is discussed in Section 2. Unlike the usual ERW, the additional amnesia-reinforcement induces
+two discrete-time martingales, instead of a single martingale, which are strongly correlated in a
+nontrivial fashion. Such strong correlation of martingales will eventually pose some computational
+difficulties when we analyze the limiting behavior of the MARW in Section 4. For instance, when
+we compute the pointwise limit and the scaling limit of (Sn)n∈N in the diffusive regime, the two
+strongly correlated martingales have to be dealt with separately, see [8, 31, 32] for the same
+methodology.
+As a courtesy to our readers, we give a preview of some of our main results, whose proofs will
+be deferred to Theorem 4.1, Theorem 4.2, and Theorem 4.3. In the diffusive regime, we have the
+almost sure convergence,
+1
+nSn → 0
+as
+n → ∞
+P − a.s.
+Another logarithmic scaling to the MARW yields the quadratic strong law,
+1
+log n
+n
+�
+k=1
+SkST
+k
+k2
+→ C(p, (βn)n∈N) · 1
+dId
+as
+n → ∞
+P-a.s.
+where the constant C(p, (βn)n∈N) > 0 depends only on the parameter p and the control sequence
+(βn)n∈N of the amnesia-reinforcement.
+Using square-root scaling factor, we observe that the
+MARW also admits a scaling limit in the diffusive regime, or convergence in distribution, in the
+Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+� 1
+√nS⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+Wt, t ≥ 0
+�
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with
+covariance structure given in (4.6).
+It is also of interest to look at the barycenter process (Gn)n∈N of the MARW. Its definition as
+well as its limiting behavior are discussed in Section 5. Similar to the discussion of the MARW,
+we obtain its pointwise convergence, quadratic strong law, and its scaling limit. In particular,
+Theorem 5.5 states that the barycenter process admits a scaling limit at the diffusive regime, or
+convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+� 1
+√nG⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+Wtr dr, t ≥ 0
+�
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process defined in Theorem 4.3 with
+its covariance structure defined in (4.6).
+A natural question to ask is how fast the limiting Theorems in Section 4 are carried on. Section
+6 provides a quantitative estimate on the mean square convergence velocity of the pointwise limit,
+quadratic strong law, and the scaling limit of the MARW. It should be possible to derive similar
+convergence velocity to the barycenter process, which is not computed in this work. In Section
+7, we end this work with a discussion on the Cram´er moderate deviations of the MARW in the
+diffusive and critical regimes. As a preview of our result in this Section, let (ϑn)n∈N ⊆ R be a
+non-decreasing sequence so that ϑn/√n → 0 as n → ∞, and wn the sequence with asymptotic
+
+4
+JIAMING CHEN AND LUCILE LAULIN
+behavior described in Lemma A.1. Take any non-empty Borel set B ⊆ Rd, then we have
+−
+inf
+x∈int B
+1
+2∥x∥2 ≤ lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ lim sup
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ − inf
+x∈cl B
+1
+2∥x∥2,
+(1.1)
+where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively. This is the
+Cram´er moderate deviations for the MARW in the diffusive and critical regimes.
+Moreover, we chose to postpone some technicalities regarding the analysis of the random walk
+to the Appendix A. That way, the reader can focus on the main Theorems and the ideas of their
+proofs. However, some analogous technicalities are displayed in the proof of the Theorems on the
+barycenter such that the reader can also have a complete overview of the work needed.
+Other probabilistic aspects of interest to the MARW include the statistical inference and an
+analysis on the Fisher information, see [7], as well as the Wasserstein distance of the reinforced
+random walk, see [21]. Perturbations of the amnesia intensity and its stability for the MARW is
+also of independent interest. A similar topic for another type of stochastic process, the Schramn-
+Loewner evolution, has been considered in [2, 15]. The transience and recurrence property of the
+MARW remains unknown, to the best of our knowledge. Readers are referred to [11, 20] for an
+exposition on the ERW without the amnesia reinforcement mechanism.
+2. The amnesia-reinforced elephant random walk
+To begin with, let us properly introduce the MARW. It is the natural extension to higher
+dimensions of the one-dimensional MARW, defined in [31]. For an arbitrarily given dimension
+d ≥ 1, let (Sn)n∈N be a (reinforced) random walk on Zd starting from the origin at time n = 0,
+i.e. S0 = 0. At time n = 1, the reinforced random walk moves to one of the 2d nearest-neighbors
+with equal probability 1/2d. After that, at time n ≥ 1, the reinforced random walk chooses at
+random an integer 1 ≤ k ≤ n among the past times and performs the same step with probabily p,
+or goes in any of the 2d − 1 other directions with probability (1 − p)/(2d − 1). This random walk
+possesses the amnesia property, in the sense that it remembers its most recent past steps better
+than its remote past steps. Colloquially, this random walk has higher probability to choose its
+recent steps than its earlier steps.
+From a mathematical perspective, the position of this reinforced random walk at time n+1 ≥ 1
+is given by
+Sn+1 = Sn + Xn+1
+with Xn+1 being defined as the step of this random walk at time n + 1, satisfying
+Xn+1 = An+1Xβn+1.
+Here An+1 is a random d × d matrix given by
+P(An = +Id) = p,
+and, for all 1 ≤ k ≤ d − 1,
+P(An = −Id) = P(An = +Jk
+d ) = P(An = −Jk
+d ) = 1 − p
+2d − 1
+where Id is the identity matrix of order d, Id = (δi,j)d and Jd = C(0, 1, 0, . . . , 0) is the circulant
+matrix of order d such that J = (δi+1,j)d. It is easy to observe that the fixed permutation matrix
+Jd satisfied Jd
+d = Id. The distribution of the memory βn of the reinforced random walk is such
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+5
+that the probability of choosing a fixed past time k ∈ N decays approximately with rate kβ/nβ+1,
+where β ≥ 0 is the amnesia parameter.
+(a) n = 10
+(b) n = 100
+Figure 2. Evolution of the distribution of the memory β depending on the value
+of β and the time.
+To be precise, this random walk chooses βn+1 according to
+P
+�
+βn+1 = k
+�
+= (β + 1)Γ(β + k)Γ(n)
+Γ(k)Γ(β + n + 1)
+= β + 1
+n
+·
+µk
+µn+1
+for all
+1 ≤ k ≤ n,
+where
+µn =
+n−1
+�
+k=1
+�
+1 + β
+k
+�
+=
+Γ(β + n)
+Γ(n)Γ(β + 1).
+(2.1)
+(a) d = 1
+(b) d = 2
+(c) d = 3
+(d) d = 10
+Figure 3. Competition between the dimension and the amnesia.
+Figure 3 aims to give a better understanding on how amnesia affects the MARW in various
+dimensions. The horizontal axis corresponds to p (from 0 to 1) and the vertical axis corresponds to
+β (from 0 to 10, arbitrary chosen). The diffusive regime, ie. when p < 4dβ+2d+1
+4d(β+1)
+or a < 1−
+1
+2(β+1),
+is in blue while the superdiffusive regime is in red, see Lemma A.1 for the definition of the regimes.
+One can observe that when the amnesia parameter β grows, the superdiffusive regime tends to be
+less represented. It should also be noted that when the dimension grows the superdiffusive regime
+is more important. Hence, the amnesia is somehow leading the MARW to a behavior closer to
+the one in dimension 1. When β vanishes, i.e. β = 0, the MARW reduces to the multidimensional
+elephant random walk (MERW) introduced in [9].
+The two random variables An and βn are constructed to be conditionally independent. At each
+time n, define the σ-algebra Fn = σ(X1, . . . , Xn). Then (Fn)n∈N is a discrete-time filtration to
+which the MARW is clearly adapted.
+
+β= 0
+0.5
+β= 1
+β= 2
+0.4 
+β= 5
+β= 10
+0.3
+0.2
+0.1
+0.0
+2
+4
+9
+00
+100.10
+β= 0
+β= 1
+0.08
+β= 2
+β= 5
+0.06 
+β= 10
+0.04-
+0.02
+0.00
+0
+20
+40
+60
+80
+10010
+8 -
+6
+4 
+2 -
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p10
+8 -
+6
+B
+4 
++0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p10
+8 -
+6
+B
+4 
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p10
+8 -
+6
+B
+4 
++0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p6
+JIAMING CHEN AND LUCILE LAULIN
+Since An and βn are conditionally independent, we clearly have
+E
+�
+Xn+1|Fn
+�
+= E
+�
+An
+�
+E
+�
+Xβn+1|Fn
+�
+= 2dp − 1
+2d − 1 E
+�
+n
+�
+k=1
+Xk1{βn+1=k}|Fn
+�
+= 2dp − 1
+2d − 1 · β + 1
+nµn+1
+n
+�
+k=1
+µkXk.
+(2.2)
+We further denote
+a = 2dp − 1
+2d − 1
+and
+Yn =
+n
+�
+k=1
+µkXk
+(2.3)
+such that
+E
+�
+Yn+1|Fn
+�
+=
+�
+1 + a(β + 1)
+n
+�
+Yn = γnYn
+with γn = 1 + a(β + 1)/n. Hereafter, for each n ≥ 1, let
+an =
+n−1
+�
+k=1
+γ−1
+k
+= Γ(n)Γ(a(β + 1) + 1)
+Γ(a(β + 1) + n)
+and
+wn =
+n
+�
+k=1
+(akµk)2.
+(2.4)
+From a Gamma function estimate, also see in [31], we have that
+na(β+1)an → Γ(a(β + 1) + 1)
+as
+n → ∞
+(2.5)
+and
+n−βµn → Γ(β + 1)−1
+as
+n → ∞.
+(2.6)
+3. A correlated martingale approach
+Define the following two Rd-valued processes by
+Mn = anYn
+and
+Nn = Sn +
+a(β + 1)
+β − a(β + 1)µ−1
+n Yn.
+(3.1)
+Proposition 3.1. The Rd-valued processes (Mn)n∈N and (Nn)n∈N defined in (3.1) are locally
+square-integrable martingales adapted to (Fn)n∈N.
+Proof. Since, both Mn and Nn are finite sums for each n ≥ 1, the square-integrability and adapt-
+ness are immediate. By (2.3) and (2.4), we have
+E
+�
+Mn+1|Fn
+�
+= anγ−1
+n Yn + anµnγ−1
+n E
+�
+Xn+1|Fn
+�
+= anYn.
+And by (2.2), we have
+E
+�
+Nn+1|Fn
+�
+= E
+�
+Sn+1 +
+a(β + 1)
+β − a(β + 1)µ−1
+n+1Yn+1|Fn
+�
+= Sn +
+a(β + 1)
+β − a(β + 1)µ−1
+n Yn.
+Hence the assertion is verified.
+□
+Notice that via introducing the martingales (Mn)n∈N and (Nn)n∈N, we can write Sn as
+Sn = Nn −
+a(β + 1)
+β − a(β + 1)(anµn)−1Mn.
+(3.2)
+This writing is the key on which rely all of our analysis and our martingale approach.
+Moreover, the asymptotic behavior of (Mn)n∈N is closely related to wn defined in (2.4). In fact,
+we have the following asymptotic result, which states the three regimes of the MARW.
+Lemma 3.1. In the diffusive regime when p < 4dβ+2d+1
+4d(β+1)
+or a < 1 −
+1
+2(β+1), we have
+wn
+n1−2(a(β+1)−β) → l(β)
+as
+n → ∞
+(3.3)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+7
+with
+l(β) =
+1
+1 + 2(β − a(β + 1))
+�Γ(a(β + 1) + 1)
+Γ(β + 1)
+�2
+.
+In the critical regime when p = 4dβ+2d+1
+4d(β+1)
+or a = 1 −
+1
+2(β+1), we have
+wn
+log n →
+�Γ(β + 1 + 1
+2)
+Γ(β + 1)
+�2
+as
+n → ∞.
+(3.4)
+In the superdiffusive regime when p > 4dβ+2d+1
+4d(β+1)
+or a > 1 −
+1
+2(β+1), we have
+wn →
+∞
+�
+k=1
+�Γ(a(β + 1) + 1)Γ(β + k)
+Γ(a(β + 1) + k)Γ(β + 1)
+�2
+< ∞
+as
+n → ∞.
+(3.5)
+In order to investigate the asymptotic behavior of (Sn)n∈N, we first introduce an arbitrarily
+fixed test non-zero vector u ∈ Rd and we define
+Mn(u) = uT Mn
+and
+Nn(u) = uT Nn
+for each
+n ∈ N.
+It is then clear that (Mn(u))n∈N (Nn(u))n∈N are real-valued locally square-integrable martingales
+for each fixed u ∈ Rd.
+We further infer that (Sn(u))n∈N satisfies an equation analogous to (3.2). In
+this setting, we have reduced the multidimensional martingales to real-valued martingales without
+loss of generality. This technique greatly simplifies our martingale analysis. From now on, we fix
+the test vector u ∈ Rd and we introduce the two-dimensional martingale (Ln(u))n∈N defined as
+Ln(u) =
+�
+Nn(u)
+Mn(u)
+�
+for each
+n ∈ N.
+(3.6)
+Denote the martingale increment ϵn+1 = Yn+1 − γnYn for each n.
+Then (ϵn)n∈N satisfies the
+martingale difference relation E[ϵn+1|Fn] = 0. We obtain that
+∆Ln+1(u) = Ln+1(u) − Ln(u) =
+�
+Sn+1(u) − Sn(u) +
+a(β+1)
+β−a(β+1)
+�
+µ−1
+n+1Yn+1(u) − µ−1
+n Yn(u)
+�
+an+1Yn+1(u) − anYn(u)
+�
+=
+�
+βµ−1
+n+1
+β−a(β+1)
+�
+µn+1Xn+1(u) − (γn − 1)Yn(u)
+�
+an+1ϵn+1(u)
+�
+=
+�
+βµ−1
+n+1
+β−a(β+1)
+an+1
+�
+ϵn+1(u).
+(3.7)
+4. Scaling limit and convergence
+In this section, we discuss the scaling limit as well as the almost sure convergence in the
+diffusive, critical and the superdiffusive regimes, depending on the value of p with respect to
+(4dβ +2d+1)/(4d(β +1)). We also give the quadratic strong law in the diffusive regime as well as
+in the critical regime. Afterwards, the mean square convergence is established in the superdiffusive
+regime.
+4.1. The diffusive regime.
+Theorem 4.1. We have the almost sure convergence
+1
+nSn → 0
+as
+n → ∞
+P-a.s.
+
+8
+JIAMING CHEN AND LUCILE LAULIN
+Proof. We have from [18, Theorem 4.3.15] again that, for all γ > 0,
+∥Mn∥2
+λmax⟨M⟩n
+= o
+��
+log Tr⟨M⟩n
+�1+γ�
+P-a.s.
+(4.1)
+From equation (A.9) and the fact that λmax⟨M⟩n ≤ Tr⟨M⟩n ≤ wn, we get
+∥Mn∥2 = o
+�
+wn
+�
+log wn
+�1+γ�
+P-a.s.
+(4.2)
+By (3.3), we observe
+∥Mn∥2 = o
+�
+n1−2(a(β+1)−β)�
+log n
+�1+γ�
+P-a.s.
+Since Mn = anYn, we have from equations (2.5) and (2.6)
+∥Yn∥2
+(nµn+1)2 = o
+�
+n−1�
+log n
+�1+γ�
+P-a.s.
+which implies
+Yn
+nµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+By (A.10) and [18, Theorem 4.3.15] again, we find that
+∥Nn∥2 = o
+�
+n
+�
+log n
+�1+γ�
+P-a.s.
+(4.3)
+Moreover, we obtain from equation (3.2)
+1
+n2
+����Sn +
+a(β + 1)
+(β − a(β + 1))µn+1
+Yn
+����
+2
+= o
+�
+n−1�
+log n
+�1+γ�
+P-a.s.
+Hence, we conclude that
+Sn
+n +
+a(β + 1)
+β − a(β + 1) ·
+Yn
+nµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+and the proof is complete.
+□
+Theorem 4.2. We have the quadratic strong law
+1
+log n
+n
+�
+k=1
+SkST
+k
+k2
+→
+2β + 1 − a
+(1 − a)(1 − 2(a(β + 1) − β)) · 1
+dId
+as
+n → ∞
+P-a.s.
+Proof. We will check that all the conditions of [32, Theorem A.3] are satisfied, see also [14, 41].
+The condition (H.1) is satisfied thanks to Lemma A.4 while the condition (H.2) directly follows
+from Lemma A.5 and the condition (H.4) is exactly the statement of Lemma A.7. Therefore,
+1
+log
+�
+det V −1
+n
+�2
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkLk(u)Lk(u)T V T
+k → 1
+duT uVt=1
+as n → ∞ P-a.s. On the one hand, we have from (A.24) that
+1
+log n
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkLk(u)Lk(u)T V T
+k → 2(1 − a)(β + 1)
+d
+uT uVt=1
+(4.4)
+as n → ∞ P-a.s. On the other hand, by (2.5), (2.6) and (A.24), we have
+n
+�(det Vn)2 − (det Vn+1)2
+(det Vn)2
+�
+→ 2(1 − a)(β + 1)
+as
+n → ∞
+P-a.s.
+Finally, we obtain from (A.17) and (4.4) that
+1
+log n
+n
+�
+k=1
+uT SkST
+k u
+k2
+=
+1
+log n
+n
+�
+k=1
+vT VkLk(u)Lk(u)T V T
+k v
+k
+→ vT Vt=1v · 1
+duT u
+(4.5)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+9
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (4.5).
+□
+Theorem 4.3. The MARW admits a scaling limit at the diffusive regime, or convergence in
+distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+� 1
+√nS⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+Wt, t ≥ 0
+�
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with
+covariance
+E
+�
+WsW T
+t
+�
+=
+a(β + 1)(1 − a) + aβ
+(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)s
+� t
+s
+�a−β(1−a)
+· 1
+dId
++
+β
+(β(1 − a) − a)(1 − a)s · 1
+dId
+for all
+0 ≤ s ≤ t < ∞.
+(4.6)
+Proof. We will check that all the conditions of [32, Theorem A.2] are satisfied, see also [14, 41].
+The condition (H.1) is satisfied thanks to Lemma A.4 while the condition (H.2) directly follows
+from Lemma A.5 and the condition (H.3) is exactly the statement of Lemma A.6. Consequently,
+we have the convergence in distribution in the Skorokhod space D(R+) such that
+�
+VnL⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+Wt(u), t ≥ 0
+�
+where (Wt(u))t≥0 is a continuous R2-valued centered Gaussian process such that W0 = 0 and with
+covariance
+E
+�
+Ws(u)Wt(u)T �
+= 1
+duT uVs
+for all
+0 ≤ s ≤ t < ∞.
+From (2.5), (2.6), and (3.2), we see that S⌊nt⌋(u) is asymptotically equivalent to
+N⌊nt⌋(u) + tβ−a(β+1)
+a(β + 1)
+β − a(β + 1)(anµn)−1M⌊nt⌋(u)
+P-a.s.
+Multiplying on the left side by vt = (1, ta(β+1)−β)T , we obtain
+� 1
+√nS⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+Wt(u), t ≥ 0
+�
+with Wt(u) = vT
+t Wt(u). Hereafter, when 0 ≤ s ≤ t < ∞, we have the covariance
+E
+�
+Ws(u)Wt(u)T �
+= vT
+s E
+�
+Ws(u)Wt(u)T �
+vt = 1
+d(uT u)vT
+s Vsvt.
+(4.7)
+Solving (4.7), we have
+E
+�
+WsW T
+t
+�
+= 1
+dvT
+s Vsvt
+for all
+0 ≤ s ≤ t < ∞
+and the assertion (4.6) is verified.
+□
+4.2. The critical regime.
+Theorem 4.4. We have the almost sure convergence
+1
+√n log nSn → 0
+as
+n → ∞
+P-a.s.
+Proof. We still have (4.1) and (4.2) such that
+∥Mn∥2 = o
+�
+wn
+�
+log wn
+�1+γ�
+for all
+γ > 0
+P-a.s.
+However, in the critical regime, we have (3.4) rather than (3.3), and
+wn
+log n →
+�Γ(β + 1 + 1
+2)
+Γ(β + 1)
+�2
+as
+n → ∞.
+
+10
+JIAMING CHEN AND LUCILE LAULIN
+Since (2.5), (2.6), and since Mn = anYn, we observe for all γ > 0 that
+∥Yn∥2
+n(log n)2µ2n
+= o
+�
+(log n)−1�
+log log n
+�1+γ�
+P-a.s.
+In this regard
+Yn
+√n log nµn
+→ 0
+as
+n → ∞
+P-a.s.
+(4.8)
+Similarly, we still have (A.10) and
+∥Nn∥2 = o
+�
+n
+�
+log n
+�1+γ�
+for all
+γ > 0
+P-a.s.
+Then
+∥Nn∥2
+n(log n)2 = o
+�
+(log n)γ−1�
+for all
+γ ∈ (0, 1)
+P-a.s.
+and therefore
+Nn
+√n log n → 0
+as
+n → ∞
+P-a.s.
+By (3.2), we can hereafter conclude that
+Sn
+√n log n +
+a(β + 1)
+β − a(β + 1) ·
+Yn
+√n log nµn
+→ 0
+as
+n → ∞
+P-a.s.
+Combining with (4.8), the assertion is verified.
+□
+Theorem 4.5. We have the quadratic strong law
+1
+log log n
+n
+�
+k=1
+SkST
+k
+(k log k)2 → (2β + 1)2 · 1
+dId
+as
+n → ∞
+P-a.s.
+Proof. We will check that all the conditions of [32, Theorem A.3] are satisfied. The condition
+(H.1) is satisfied thanks to Lemma A.8 while the condition (H.2) directly follows from Lemma
+A.9 and the condition (H.4) is exactly the statement of Lemma A.10. Therefore,
+1
+log
+�
+det W −1
+n
+�2
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkLk(u)Lk(u)T W T
+k → 1
+duT uW
+(4.9)
+as n → ∞ P-a.s. On the one hand, we have from (A.34)
+1
+log log n
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkLk(u)Lk(u)T W T
+k → 1
+duT uW
+as n → ∞ P-a.s. On the other hand, by (2.5), (2.6), and (A.33), we have
+n log n
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+→ (2β + 1)2
+as
+n → ∞
+P-a.s.
+Then, we obtain from (A.17) and (4.9) that
+1
+log log n
+n
+�
+k=1
+uT SkST
+k u
+(k log k)2 =
+1
+log log n
+n
+�
+k=1
+wT WkLk(u)Lk(u)T W T
+k w
+k log k
+→ (2β + 1)2
+d
+uT u
+(4.10)
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (4.10).
+□
+Theorem 4.6. The MARW admits a scaling limit at the critical regime, or convergence in dis-
+tribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+�
+1
+�
+nt log n
+S⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+(2β + 1)Bt, t ≥ 0
+�
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+11
+where (Bt)t≥0 is a continuous d-dimensional canonical Brownian motion with covariance
+E
+�
+BsBT
+t
+�
+= s · 1
+dId
+for all
+0 ≤ s ≤ t < ∞.
+Proof. We will check that all the three conditions of [32, Theorem A.2] are satisfied, see also [42,
+Theorem 1]. First of all, by (3.4) and (A.7) we know that
+w−1/2
+n
+⟨M(u)⟩⌊nt⌋w−1/2
+n
+→ t
+d · uT u
+as
+n → ∞
+P-a.s.
+(4.11)
+Hence the condition (H.1) is satisfied. Notice that
+⌊nt⌋
+�
+k=1
+1
+wn
+E
+�
+∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1
+�
+≤
+⌊nt⌋
+�
+k=1
+�w⌊nt⌋
+wn
+�2
+1
+ϵ2w2
+⌊nt⌋
+E
+�
+∆Mk(u)4|Fk−1
+�
+,
+(4.12)
+since (2.5), (2.6), and (A.21), we observe that
+⌊nt⌋
+�
+k=1
+��∆Mk(u)4�� ≤ C1(β)∥u∥4
+⌊nt⌋
+�
+k=1
+(akµk)4 ≤ C2(β)∥u∥4
+⌊nt⌋
+�
+k=1
+1
+k2
+P-a.s.
+(4.13)
+with constants C1(β), C2(β) > 0. Therefore, by (4.12) and (4.13), we have
+⌊nt⌋
+�
+k=1
+1
+wn
+E
+�
+∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1
+�
+≤ C3(β)∥u∥4 · t2
+ϵ2 ·
+1
+nt(log nt)2
+P-a.s.
+Simplifying the above expression, we obtain
+⌊nt⌋
+�
+k=1
+1
+wn
+E
+�
+∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+(4.14)
+Then the condition (H.2), or the Lindeberg condition, is satisfied by (4.14). In this particular case
+at critical regime, (4.11) implies that the condition (H.3) is satisfied. Hence
+�
+1
+√wn
+M⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+Bt(u), t ≥ 0
+�
+where (Bt(u))t≥0 is a continuous real-valued centered Gaussian process such that B0(u) = 0 and
+with covariance
+E
+�
+Bs(u)Bt(u)
+�
+= s
+d · uT u
+for all
+0 ≤ s ≤ t < ∞.
+In the critical regime, from (3.2) we can write
+S⌊nt⌋(u) = N⌊nt⌋(u) + (2β + 1) M⌊nt⌋(u)
+a⌊nt⌋µ⌊nt⌋
+.
+(4.15)
+From (A.8) we know that
+⟨N(u)⟩⌊nt⌋
+nt log n
+→ 0
+and
+N⌊nt⌋(u)
+�
+nt log n
+→ 0
+as
+n → ∞
+P-a.s.
+(4.16)
+Using (2.5), (2.6), and (3.4) again, we conclude that
+�
+1
+�
+nt log n
+S⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+(2β + 1)Bt(u), t ≥ 0
+�
+with
+E
+�
+Bs(u)Bt(u)
+�
+= s · uT u
+d
+for all
+0 ≤ s ≤ t.
+(4.17)
+Solving (4.17), we get
+E
+�
+BsBT
+t
+�
+= s · 1
+dId
+for all
+0 ≤ s ≤ t.
+
+12
+JIAMING CHEN AND LUCILE LAULIN
+which completes the proof.
+□
+4.3. The superdiffusive regime.
+Theorem 4.7. We have the almost sure convergence
+1
+na(β+1)−β Sn → Lβ
+as
+n → ∞
+P-a.s.
+where the limiting Lβ is an Rd-valued random variable.
+Remark 4.1. In fact, from Theorem 4.8 below, we will see the random vector Lβ is non-degenerate.
+Proof. From (3.5) and (A.7), in the superdiffusive regime, we have
+Tr⟨M⟩n ≤ wn ≤
+∞
+�
+k=1
+�Γ(a(β + 1) + 1)Γ(β + k)
+Γ(a(β + 1) + k)Γ(β + 1)
+�2
+< ∞
+for all
+n ∈ N.
+By [18, Theorem 4.3.15], this leads to
+Mn → M
+as
+n → ∞
+P-a.s.
+with
+M =
+∞
+�
+k=1
+akϵk.
+By (3.1), Mn = anYn, and by (2.5), we observe that
+Yn
+na(β+1) →
+1
+Γ(a(β + 1) + 1)M
+as
+n → ∞
+P-a.s.
+(4.18)
+Moreover, equations (4.3) still holds and, as 2a(β + 1) > 2β + 1 in the superdiffusive regime, we
+find that
+1
+n2(a(β+1)−β)
+����Sn +
+a(β + 1)
+(β − a(β + 1))µn+1
+Yn
+����
+2
+= o
+�
+n−(1−2a(β+1)+2β)�
+log n
+�1+γ�
+P-a.s.
+Thanks to (2.6), we obtain
+Sn
+na(β+1)−β +
+a(β + 1)
+β − a(β + 1) · Γ(β + 1)Yn
+na(β+1)
+→ 0
+as
+n → ∞
+P-a.s.
+(4.19)
+Combining (4.18), it yields
+Sn
+na(β+1)−β → Lβ
+as
+n → ∞
+P-a.s.
+where
+Lβ =
+a(β + 1)
+a(β + 1) − β ·
+Γ(β + 1)
+Γ(a(β + 1) + 1)M
+(4.20)
+and the assertion follows.
+□
+Theorem 4.8. We have the following mean square convergence
+E
+�����
+1
+na(β+1)−β Sn − Lβ
+����
+2�
+→ 0
+as
+n → ∞.
+(4.21)
+Proof. For each test vector u ∈ Rd, we have
+E
+�
+Mn(u)2�
+= E
+�
+⟨M(u)⟩n
+�
+≤ 1
+dwnuT u
+for all
+n ∈ N.
+From (3.5), we obtain
+sup
+n≥1
+E
+�
+Mn(u)2�
+< ∞
+which implies that (Mn(u))n∈N is a martingale bounded in L2. Therefore
+E
+�
+|Mn(u) − M(u)|2�
+→ 0
+as
+n → ∞.
+(4.22)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+13
+Moreover, on the one hand (4.22) together with (4.18) implies that
+E
+�����
+1
+na(β+1) Yn(u) − Y (u)
+����
+2�
+→ 0
+as
+n → ∞.
+(4.23)
+On the other hand, from (A.8) we know that
+E
+�
+Nn(u)2�
+= E
+�
+⟨N(u)⟩n
+�
+≤ 1
+d
+�
+β
+β − a(β + 1)
+�2
+nuT u
+for all
+n ∈ N.
+Since a(β + 1) > β + 1
+2 in the superdiffusive regime, we have
+E
+�����
+1
+na(β+1)−β Nn(u)
+����
+2�
+→ 0
+as
+n → ∞.
+(4.24)
+The proof is complete by combining (4.23) and (4.24).
+□
+Remark 4.2. The expected value of Lβ is
+E
+�
+Lβ
+�
+= 0
+(4.25)
+whereas its quadratic deviation is
+E
+�
+LβLT
+β
+�
+=
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(β + 1)2Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2
+· 1
+dId.
+(4.26)
+Theorem 4.9. The MARW admits a scaling limit at the superdiffusive regime, or convergence in
+distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+�
+1
+na(β+1)−β S⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+Qt, t ≥ 0
+�
+(4.27)
+with the limiting Qt = ta(β+1)−βLβ for all t ≥ 0.
+Proof. For all t ≥ 0 and from (4.19), we observe that
+S⌊nt⌋
+⌊nt⌋a(β+1)−β +
+a(β + 1)
+β − a(β + 1) ·
+Y⌊nt⌋
+⌊nt⌋a(β+1) → 0
+as
+n → ∞
+P-a.s.
+which implies
+1
+na(β+1)−β Sn → ta(β+1)−βLβ
+as
+n → ∞
+P-a.s.
+(4.28)
+The P-a.s. convergence in (4.28)holds in all finite-dimensional distributions which characterizes
+the Skorokhod space topology. Hence, we have (4.27) and the assertion is verified.
+□
+5. Scaling limit of the barycenter process
+The study of the scaling limit of the MARW (Sn)n∈N gives us some information on its asymptotic
+behavior.
+Nonetheless, to understand its pathwise geometric features, we need to discuss its
+barycenter, or center of mass process. Such topics have been raised and discussed in [36, 43]. In
+this Section, we turn our attention to the above-mentioned barycenter process (Gn)n∈N defined
+by
+Gn := 1
+n
+n
+�
+k=1
+Sk
+(5.1)
+Our work contains the discussion on the scaling limit and the almost sure convergence in the
+diffusive, critical and superdiffusive regimes. The quadratic strong law in the diffusive and crit-
+ical regimes is also discussed while the mean square convergence in the superdiffusive regime is
+established.
+
+14
+JIAMING CHEN AND LUCILE LAULIN
+5.1. Almost sure convergence. The barycenter process was discussed in [8] for the elephant
+random walk in dimension d, which is a special case of the process we study here when β = 0. We
+first begin with the almost sure convergence.
+Theorem 5.1. We have the almost sure convergence, in the diffusive regime,
+1
+nGn → 0
+as
+n → ∞
+P-a.s.
+(5.2)
+while in the critical regime,
+1
+√n log nGn → 0
+as
+n → ∞
+P-a.s.
+(5.3)
+and, in the superdiffusive regime,
+1
+na(β+1)−β Gn →
+1
+1 + a(β + 1) − β Lβ
+as
+n → ∞
+P-a.s.
+(5.4)
+where Lβ was characterized in Theorems 4.7 and 4.2.
+Proof. In the diffusive regime, from (5.1) we observe that
+1
+nGn =
+n
+�
+k=1
+k
+n2 · 1
+k Sk =
+n
+�
+k=1
+1
+k Ska′
+n,k
+with
+a′
+n,k = k
+n2 .
+Since �n
+k=1 a′
+n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.1, from Lemma
+A.12 we can conclude that
+1
+nGn =
+n
+�
+k=1
+1
+k Ska′
+n,k → 0
+as
+n → ∞
+P-a.s.
+such that (5.2) is verified. In the critical regime, we have from (5.1) that
+1
+√n log nGn =
+1
+n3/2 log n
+n
+�
+k=1
+Sk =
+n
+�
+k=1
+1
+√
+k log k
+Ska′′
+n,k
+with
+a′′
+n,k = k1/2 log k
+n3/2 log n.
+Since �n
+k=1 a′′
+n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.4 holds, we get
+from Lemma A.12 hat
+1
+√n log nGn =
+n
+�
+k=1
+1
+√
+k log k
+Ska′′
+n,k → 0
+as
+n → ∞
+P-a.s.
+and we obtain (5.3). Finally, in the superdiffusive regime, we also get from (5.1) that
+1
+na(β+1)−β Gn =
+1
+n1+a(β+1)−β
+n
+�
+k=1
+Sn =
+n
+�
+k=1
+1
+ka(β+1)−β Ska′′′
+n,k
+with
+a′′′
+n,k =
+ka(β+1)−β
+n1+a(β+1)−β .
+Since
+n
+�
+k=1
+a′′′
+n,k →
+1
+1 + a(β + 1) − β
+as
+n → ∞
+by a simple calculation, and because of the almost sure convergence in Theorem 4.7, we can
+conclude using Lemma A.13
+1
+na(β+1)−β Gn →
+1
+1 + a(β + 1) − β Lβ
+as
+n → ∞
+P-a.s.
+and (5.4) is verified.
+□
+5.2. Quadratic strong law.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+15
+Theorem 5.2. In the diffusive regime, we have the quadratic strong law
+1
+log n
+n
+�
+k=1
+GkGT
+k
+k2
+→ 4I(a, β) · 1
+dId
+as
+n → ∞
+P-a.s.
+where I(a, β) is given explicitly
+I(a, β) =
+1
+Γ(a(β + 1) + 1)2Γ(β + 1)2 ·
+2a2(1 − a)(β + 1)3
+3(β − a(β + 1))2(1 − a(β + 1) + β).
+Proof. We will check that all the three conditions of [32, Theorem A.2] are satisfied. Looking back
+to (5.1), we observe that
+Gn = 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+1
+akµk
+Mk = 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+1
+akµk
+k
+�
+l=1
+alϵl.
+Then, changing the order of summation, we have
+Gn = 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+akϵk
+n
+�
+l=k
+1
+alϵl
+= 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+ak(δn − δk−1)ϵk
+where we define δn = �n
+k=1(akµk)−1 for all n ∈ N. Moreover, we denote
+Zn =
+n
+�
+k=1
+Nk −
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+akδk−1ϵk.
+such that we have
+Gn = 1
+nZn − δn
+n ·
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+akϵk = 1
+n
+�
+Zn −
+a(β + 1)
+β − a(β + 1)δnMn
+�
+.
+For a fixed text vector u ∈ Rd, we define
+Hn(u) =
+�
+Zn(u)
+Mn(u)
+�
+for all
+n ∈ N.
+(5.5)
+which implies
+∆Hn(u) = Hn+1(u) − Hn(u) =
+�
+Nn+1(u)ϵn+1(u)−1 −
+a(β+1)
+β−a(β+1)an+1δn
+an+1
+�
+ϵn+1(u).
+Then, let
+Vn =
+1
+n3/2
+�
+1
+0
+0
+a(β+1)
+β−a(β+1)δn
+�
+and
+v =
+�
+1
+−1
+�
+.
+Then it is immediate that
+vT VnHn(u) =
+1
+√nGn
+for all
+n ∈ N
+(5.6)
+and that
+lim
+n→∞ Vn⟨H(u)⟩nV T
+n = lim
+n→∞
+1
+n3
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+�
+a(β + 1)
+β − a(β + 1)
+�2
+δ2
+ka2
+k+1E
+�
+ϵk+1(u)2|Fk
+�
+= lim
+n→∞
+1
+n3 ·
+a2(1 − a)(β + 1)3uT u
+d(β − a(β + 1))2(1 − a(β + 1) + β)
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+δ2
+ka2
+k+1µ2
+k+1
+P-a.s.
+
+16
+JIAMING CHEN AND LUCILE LAULIN
+By (2.5) and (2.6), we know that
+n−(1+a(β+1)−β)δn →
+1
+1 + a(β + 1) − β ·
+1
+Γ(a(β + 1) + 1)Γ(β + 1)
+as
+n → ∞.
+Hence the above calculation leads us to
+Vn⟨H(u)⟩nV T
+n → I(a, β)uT u · 1
+d
+�
+1
+−1
+−1
+1
+�
+as
+n → ∞
+P-a.s.
+(5.7)
+where
+I(a, β) =
+1
+1 − 2(a(β + 1) − β) ·
+a2(1 − a)(β + 1)3
+(β − a(β + 1))2(1 − a(β + 1) + β).
+(5.8)
+Consequently, (5.7) ensures that the condition (H.1) is satisfied. Notice that by (2.3) and (3.1),
+there exists some constant C1(a, β) > 0 and similarly, by (2.5), (2.6), (A.22), there exists some
+other constant C2(a, β) > 0 such that
+∥Nn∥2 ≤ C1(a, β)n2
+and
+a2
+kϵk(u)2 ≤ C2(a, β)n2δ−2
+n
+for all
+1 ≤ k ≤ n.
+Moreover, notice that for all 1 ≤ k ≤ n,
+Vn∆Hk(u) =
+1
+n3/2
+�
+Nk(u)ϵk(u)−1 −
+a(β+1)
+β−a(β+1)akδk−1
+a(β+1)
+β−a(β+1)akδn
+�
+ϵk(u).
+Hence, for all 1 ≤ k ≤ n, we observe that
+∥Vn∆Hk(u)∥2 ≤ 4a2
+k
+n3
+�
+a(β + 1)
+β − a(β + 1)
+�2��β − a(β + 1)
+aka(β + 1)
+Nk(u)
+ϵk(u)
+�2
++ δ2
+k−1 + δ2
+n
+�
+ϵk(u)2 ≤ C(a, β)
+n
+(5.9)
+for some constant C(a, β) > 0. Consequently, we
+n
+�
+k=1
+E
+�
+∥Vn∆Hk(u)∥4�
+≤ 1
+nC(a, β) → 0
+as
+n → ∞
+P-a.s.
+since, for all ϵ > 0,
+n
+�
+k=1
+E
+�
+∥Vn∆Hk(u)∥21{∥Vn∆Hk(u)∥>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Vn∆Hk(u)∥4�
+→ 0
+as
+n → ∞
+P-a.s.
+(5.10)
+Then the condition (H.2), or the Lindeberg condition, is satisfied by (5.10). Hereafter, by (2.5),
+(2.6), and by the definition of δn, we know there exists some constant C′(a, β) ̸= 0 such that
+log
+�
+det V −1
+n
+�2
+log n
+→ C′(a, β)
+as
+n → ∞.
+This ensures that there exists some other constant C′′(a, β) > 0 such that
+∞
+�
+n=1
+1
+�
+log
+�
+det V −1
+n
+�2�2 E
+�
+∥Vn∆Hn(u)∥4|Fn−1
+�
+≤ C2(a, β)
+∞
+�
+n=1
+1
+(log n)2 E
+�
+∥Vn∆Hn(u)∥4|Fn−1
+�
+.
+Finally, using (5.9) leads to
+∞
+�
+n=1
+1
+(log n)2 ∥Vn∆Hn(u)∥4 ≤ C(a, β)
+∞
+�
+n=1
+1
+(n log n)2 < ∞
+P-a.s.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+17
+for some constant C(a, β) > 0 depending only on a and β. The condition (H.4) is satisfied by
+combining the above with (5.10). On the one hand,
+1
+log
+�
+det V −1
+n
+�2
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkHk(u)Hk(u)T V T
+k → 1
+duT uV
+(5.11)
+as n → ∞ P-a.s. where
+V =
+�
+1
+−1
+−1
+1
+�
+I(a, β)
+and I(a, β) has been specified in (5.8). Then, we have
+1
+log n
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkHk(u)Hk(u)T V T
+k → 4 − 2(a(β + 1) − β)
+d
+uT uV
+as n → ∞ P-a.s. since
+log n
+log
+�
+det V −1
+n
+�2 → 4 − 2(a(β + 1) − β)
+as
+n → ∞.
+On the other hand, by (2.5) and (2.6), we have
+n
+�(det Vn)2 − (det Vn+1)2
+(det Vn)2
+�
+→ 4 − 2
+�
+a(β + 1) − β
+�
+as
+n → ∞
+P-a.s.
+Using (5.6) and (5.11), we observe that
+1
+log n
+n
+�
+k=1
+uT GkGT
+k u
+k2
+=
+1
+log n
+n
+�
+k=1
+vT VkHk(u)Hk(u)T V T
+k v
+k
+→ vT V v · 1
+duT u
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (4.5).
+□
+Theorem 5.3. In the critical regime, we have the quadratic strong law
+1
+log log n
+n
+�
+k=1
+GkGT
+k
+(k log k)2 → 4(2β + 1)2
+9
+· 1
+dId
+as
+n → ∞
+P-a.s.
+Proof. We will check that all the three conditions of [32, Theorem A.2] are satisfied. Denote
+Wn =
+1
+n√n log n
+�
+1
+0
+0
+a(β+1)
+β−a(β+1)δn
+�
+and
+w =
+�
+1
+−1
+�
+.
+Then, for H defined in (5.5), it is clear that
+wT WnHn(u) =
+1
+√n log nGn
+for all
+n ∈ N
+and that
+lim
+n→∞ Wn⟨H(u)⟩nW T
+n = lim
+n→∞
+1
+n3 log n
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+(2β + 1)2δ2
+ka2
+k+1E
+�
+ϵk+1(u)2|Fk
+�
+= lim
+n→∞
+(2β + 1)2
+n3 log n · uT u
+d
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+δ2
+ka2
+k+1µ2
+k+1
+P-a.s.
+By (2.5) and (2.6), we know that
+n−3/2δn → 2
+3 ·
+Γ(β + 1)
+Γ(β + 1 + 1
+2)
+as
+n → ∞.
+
+18
+JIAMING CHEN AND LUCILE LAULIN
+Hence, the above calculation leads us to
+Wn⟨H(u)⟩nW T
+n → I(β)uT u · 1
+d
+�
+1
+−1
+−1
+1
+�
+as
+n → ∞
+P-a.s.
+with
+I(β) = 4(2β + 1)2
+9
+.
+(5.12)
+Consequently, the condition (H.1) is satisfied thanks to (5.12). Notice that by (2.3) and (3.1),
+there exists some constant C1(β) > 0 and similarly, there exists some constant C2(β) > 0 such
+that
+∥Nn∥2 ≤ C1(β)n2
+and
+a2
+kϵk(u)2 ≤ C2(β)n2δ−2
+n log n
+for all
+1 ≤ k ≤ n.
+Then, notice for all 1 ≤ k ≤ n that
+Wn∆Hk(u) =
+1
+n√n log n
+�
+Nk(u)ϵk(u)−1 −
+a(β+1)
+β−a(β+1)akδk−1
+a(β+1)
+β−a(β+1)akδn
+�
+ϵk(u).
+The ensures that, for all 1 ≤ k ≤ n,
+∥Wn∆Hk(u)∥2 ≤
+4a2
+k
+n3 log n(2β + 1)2
+��
+(2β + 1)−2 Nk(u)
+ϵk(u)
+�2 + δ2
+k−1 + δ2
+n
+�
+ϵk(u)2 ≤ C(β)
+n
+(5.13)
+for some constant C(β) > 0. Hence,
+n
+�
+k=1
+E
+�
+∥Wn∆Hk(u)∥4�
+≤ 1
+nC(β) → 0
+as
+n → ∞
+P-a.s.
+since, for all ϵ > 0,
+n
+�
+k=1
+E
+�
+∥Wn∆Hk(u)∥21{∥Wn∆Hk(u)∥>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Wn∆Hk(u)∥4�
+→ 0
+as
+n → ∞.
+(5.14)
+Therefore, the condition (H.2), or the Lindeberg condition, is satisfied using (5.14). Hereafter, we
+know that
+log
+�
+det W −1
+n
+�2
+log log n
+→ 4
+as
+n → ∞.
+This ensures that there exists some constant C2(β) > 0 such that
+∞
+�
+n=1
+1
+�
+log
+�
+det W −1
+n
+�2�2 E
+�
+∥Wn∆Hn(u)∥4|Fn−1
+�
+≤
+∞
+�
+n=1
+C2(β)
+(log log n)2 E
+�
+∥Wn∆Hn(u)∥4|Fn−1
+�
+.
+(5.15)
+We get from (5.13) that
+∞
+�
+n=1
+1
+(log log n)2 ∥Wn∆Hn(u)∥4 ≤ C(β)
+∞
+�
+n=1
+1
+(n log n log log n)2 < ∞
+P-a.s.
+for some constant C(β) > 0 depending only onβ. The condition (H.4) is satisfied using the above
+together with (5.15). Then,
+1
+log
+�
+det W −1
+n
+�2
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkHk(u)Hk(u)T W T
+k → 1
+duT uW
+as n → ∞ P-a.s. where
+W = 4(2β + 1)2
+9
+�
+1
+−1
+−1
+1
+�
+.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+19
+Furthermore, on the one hand we have
+1
+log log n
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkHk(u)Hk(u)T W T
+k → 1
+duT uW
+as n → ∞ P-a.s. since
+log log n
+log
+�
+det W −1
+n
+�2 → 1
+4
+as
+n → ∞.
+On the other hand, we have
+n log n
+�(det Wn)2 − (det Wn+1)2
+(det Wn)2
+�
+→ 1
+as
+n → ∞
+P-a.s.
+By (5.6) and (5.11), we observe that
+1
+log log n
+n
+�
+k=1
+uT GkGT
+k u
+(k log k)2 =
+1
+log log n
+n
+�
+k=1
+wT WkHk(u)Hk(u)T W T
+k w
+4k log k
+→ wT Ww · 1
+4duT u
+(5.16)
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (5.16).
+□
+Theorem 5.4. In the superdiffusive regime, we have the mean square convergence, given by
+E
+�����
+1
+na(β+1)−β Gn −
+1
+1 + a(β + 1) − β Lβ
+����
+2�
+→ 0
+as
+n → ∞.
+(5.17)
+Proof. For all test vector u ∈ Rd, it is immediate that
+E
+�����
+1
+na(β+1)−β Gn(u) −
+1
+1 + a(β + 1) − β Lβ(u)
+����
+2�
+≤ 2E
+�����
+1
+n1+a(β+1)−β Zn(u)
+����
+2�
++ 2E
+�����
+1
+n1+a(β+1)−β ·
+a(β + 1)
+a(β + 1) − β δnMn −
+1
+1 + a(β + 1) − β Lβ
+����
+2�
+.
+(5.18)
+By (4.20) and (5.7), the second term converges to zero. Looking back to the first term in (5.18),
+we observe
+E
+�����
+1
+n1+a(β+1)−β Zn(u)
+����
+2�
+≤
+4
+n1+2(a(β+1)−β)
+n
+�
+k=1
+E
+�
+Nk(u)2�
++
+4
+n1+2(a(β+1)−β)
+�
+a(β + 1)
+a(β + 1) − β
+�2
+E
+������
+n
+�
+k=1
+akδk−1ϵk(u)
+�����
+2�
+.
+(5.19)
+The first term in (5.19) converges to zero because E[Nk(u)] ≤ (uT u)n for all 1 ≤ k ≤ n, and
+moreover, in the superdiffusive regime we have a(β +1) > β +1/2. The second term in (5.19) also
+converges to zero because
+n−(1+a(β+1)−β)δn →
+1
+1 + a(β + 1) − β ·
+1
+Γ(1 + a(β + 1))Γ(β + 1)
+as
+n → ∞.
+Finally, using the above and that M(u) = �∞
+k=1 akϵk(u) is bounded in L2, the assertion follows.
+□
+5.3. Scaling limit.
+Theorem 5.5. The barycenter process admits a scaling limit at the diffusive regime, or conver-
+gence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+� 1
+√nG⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+Wtr dr, t ≥ 0
+�
+
+20
+JIAMING CHEN AND LUCILE LAULIN
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.3 with its
+covariance defined in (4.6). In particular,
+E
+��
+1
+�
+0
+Wsv dv
+��
+1
+�
+0
+Wtu du
+�T �
+=
+β
+3(β(1 − a) − a)(1 − a)s · 1
+dId
++
+2(a(β + 1)(1 − a) + aβ)
+3(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)(1 + (1 − a)(β + 1))ta−β(1−a)s1−a+β(1−a) · 1
+dId
+(5.20)
+for all 0 ≤ s ≤ t < ∞.
+Proof. An easy calculation leads to
+lim
+n→∞
+1
+��nG⌊nt⌋ = lim
+n→∞
+1
+�
+0
+1
+√nS⌊ntr⌋ dr =⇒
+1
+�
+0
+Wtr dr
+which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]). Then, the last convergence
+in law is due to the functional central limit Theorem 4.3, with (Wt)t≥0 defined there. Hence, the
+barycenter process (Gn)n∈N admits a Gaussian scaling limit in the diffusive regime as well, with
+covariance
+E
+��
+1
+�
+0
+Wsv dv
+��
+1
+�
+0
+Wtu du
+�T �
+= 2
+1
+�
+0
+u
+�
+0
+E
+�
+WsvW T
+tu
+�
+dv du.
+Using (4.6), the formula (5.20) and the assertion follows.
+□
+Theorem 5.6. The barycenter process admits a scaling limit at the critical regime, or convergence
+in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+�
+1
+�
+nt log n
+G⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+(2β + 1)Btr dr, t ≥ 0
+�
+where (Bt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.6 with its
+covariance defined in (4.17).
+Proof. For each r ∈ [0, 1], (3.2) and (4.11) implies that
+lim
+n→∞
+1
+�
+nt log n
+· M⌊ntr⌋(u)
+a⌊ntr⌋µ⌊ntr⌋
+= lim
+n→∞
+1
+�
+nt log n
+�
+ntr(log n + r
+t log r)
+�1/2Btr(u)
+P-a.s.
+for all u ∈ Rd. Moreover, (4.16) yields
+lim
+n→∞
+1
+�
+nt log n
+N⌊ntr⌋(u) = lim
+n→∞ r1/2 ·
+1
+�
+ntr log n
+N⌊ntr⌋(u) = 0
+P-a.s.
+for all u ∈ Rd. By (4.15), we have
+�
+1
+�
+nt log n
+S⌊ntr⌋(u), t ≥ 0
+�
+=⇒
+�
+(2β + 1)Btr(u), t ≥ 0
+�
+for all u ∈ Rd and r ∈ [0, 1]. Hence, we use again
+lim
+n→∞
+1
+�
+nt log n
+G⌊nt⌋ = lim
+n→∞
+1
+�
+0
+1
+�
+nt log n
+S⌊ntr⌋ dr =⇒
+1
+�
+0
+(2β + 1)Btr dr
+and the assertion is verified.
+□
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+21
+Theorem 5.7. The barycenter process admits a scaling limit at the superdiffusive regime, or
+convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+�
+1
+na(β+1)−β G⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+Qtr dr, t ≥ 0
+�
+with the covariance specified in (5.3) and the limiting Lβ characterized in Theorem 4.8 and Qt =
+ta(β+1)−βLβ characterized in Theorem 4.9 for all t ≥ 0.
+Proof. Again, we find that
+lim
+n→∞
+1
+na(β+1)−β G⌊nt⌋ =
+1
+�
+0
+1
+na(β+1)−β S⌊ntr⌋ dr =⇒
+1
+�
+0
+Qtr dr
+which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]). Then, the last convergence
+in law is due to the functional central limit Theorem 4.9. Hence the barycenter process (Gn)n∈N
+admits a non-degenerate scaling limit in the superdiffusive regime as well, with covariance
+E
+��
+1
+�
+0
+Qsv dv
+��
+1
+�
+0
+Qtu du
+�T �
+= 2
+1
+�
+0
+u
+�
+0
+E
+�
+QsvQT
+tu
+�
+dv du = ta(β+1)−βsa(β+1)−β
+(1 + a(β + 1) − β)2 E
+�
+LβLT
+β
+�
+= ta(β+1)−βsa(β+1)−β
+(1 + a(β + 1) − β)2
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2 · 1
+dId
+for all 0 ≤ s ≤ t < ∞.
+□
+6. Velocity of quadratic mean displacement
+In this Section, we investigate the velocity of the mean square displacement of the MARW.
+This quantitative estimates give us the information on how fast the limit Theorems in Section 4
+are carried on. Similar convergence velocities have been discussed in [20, 26], where the authors
+analyzed the convergence velocity of the moments of a one-dimensional elephant random walk of
+all orders. In the superdiffusive regime, the convergence velocity was discussed in [6]. Here, only
+the rate of quadratic moment convergence for the MARW in all of the three (diffusive, critical,
+and superdiffusive) regimes are discussed.
+Following the limit Theorems in Section 4, we expect the asymptotic behavior of the mean
+square displacement is as follows,
+E
+�
+SnST
+n
+�
+∼
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+n ·
+(a−2β)(1−a)(β+1)+β(a+1)
+(2(β+1)(1−a)−1)(a−β(1−a))(1−a) · 1
+dId
+when
+a < 1 −
+1
+2(β+1)
+n log n · (2β + 1)2 · 1
+dId
+when
+a = 1 −
+1
+2(β+1)
+n2(a(β+1)−β) ·
+�
+a(β+1)
+β−a(β+1)
+�2
+Γ(2(a−1)(β+1)+1)
+Γ((2a−1)(β+1)+1)2 · 1
+dId
+when
+a > 1 −
+1
+2(β+1),
+(6.1)
+where the notation ∼ indicates two sequences an ∼ bn if and only if an/bn → 1 as n → ∞.
+The aim of this Section is not only to show that the above estimates (6.1) are valid, but also
+to investigate the exact velocity of their convergence in the diffusive and critical regime.
+6.1. Diffusive regime.
+Theorem 6.1. For all p < (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,
+1
+nE
+�
+SnST
+n
+�
+−
+(a − 2β)(1 − a)(β + 1) + β(a + 1)
+(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1
+dId
+∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1
+dId.
+
+22
+JIAMING CHEN AND LUCILE LAULIN
+Proof. Take the vector v = (1, −1)T and Vn ∈ R2×2 as in (A.16). Then,
+1
+√nSn(u) = vT VnLn(u),
+where Ln(u) = (Nn(u), Mn(u))T is as in (3.6). In particular,
+1
+nuT E
+�
+SnST
+n
+�
+u = vT VnE
+�
+Ln(u)Ln(u)T �
+V T
+n v
+= vT VnE
+� �
+E
+�
+Nn(u)2�
+E
+�
+Nn(u)Mn(u)
+�
+E
+�
+Mn(u)Nn(u)
+�
+E
+�
+Mn(u)2�
+� �
+V T
+n v
+= vT VnE
+� �
+E
+�
+⟨N(u)⟩n
+�
+E
+�
+⟨N(u), M(u)⟩n
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+E
+�
+⟨M(u)⟩n
+�
+� �
+V T
+n v.
+Therefore,
+1
+nuT E
+�
+SnST
+n
+�
+u = 1
+nE
+�
+⟨N(u)⟩n
+�
++
+1
+na2nµ2n
+�
+a(β + 1)
+β − a(β + 1)
+�2
+E
+�
+⟨M(u)⟩n
+�
+−
+2
+nanµn
+�
+a(β + 1)
+β − a(β + 1)
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+.
+Since the test vector u ∈ Rd is taken arbitrarily, we get from Lemmas A.15 and A.16 that
+1
+nE
+�
+SnST
+n
+�
+−
+(a − 2β)(1 − a)(β + 1) + β(a + 1)
+(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1
+dId
+∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1
+dId
+as
+n → ∞.
+□
+6.2. Critical regime.
+Theorem 6.2. When p = (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,
+1
+n log nE
+�
+SnST
+n
+�
+− (2β + 1)2 · 1
+dId ∼ −(C1(log n)−1 + C2n−1) · 1
+dId.
+Proof. Take w = (1, −1)T and Wn ∈ R2×2 as in (A.28). Then
+1
+√n log nSn(u) = wT WnLn(u) as in
+(A.29) for all u ∈ Rd. In particular,
+1
+n log nuT E
+�
+SnST
+n
+�
+u = wT WnE
+�
+Ln(u)Ln(u)T �
+W T
+n w.
+Hence,
+1
+n log nuT E
+�
+SnST
+n
+�
+u = wT WnE
+� �
+E
+�
+⟨N(u)⟩n
+�
+E
+�
+⟨N(u), M(u)⟩n
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+E
+�
+⟨M(u)⟩n
+�
+� �
+W T
+n w.
+Therefore, we get by (3.4) as n → ∞,
+1
+n log nuT E
+�
+SnST
+n
+�
+u =
+1
+n log n
+�
+E
+�
+⟨N(u)⟩n
+�
++ (2β + 1)2
+a2nµ2n
+E
+�
+⟨M(u)⟩n
+��
+,
+which implies
+1
+n log nE
+�
+SnST
+n
+�
+− (2β + 1)2 · 1
+dId ∼ −(C1(log n)−1 + C2n−1) · 1
+dId
+as
+n → ∞.
+□
+6.3. Superdiffusive regime.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+23
+Theorem 6.3. When p > (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,
+1
+n2(a(β+1)−β) E
+�
+SnST
+n
+�
+−
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2 · 1
+dId
+∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β)).
+Proof. Similar to previous computations for the diffusive regime, we have for all u ∈ Rd,
+1
+n2(a(β+1)−β) uT E
+�
+SnST
+n
+�
+u =
+1
+n2(a(β+1)−β) E
+�
+⟨N(u)⟩n
+�
++
+1
+n2(a(β+1)−β)a2nµ2n
+�
+a(β + 1)
+β − a(β + 1)
+�2
+E
+�
+⟨M(u)⟩n
+�
+−
+2
+n2(a(β+1)−β)anµn
+�
+a(β + 1)
+β − a(β + 1)
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+.
+Hence, by (2.5), (2.6), (3.5) and since u ∈ Rd is arbitrary,
+1
+n2(a(β+1)−β) E
+�
+SnST
+n
+�
+−
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2 · 1
+dId
+∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β))
+as
+n → ∞.
+□
+7. Cram´er moderate deviations
+In this Section, we discuss the Cram´er moderate deviations for the multidimensional reinforced
+random walk (Sn)n∈N. The similar statistical quantity as well as the Berry-Esseen bound for the
+one-dimensional elephant random walk (ERW) without amnesia-reinforcement has been given in
+[20]. Our derivation of Cram´er moderate deviations for the MARW does not rely on a Berry-
+Esseen bound. The discussion of such statistical quantities is expected to reveal the transience
+property and the central limit Theorems for the MARW. For this direction, readers are refereed
+to [3, 16]. Thanks to Lemma A.21 and Lemma A.22, we can properly state the Cram´er moderate
+deviations principles for the MARW.
+Theorem 7.1. In the diffusive and critical regimes, we have the following Cram´er moderate
+deviations for the MARW. Let (ϑn)n∈N ⊆ R be a non-decreasing sequence so that ϑn/√n → 0 as
+n → ∞. Take any non-empty Borel set B ⊆ Rd, then we have
+−
+inf
+x∈int B
+1
+2∥x∥2 ≤ lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ lim sup
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ − inf
+x∈cl B
+1
+2∥x∥2,
+where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively.
+Proof. Our proof will only present the Cram´er moderate deviations for the MARW in the diffusive
+regime. The same property for the critical regime follows from exactly the same steps. First, take
+xB = infx∈B ∥x∥.
+Then it is obvious that infx∈cl B ∥x∥ ≤ xB and infx∈cl B ∥x∥2/2 ≤ x2
+B/2.
+Henceforth,
+P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤
+d
+�
+j=1
+P
+�����
+anµnSj
+n
+√wn
+���� ≥ ϑnxB
+d
+�
+≤
+�
+1 − Φ(ϑnxB)
+�
+F(B, ϑ, n),
+(7.1)
+
+24
+JIAMING CHEN AND LUCILE LAULIN
+where we write
+F(B, ϑ, n) := 2Cd · exp
+�
+1
+√n
+� ϑnxB
+2d
+�3 + 1
+n
+� ϑnxB
+2d
+�2 +
+1
+√n(1 + 1
+2 log n)(1 + ϑnxB
+2d )
+�
++ 2Cd · exp
+�
+1
+√n
+� ϑnxB
+2d
+�3 +
+1
+n2(1−a)(β+1)
+� ϑnxB
+2d
+�2 +
+1
+√n(1 + 1
+2 log n)(n1/2−(1−a)(β+1) + ϑnxB
+2d )
+�
+.
+Hence,
+lim sup
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ −1
+2x2
+B ≤ − inf
+x∈cl B
+1
+2∥x∥2.
+To achieve the asymptotic lower bound, we first notice that this assertion automatically holds if
+int B = ∅, whence − infx∈∅ ∥x∥2/2 = −∞. Consequently, we assume that int B ̸= ∅. Notice that
+int B is open in Rd. Hence, for all ϵ∗ > 0 sufficiently small, we could find x∗ ∈ int B with
+0 < 1
+2∥x∗∥2 <
+inf
+x∈int B
+1
+2∥x∥2 + ϵ∗
+and
+0 < min
+���xj
+∗
+�� : 1 ≤ j ≤ d
+�
+.
+Choose ϵ∗∗ sufficient small such that 0 < ϵ∗∗ <
+���xj
+∗
+��� for each j = 1, . . . , d. Then,
+U(x∗, ϵ∗∗) ⊆ int B ⊆ B,
+where
+U(x∗, ϵ∗∗) :=
+�
+x ∈ Rd :
+��xj − xj
+∗
+�� < ϵ∗∗ for all j
+�
+.
+On the other hand,
+P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≥ P
+�anµnSn
+√wn
+∈ ϑn · U(x∗, ϵ∗∗)
+�
+≥
+d
+�
+j=1
+P
+�
+ϑn(xj
+∗ + ϵ∗∗) ≥ anµnSj
+n
+√wn
+≥ ϑn(xj
+∗ − ϵ∗∗)
+�
+.
+From Lemma A.21 and Lemma A.22, we know that
+lim
+n→∞ P
+�anµnSj
+n
+√wn
+≥ ϑn(xj
+∗ + ϵ∗∗)
+��
+P
+�anµnSj
+n
+√wn
+≥ ϑn(xj
+∗ − ϵ∗∗)
+�
+= 0
+for each
+j.
+Similar to (7.1),
+lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≥ −1
+2∥x∗ − ϵ∗∗∥2.
+Letting ϵ∗∗ → 0, we observe that
+lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≥ −1
+2∥x∗∥2 ≥ −
+inf
+x∈int B
+1
+2∥x∥2 − ϵ∗.
+Since ϵ∗ > 0 was take arbitrarily, letting ϵ∗ → 0, we verify the assertion.
+□
+Appendix A. Technical Lemmas
+A.1. Asymptotics of the processes. We start by introducing the following processes that are
+of great influence on the behavior of the random walk. Let (e1, e2, . . . , ed) denote a canonical
+Euclidean basis of Rd. For each n ∈ N and 1 ≤ j ≤ d, define
+N X
+n (j) =
+n
+�
+k=1
+1{Xj
+k̸=0}µk
+and
+Σn =
+d
+�
+j=1
+N X
+n (j)ejeT
+j ,
+(A.1)
+such that (Σn)n∈N is a matrix-valued process.
+Lemma A.1. We have the following almost sure convergence in the three regimes.
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+(A.2)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+25
+Proof. For each n ∈ N and 1 ≤ j ≤ d, define
+ΛX
+n (j) = N X
+n (j)
+n
+.
+(A.3)
+It follows from (A.1) that
+ΛX
+n+1(j) =
+n
+n + 1ΛX
+n (j) +
+1
+n + 11{Xj
+n+1̸=0}µn+1.
+Moreover, we observe thanks to (A.12) that
+ΛX
+n+1(j) =
+n
+n + 1 · γnΛX
+n (j) +
+1
+n + 11{Xj
+n+1̸=0}µn+1 − a(β + 1)
+n + 1 ΛX
+n (j)
+=
+n
+n + 1 · γnΛX
+n (j) + µn+1
+n
+δX
+n+1(j) + (1 − a)µn+1
+d(n + 1)
+with
+δX
+n+1(j) = 1{Xj
+n+1̸=0} − P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+.
+Then, by (2.4) we know
+ΛX
+n (j) =
+1
+nan
+�
+ΛX
+1 (j) + 1 − a
+d
+n
+�
+k=2
+akµk + HX
+n (j)
+�
+(A.4)
+with
+HX
+n (j) =
+n
+�
+k=2
+akµkδX
+k (j).
+It is clear that for a fixed 1 ≤ j ≤ d, the real-valued process (HX
+n (j))n∈N is locally square-integrable
+since it is a finite sum. Afterwards, this process appears to be a martingale adapted to (Fn)n∈N
+because (δX
+n (j))n∈N satisfied the martingale difference relation E[δX
+n+1(j)|Fn] = 0. It is obvious
+that
+⟨HX(j)⟩n ≤ wn =
+n
+�
+k=1
+(akµk)2
+P-a.s.
+Hence, we get by [18, Theorem 4.3.15] that for all γ > 0
+HX
+n (j)2
+⟨HX(j)⟩n
+= o
+��
+log⟨HX(j)⟩n
+�1+γ�
+P-a.s.
+(A.5)
+Since ⟨HX(j)⟩n ≤ wn and by (A.5), we obtain that
+HX
+n (j)2 = o
+�
+wn
+�
+log wn
+�1+γ�
+P-a.s.
+In the diffusive regime, by Lemma A.1 and (3.3), we have
+HX
+n (j)2 = o
+�
+n1−2(a(β+1)−β)�
+log n
+�1+γ�
+P-a.s.
+By (2.5) and (2.6), we observe that
+� HX
+n (j)
+nanµn+1
+�2
+= o
+�
+n−1�
+log n
+�1+γ�
+P-a.s.
+Hence
+HX
+n (j)
+nanµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+By (2.5) and (2.6) again, we observe further
+1
+nanµn+1
+n
+�
+k=1
+akµk →
+1
+(1 − a)(β + 1)
+as
+n → ∞.
+(A.6)
+
+26
+JIAMING CHEN AND LUCILE LAULIN
+Hence, we have
+µ−1
+n+1ΛX
+n (j) →→
+1
+β + 1
+as
+n → ∞.
+By (A.3) and (A.4), we can then conclude that
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+in the diffusive regime. In the critical regime, where a = 1 −
+1
+2(β+1), we have from (3.4))
+HX
+n (j)2 = o
+�
+log n
+�
+log log n
+�1+γ�
+P-a.s.
+Hence
+� HX
+n (j)
+nanµn+1
+�2
+= o
+�
+n−1 log n
+�
+log log n
+�1+γ�
+P-a.s.
+which implies that
+HX
+n (j)
+nanµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+Similar to the convergence in (A.6), in the critical regime, we observe
+1
+nanµn+1
+n
+�
+k=1
+akµk → 1
+2
+P-a.s.
+Hence, we conclude that
+µ−1
+n+1ΛX
+n (j) →
+1
+d(β + 1)
+and
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+which proves (A.2). In the superdiffusive regime, we have
+HX
+n (j)2 = o
+�
+1
+�
+P-a.s.
+and then
+� HX
+n (j)
+nanµn+1
+�2
+= o
+�
+n−2(1−a)(β+1)�
+P-a.s.
+which implies
+HX
+n (j)
+nanµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+We can similarly show that
+µ−1
+n+1ΛX
+n (j) →
+1
+β + 1
+as
+n → ∞.
+which then ensures that
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+Consequently, the assertion is verified.
+□
+The next result follows directly from the definition of Mn and Nn
+Lemma A.2. We have the following formulas for the predictable matrix-valued quadratic varia-
+tions
+⟨M⟩n = (a1µ1)2E
+�
+X1XT
+1
+�
++
+n−1
+�
+k=1
+a(β + 1)
+ka−2
+k+1
+µk+1Σk + 1 − a
+da−2
+k+1
+µ2
+k+1Id −
+�γk − 1
+a−1
+k+1
+�2
+YkY T
+k ,
+(A.7)
+and
+⟨N⟩n =
+�
+β
+β − a(β + 1)
+�2
+E
+�
+X1XT
+1
+�
++
+n−1
+�
+k=1
+a(β + 1)
+kµk+1
+Σk + 1 − a
+d
+Id −
+�γk − 1
+µk+1
+�2
+YkY T
+k .
+(A.8)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+27
+In particular, we have
+Tr⟨M⟩n = wn −
+n
+�
+k=1
+(γk − 1)2a2
+k+1∥Yk∥2,
+(A.9)
+and
+Tr⟨N⟩n =
+�
+β
+β − a(β + 1)
+�2
+n −
+n−1
+�
+k=1
+�a(β + 1)
+kµk+1
+�2
+∥Yk∥2.
+(A.10)
+Lemma A.3. We have the following estimate for the matrix-valued conditional expectation.
+E
+�
+ϵn+1ϵT
+n+1|Fn
+�
+= a(β + 1)
+n
+µn+1Σn + 1 − a
+d
+µ2
+n+1Id − (γn − 1)2YnY T
+n .
+And as a consequence
+E
+�
+∥ϵn+1∥2|Fn
+�
+= µ2
+n+1 − (γn − 1)2∥Yn∥2.
+Proof. Observe that
+E
+�
+ϵn+1ϵT
+n+1|Fn
+�
+= E
+�
+Yn+1Y T
+n+1|Fn
+�
+− γ2
+nYnY T
+n
+with
+E
+�
+Yn+1Y T
+n+1|Fn
+�
+= YnY T
+n + 2µn+1YnE
+�
+XT
+n+1|Fn
+�
++ µ2
+n+1E
+�
+Xn+1XT
+n+1|Fn
+�
+=
+�
+1 + 2a(β + 1)
+n
+�
+YnY T
+n + µ2
+n+1E
+�
+Xn+1XT
+n+1|Fn
+�
+.
+(A.11)
+For all k ≥ 1, we know that XkXT
+k = �d
+j=1 1{Xj
+k̸=0}ejeT
+j . Then
+P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+=
+n
+�
+k=1
+P
+�
+βn+1 = k
+�
+· P
+�
+(AnXk)j ̸= 0|Fn
+�
+=
+n
+�
+k=1
+1{Xj
+k̸=0}P
+�
+An = ±Id
+�
+· (β + 1)µk
+nµn+1
++
+n
+�
+k=1
+�
+1 − 1{Xj
+k̸=0}
+�
+P
+�
+An = ±Jd
+�
+· (β + 1)µk
+nµn+1
+.
+Hence
+P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+= β + 1
+nµn+1
+·
+�
+P
+�
+An = +Id
+�
+− P
+�
+An = +Jd
+��
+N X
+n (j) + 2P
+�
+An = +Jd
+�
+= a(β + 1)
+nµn+1
+N X
+n (j) + 1 − a
+d
+.
+(A.12)
+Therefore
+E
+�
+Xn+1XT
+n+1|Fn
+�
+=
+d
+�
+j=1
+P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+ejeT
+j = a(β + 1)
+nµn+1
+Σn + 1 − a
+d
+Id.
+(A.13)
+And from (A.11) and (A.13) we can conclude that
+E
+�
+ϵn+1ϵT
+n+1|Fn
+�
+= E
+�
+Yn+1Y T
+n+1|Fn
+�
+− γ2
+nYnY T
+n
+=
+�
+1 + 2a(β + 1)
+n
+�
+YnY T
+n + a(β + 1)
+n
+µn+1Σn + 1 − a
+d
+µ2
+n+1Id − γ2
+nYnY T
+n
+= a(β + 1)
+n
+µn+1Σn + 1 − a
+d
+µ2
+n+1Id − (γn − 1)2YnY T
+n .
+(A.14)
+On the other hand
+Tr(Σn) = nµn+1
+β + 1 .
+(A.15)
+Taking traces in (A.14) and by (A.15), we have
+E
+�
+∥ϵn+1∥2|Fn
+�
+= µ2
+n+1 − (γn − 1)2∥Yn∥2
+which ensures that the assertion is verified.
+□
+
+28
+JIAMING CHEN AND LUCILE LAULIN
+A.2. Scaling limits of the random walk and the barycenter.
+A.2.1. The diffusive regime.
+Lemma A.4. For each n ∈ N and test vector u ∈ Rd, let
+Vn =
+1
+√n
+�
+1
+0
+0
+a(β+1)
+β−a(β+1)(anµn)−1
+�
+and
+v =
+�
+1
+−1
+�
+.
+(A.16)
+Then
+vT VnLn(u) =
+1
+√nSn(u).
+(A.17)
+And for all t ≥ 0, we have
+Vn⟨L(u)⟩⌊nt⌋V T
+n → uT u
+d Vt
+as
+n → ∞
+P-a.s.
+(A.18)
+where
+Vt =
+1
+(β − a(β + 1))2
+�
+β2t
+aβ
+1−at1+β−a(β+1)
+aβ
+1−at1+β−a(β+1)
+a2(β+1)2
+1−2a(β+1)+2β t1+2β−2a(β+1)
+�
+.
+(A.19)
+Proof. From Lemma A.3 and the fact that ⟨M(u)⟩n = uT ⟨M⟩nu, we see that
+⟨M(u)⟩⌊nt⌋ = a2
+1µ2
+1uT E
+�
+X1XT
+1
+�
+u
++
+⌊nt⌋−1
+�
+k=1
+a(β + 1)
+k
+a2
+k+1µk+1uT Σku + 1 − a
+d
+a2
+k+1µ2
+k+1uT u − (γk − 1)2a2
+k+1uT YkY T
+k u
+and
+⟨N(u)⟩⌊nt⌋ =
+�
+β
+β − a(β + 1)
+�2
+uT E
+�
+X1XT
+1
+�
+u
++
+�
+β
+β − a(β + 1)
+�2 ⌊nt⌋−1
+�
+k=1
+a(β + 1)
+kµk+1
+uT Σku + 1 − a
+d
+uT u −
+�γk − 1
+µk+1
+�2
+uT YkY T
+k u.
+Using a similar token and Lemma A.1, we can work out the off-diagonal entries in ⟨L(u)⟩⌊nt⌋, and
+we obtain that
+lim
+n→∞ Vn⟨L(u)⟩⌊nt⌋V T
+n
+= lim
+n→∞
+uT u
+nd(β − a(β + 1))2
+�
+�
+�
+β2⌊nt⌋
+a(β+1)β
+anµn
+�⌊nt⌋−1
+k=0
+ak+1µk+1
+a(β+1)β
+anµn
+�⌊nt⌋−1
+k=0
+ak+1µk+1
+�
+a(β+1)
+anµn
+�2 �⌊nt⌋−1
+k=0
+(ak+1µk+1)2
+�
+�
+�
+=
+uT u
+d(β − a(β + 1))2
+�
+β2t
+aβ
+1−at1−(a(β+1)−β)
+aβ
+1−at1−(a(β+1)−β)
+a2(β+1)2
+1−2(a(β+1)−β)t1−2(a(β+1)−β)
+�
+= uT u
+d Vt
+P-a.s.
+where the last equality is due to (2.5) and (2.6). Thus, it implies that
+1
+nanµn
+n
+�
+k=1
+akµk →
+1
+1 − (a(β + 1) − β)
+and
+1
+n(anµn)2
+n
+�
+k=1
+(akµk)2 →
+1
+1 − 2(a(β + 1) − β)
+as n → ∞. Hence, equation (A.18) holds and the assertion is then verified.
+□
+Lemma A.5. The MARW satisfies the Lindeberg condition in the diffusive regime. That is, for
+all t ≥ 0 and all ϵ > 0,
+⌊nt⌋
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+29
+Proof. On the one hand, it is easy to compute from (3.7) and (A.16) that, for all 1 ≤ k ≤ n,
+Vn∆Lk(u) =
+1
+√n(β − a(β + 1))µn
+�
+β µn
+µk
+a ak
+an
+�
+ϵk(u)
+which implies
+∥Vn∆Lk(u)∥2 =
+1
+n(β − a(β + 1))2
+�β2
+µ2
+k
++
+a2a2
+k
+(anµn)2
+�
+ϵk(u)2.
+Hence
+∥Vn∆Lk(u)∥4 ≤
+2
+n2(β − a(β + 1))4
+�β4
+µ4
+k
++
+a4a4
+k
+(anµn)4
+�
+ϵk(u)4.
+(A.20)
+On the other hand, from (2.5) we observe that
+1
+na2n
+n
+�
+k=1
+a2
+k ≤ C1(a, β)−1
+and
+1
+na4n
+n
+�
+k=1
+a4
+k ≤ C2(a, β)−1
+for all
+n ∈ N
+(A.21)
+and where C1(a, β), C2(a, β) > 0 are constants depending only on a and β. Moreover, we get that
+sup
+1≤k≤n
+|ϵk(u)| ≤
+sup
+1≤k≤n
+∥ϵk∥∥u∥ ≤
+sup
+1≤k≤n
+(β + 2)µk∥u∥ ≤ (β + 2)µn∥u∥.
+(A.22)
+Hence, we deduce from (A.21) and (A.22)
+n
+�
+k=1
+∥Vn∆Lk(u)∥4 ≤
+2
+n2(β − a(β + 1))4
+��
+β(β + 2)
+�4∥u∥4 +
+�
+a(β + 2)
+�4∥u∥4
+C2(a, β)
+�
+→ 0
+(A.23)
+as n → ∞ P-a.s. This implies that
+n
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Therefore, for all ϵ > 0, we obtain
+n
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as n → ∞ P-a.s. This yields finally
+⌊nt⌋
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+⌊nt⌋
+�
+k=1
+E
+����(VnV −1
+⌊nt⌋)V⌊nt⌋∆Lk(u)
+���
+4
+|Fk−1
+�
+→ 0
+as n → ∞ P-a.s. since VnV −1
+⌊nt⌋ converges as n → ∞.
+□
+Lemma A.6. The deterministic matrix Vt defined in (A.19) can be rewritten as
+Vt = tα1K1 + tα2K2 + · · · + tαqKq
+with q ∈ N, αj > 0 and each Kj is a symmetric matrix for all 1 ≤ j ≤ 1.
+Proof. A direct computation analoguous to the one in [32] shows that Vt = tα1K1+tα2K2+tα3K3,
+where
+α1 = 1,
+α2 = 1 − a(β + 1) > 0,
+α3 = 1 − 2a(β + 1) > 0
+since a < 1 −
+1
+2(β+1) is in the diffusive regime. Moreover
+K1 =
+β2
+(a(β + 1) − β)2
+�
+1
+0
+0
+0
+�
+,
+K2 =
+aβ
+(1 − a)(a(β + 1) − β)2
+�
+0
+1
+1
+0
+�
+,
+K3 =
+a2(β + 1)2
+(1 − 2a(β + 1) + 2β)(a(β + 1) − β)2
+�
+0
+0
+0
+1
+�
+.
+
+30
+JIAMING CHEN AND LUCILE LAULIN
+□
+Lemma A.7. Given the matrix-valued process (Vn)n∈N define in (A.16), we have
+∞
+�
+n=1
+1
+�
+log
+�
+det V −1
+n
+�2�2 E
+�
+∥Vn∆Ln(u)∥4|Fn−1
+�
+< ∞
+P-a.s.
+Proof. From (A.16), it is immediate that
+det V −1
+n
+= β − a(β + 1)
+a(β + 1)
+nanµn.
+(A.24)
+By (2.5) and (2.6), we obtain
+log
+�
+det V −1
+n
+�2
+log n
+→ 2(1 − a)(β + 1)
+as
+n → ∞
+P-a.s.
+(A.25)
+Hence there exists a constant C(a, β) > 0 depending only on a and β such that
+∞
+�
+n=1
+1
+�
+log
+�
+det V −1
+n
+�2�2 E
+�
+∥Vn∆Ln(u)∥4|Fn−1
+�
+≤ C(a, β)
+∞
+�
+n=1
+1
+(log n)2 E
+�
+∥Vn∆Ln(u)∥4|Fn−1
+�
+.
+(A.26)
+Hereafter, equations (A.20), (A.22), (A.23) together imply that
+∞
+�
+n=1
+1
+(log n)2 ∥Vn∆Ln(u)∥4 ≤ C′(a, β)
+∞
+�
+n=1
+1
+(n log n)2 < ∞
+P-a.s.
+(A.27)
+for some other constant C′(a, β) > 0 depending only on a and β. Consequently, equation (A.27)
+together (A.26) ensures that the assertion is verified.
+□
+A.2.2. The critical regime.
+Lemma A.8. For each n ∈ N and test vector u ∈ Rd, let
+Wn =
+1
+√n log n
+�
+1
+0
+0
+2β+1
+anµn
+�
+and
+w =
+�
+1
+−1
+�
+.
+(A.28)
+Then for all t ≥ 0, we have
+wT WnLn(u) =
+1
+√n log nSn(u)
+(A.29)
+and
+Wn⟨L(u)⟩nW T
+n → uT u
+d W
+as
+n → ∞
+P-a.s.
+where
+Wt = (2β + 1)2
+�
+0
+0
+0
+1
+�
+.
+(A.30)
+Proof. It is clear that (A.29) follows from (3.2). Using a similar token than for the proof Lemma
+A.4, we have
+lim
+n→∞ Wn⟨L(u)⟩nW T
+n
+= lim
+n→∞
+4uT u
+(n log n)d
+�
+�
+�
+β2n
+β(β+ 1
+2 )
+anµn
+�n−1
+k=0 ak+1µk+1
+β(β+ 1
+2 )
+anµn
+�n−1
+k=0 ak+1µk+1
+�
+β+ 1
+2
+anµn
+�2 �n−1
+k=0(ak+1µk+1)2
+�
+�
+�
+= 4uT u
+d
+�
+0
+0
+0
+�
+β + 1
+2
+�2
+�
+= uT u
+d W
+P-a.s.
+and the proof is complete.
+□
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+31
+Lemma A.9. The MARW satisfies the Lindeberg condition in the critical regime. That is, for all
+t ≥ 0 and all ϵ > 0, given the (Wn)n∈N defined in (A.16), it satisfies
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Proof. We state that equations (A.20) and (A.21) remain true with Vn replaced by Wn. More
+precisely, they can be rewritten as
+∥Wn∆Lk(u)∥4 ≤
+32
+(n log n)2
+�β4
+µ4
+k
++
+a4a4
+k
+(anµn)4
+�
+ϵk(u)4
+(A.31)
+and
+1
+na4n
+n
+�
+k=1
+a4
+k ≤ C(a, β)−1
+for all
+n ∈ N
+where C(a, β) > 0 is a constant depending only on t, a, and β. Since (A.22) is not affected by
+switching regimes, we have that
+n
+�
+k=1
+∥Wn∆Lk(u)∥4 ≤
+32
+(n log n)2
+��
+β(β + 2)
+�4∥u∥4 +
+�
+a(β + 2)
+�4∥u∥4
+C(t, a, β)
+�
+→ 0
+(A.32)
+as n → ∞ P-a.s. This implies
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Therefore, for all ϵ > 0, we obtain
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as n → ∞ P-a.s. and the assertion is verified.
+□
+Lemma A.10. Given the matrix-valued sequence (Wn)n∈N define in (A.28), we have
+∞
+�
+n=1
+1
+�
+log
+�
+det W −1
+n
+�2�2 E
+�
+∥Wn∆Ln(u)∥4|Fn−1
+�
+< ∞
+P-a.s.
+Proof. From (A.28), it is immediate that
+det W −1
+n
+=
+1
+2β + 1
+�
+n log n · anµn.
+(A.33)
+Then, we obtain by (2.5) and (2.6) that
+log
+�
+det W −1
+n
+�2
+log log n
+→ 1
+as
+n → ∞
+P-a.s.
+(A.34)
+Hence, there exists a constant C(a, β) > 0 depending only on a and β such that
+∞
+�
+n=1
+1
+�
+log
+�
+det W −1
+n
+�2�2 E
+�
+∥Wn∆Ln(u)∥4|Fn−1
+�
+≤
+∞
+�
+n=1
+C(a, β)
+(log log n)2 E
+�
+∥Wn∆Ln(u)∥4|Fn−1
+�
+.
+(A.35)
+Hereafter, (A.31) together with (A.32) imply that
+∞
+�
+n=1
+1
+(log log n)2 ∥Wn∆Ln(u)∥4 ≤ C′(a, β)
+∞
+�
+n=1
+1
+(n log n log log n)2 < ∞
+P-a.s.
+for some other constant C′(a, β) > 0 depending only on a and β. Finally, using the above equation
+together with (A.35) completes the proof.
+□
+
+32
+JIAMING CHEN AND LUCILE LAULIN
+Lemma A.11. Fix the test vector u ∈ Rd. The growth rate of the compensator of the partial sum
+of (Nn(u)2)n∈N is less than cubic growth, in the sense that
+1
+n3
+n−1
+�
+k=1
+E
+�
+Nk+1(u)2|Fn
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Proof. The law of iterated expectations and (A.8) yields
+1
+nE
+�
+E
+�
+Nn+1(u)2|Fn
+��
+= 1
+nE
+�
+⟨N(u)⟩n
+�
+→
+�
+β
+β − a(β + 1)
+�2
+uT u
+as
+n → ∞
+P-a.s.
+The strong law of large numbers then yields
+1
+n
+n−1
+�
+k=1
+1
+k E
+�
+Nk+1(u)2|Fk
+�
+→
+�
+β
+β − a(β + 1)
+�2
+uT u
+as
+n → ∞
+P-a.s.
+Hence
+1
+n3
+n−1
+�
+k=1
+E
+�
+Nk+1(u)2|Fn
+�
+≤ 1
+n2
+n−1
+�
+k=1
+1
+k E
+�
+Nk+1(u)2|Fk
+�
+→ 0
+as
+n → ∞
+P-a.s.
+□
+A.2.3. The barycenter process. For the following Toeplitz Lemmas, see [18] and [33].
+Lemma A.12. [33, Theorem 1.1 Part I] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers
+such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N
+�kn
+k=1 |an,k| < ∞. Let (xn)n∈N
+be a real sequence. If xn → 0 as n → ∞, then �kn
+k=1 an,kxk → 0 as n → ∞.
+Lemma A.13. [33, Theorem 1.1 Part II] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers
+such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N
+�kn
+k=1 |an,k| < ∞. Let (xn)n∈N
+be a real sequence. If xn → x as n → ∞ with x ∈ R and �kn
+k=1 an,k = 1, then �kn
+k=1 an,kxk → x
+as n → ∞.
+A.3. Quadratic rate estimates. Our first result is about the convergence rate of the process
+(Yn)n∈N defined in (2.3).
+Lemma A.14. For all p ∈ (0, 1), then we have, as n → ∞,
+E[YnY T
+n ] ∼
+n2a(β+1)
+Γ(1 + 2a(β + 1)) · 1
+dId +
+n1+2β
+Γ(β + 1)2(1 + 2β − 2a(β + 1))(β + 1) · 1
+dId.
+Proof. From (A.11) and (A.13), we see
+E
+�
+Yn+1Y T
+n+1|Fn
+�
+=
+�
+1 + 2a(β + 1)
+n
+�
+YnY T
+n + µ2
+n+1
+�a(β + 1)
+nµn+1
+Σn + 1 − a
+d
+Id
+�
+.
+Then, remember that
+E
+�
+Σn
+�
+=
+d
+�
+j=1
+E
+�
+N X
+n (j)
+�
+ejeT
+j =
+d
+�
+j=1
+n
+�
+k=1
+P
+�
+Xj
+k ̸= 0
+�
+µk · ejeT
+j .
+Lemma A.1 yields E[(nµn+1)−1Σn] ∼ (β + 1)−1 · 1
+dId. Hence,
+E
+�
+Yn+1Y T
+n+1
+�
+∼
+�
+1 + 2a(β + 1)
+n
+�
+E
+�
+YnY T
+n
+�
++ µ2
+n+1
+β + 1 · 1
+dId.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+33
+A recursive argument then gives
+E
+�
+YnY T
+n
+�
+∼
+Γ(n + 2a(β + 1))
+Γ(n)Γ(1 + 2a(β + 1))E
+�
+Y1Y T
+1
+�
++
+n−1
+�
+j=1
+µ2
+j
+β + 1 ·
+�n−1
+k=1(1 + k−12a(β + 1))
+�j−1
+k=1(1 + k−12a(β + 1))
+· 1
+dId
+∼
+Γ(n + 2a(β + 1))
+Γ(n)Γ(1 + 2a(β + 1)) · 1
+dId +
+n−1
+�
+j=1
+µ2
+j
+β + 1 · Γ(n + 2a(β + 1))Γ(j)
+Γ(j + 2a(β + 1))Γ(n) · 1
+dId.
+Employing the asymptotics in (2.1) and (2.6), the assertion follows.
+□
+The process Yn = �n
+k=1 µkXk differs from Sn by a multiplicative factor at each step. When
+there is no amnesia, the asymptotics of these two processes coincide. However, when β ≥ 0, we
+have to treat the general case in another way.
+Lemma A.15. For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞,
+E
+�
+⟨M(u)⟩n
+�
+∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u,
+and
+E
+�
+⟨N(u)⟩n
+�
+∼
+�
+β
+β − a(β + 1)
+�2
+nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.
+Proof. By Lemma A.2
+E
+�
+⟨M(u)⟩n
+�
+= E
+�
+Tr⟨M⟩n
+�
+uT u = wnuT u −
+n
+�
+k=1
+(γk − 1)2a2
+k+1uT E
+�
+YkY T
+k
+�
+u.
+By Lemma A.14 and a finite summation,
+E
+�
+⟨M(u)⟩n
+�
+∼ wnuT u −
+n−1
+�
+k=1
+a2(β + 1)2
+k2
+(k + 1)−2a(β+1)(C1k2a(β+1) + C2k1+2β)uT u
+∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u.
+Similarly,
+E
+�
+⟨N(u)⟩n
+�
+= E
+�
+Tr⟨N⟩n
+�
+uT u =
+�
+β
+β − a(β + 1)
+�2
+nuT u −
+n−1
+�
+k=1
+a2(β + 1)2
+k2
+µ−2
+k+1uT E
+�
+YkY T
+k
+�
+u.
+Hence, using Lemma A.14 again, we observe
+E
+�
+⟨N(u)⟩n
+�
+∼
+�
+β
+β − a(β + 1)
+�2
+nuT u −
+n−1
+�
+k=1
+a2(β + 1)2
+k2
+(k + 1)−2β(C1k2a(β+1) + C2k1+2β)uT u
+∼
+�
+β
+β − a(β + 1)
+�2
+nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.
+□
+Lemma A.16. For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞,
+E
+�
+⟨M(u), N(u)⟩n
+�
+∼
+β
+β − a(β + 1) · Γ(β + 1)Γ(a(β + 1) + 1)
+(1 − a)(β + 1)
+n(1−a)(β+1)uT u
+− (C1n−(1−a)(β+1) + C2n(1−a)(β+1)−1)uT u.
+Proof. By (3.7) and Lemma A.2, for all test vector u ∈ Rd
+∆Ln+1(u) =
+�
+βµ−1
+n+1
+β − a(β + 1)
+�T
+ϵn+1(u),
+
+34
+JIAMING CHEN AND LUCILE LAULIN
+and therefore,
+⟨M(u), N(u)⟩n =
+n
+�
+k=1
+β
+β − a(β + 1)akµ−1
+k E
+�
+ϵk(u)ϵk(u)T |Fk−1
+�
+.
+Taking the trace will give us
+Tr⟨M, N⟩n =
+β
+β − a(β + 1)
+n
+�
+k=1
+akµk −
+β
+β − a(β + 1)
+n
+�
+k=1
+akµ−1
+k (γk − 1)2∥Yk∥2.
+Taking the expectation and using Lemma A.14 completes the proof.
+□
+A.4. Moderate deviations.
+Lemma A.17. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+��∆M j
+n
+�� ≤
+�
+a(β + 1) + 1
+�
+anµn
+for all
+n ∈ N.
+(A.36)
+Proof. By (2.3) and (3.1),
+∆M j
+n = anY j
+n − an−1Y j
+n−1 = anµnXj
+n − (an − an−1)
+n−1
+�
+k=1
+µkXj
+k.
+Since ∥Xk∥ = 1 for eack k ≤ n, then by (2.4),
+��∆M j
+n
+�� ≤ anµn + (n − 1)(an−1 − an)µn−1 ≤ anµn + a(β + 1)anµn.
+And the assertion is verified.
+□
+Lemma A.18. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+��∆N j
+n
+�� ≤ 2a(β + 1) +
+β
+β − a(β + 1)
+for all
+n ∈ N.
+Proof. By (2.3) and (3.6),
+∆N j
+n =
+βµ−1
+n+1
+β − a(β + 1)ϵj
+n+1 =
+βµ−1
+n+1
+β − a(β + 1) ·
+�
+µn+1Xj
+n+1 + (1 − γn)
+n
+�
+k=1
+Xj
+kµk
+�
+.
+Taking absolute value on both sides, and the assertion is verified.
+□
+Lemma A.19. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+����
+1
+√wn
+∆M j
+k
+���� ≤
+�
+a(β + 1) + 1
+�anµn
+√wn
+for each
+1 ≤ k ≤ n,
+(A.37)
+and in the diffusive and critical regime,
+����
+1
+wn
+⟨M j⟩n − 1
+���� ≤
+�
+�
+�
+C · n−1
+when
+a < 1 −
+1
+2(β+1)
+C · (log n)−1
+when
+a = 1 −
+1
+2(β+1).
+Proof. Dividing by √wn from both sides of (A.36), we get (A.37). Moreover, by (A.9),
+��⟨M j⟩n − wn
+�� ≤
+n
+�
+k=1
+(γk − 1)2a2
+k+1∥Yk∥2 ≤ C
+n
+�
+k=1
+wk
+k2 .
+Dividing both sides by wn and following (3.3), (3.4), the assertion is verified.
+□
+Lemma A.20. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+����
+anµn
+√wn
+∆N j
+k
+���� ≤
+�
+2a(β + 1) +
+β
+β − a(β + 1)
+�anµn
+√wn
+for each
+1 ≤ k ≤ n,
+(A.38)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+35
+and in both the diffusive and critical regime,
+����
+a2
+nµ2
+n
+wn
+⟨N j⟩n − 1
+���� ≤
+�
+�
+�
+C · n−2(1−a)(β+1)
+when
+a < 1 −
+1
+2(β+1)
+C · (n log n)−1
+when
+a = 1 −
+1
+2(β+1).
+Proof. Dividing by √wn and multiplied by anµu from both sides of (A.18), we get (A.38). Then,
+by (A.10), we make use of the estimates and the inequalities hold.
+□
+Denote by Φ(·) := (2π)−1/2 � ·
+−∞ e−t2/2 dt the cumulative distribution of the standard normal
+random variable. The following lemmas are straightforward derivations from [19, Theorem 1], see
+also [22].
+Lemma A.21. There exists an absolute constant α′(p, β) > 0 depending only on p, β such that
+for all j = 1, . . . , d and all 0 ≤ x ≤ α′(p, β) · n−1/2, in the diffusive and critical regime,
+P(M j
+n/√wn ≥ x)
+1 − Φ(x)
+= P(M j
+n/√wn ≤ −x)
+1 − Φ(−x)
+=
+�
+�
+�
+C · exp
+�
+x3
+√n + x2
+n +
+1
+√n(1 + 1
+2 log n)(1 + x)
+�
+when
+a < 1 −
+1
+2(β+1)
+C · exp
+�
+x3
+√n +
+x2
+log n + (
+1
+√log n +
+1
+2√n log n)(1 + x)
+�
+when
+a = 1 −
+1
+2(β+1).
+Lemma A.22. There exists an absolute constant α′′(p, β) > 0 depending only on p, β such that
+for all j = 1, . . . , d and all 0 ≤ x ≤ α′′(p, β) · n−1/2, in the diffusive and critical regime,
+P(anµnN j
+n/√wn ≥ x)
+1 − Φ(x)
+= P(anµnN j
+n/√wn ≤ −x)
+1 − Φ(−x)
+=
+�
+�
+�
+C · exp
+�
+x3
+√n +
+x2
+n2(1−a)(β+1) +
+1
+√n(n1/2−(1−a)(β+1) + 1
+2 log n)(1 + x)
+�
+when
+a < 1 −
+1
+2(β+1)
+C · exp
+�
+x3
+√n +
+x2
+n log n + (
+1
+√n log n +
+1
+2√n log n)(1 + x)
+�
+when
+a = 1 −
+1
+2(β+1).
+Acknowledgements. The authors wish to thank Jean Bertoin and Pierre Tarres for numerous
+discussions and insightful comments.
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+Departement Mathematik, ETH Z¨urich
+Current address: 101, R¨amistrasse, CH-8092 Z¨urich, Switzerland
+Email address: jiamchen@student.ethz.ch
+Laboratoire de Math´ematiques Jean Leray, Nantes Universit´e
+Current address: 2 Chem. de la Houssini`ere, 44322 Nantes, France
+Email address: lucile.laulin@math.cnrs.fr
+

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+A compositional account of motifs, mechanisms, and dynamics in
+biochemical regulatory networks
+Rebekah Aduddell
+James Fairbanks
+Amit Kumar
+Pablo S. Ocal
+Evan Patterson
+Brandon T. Shapiro
+Abstract
+Regulatory networks depict promoting or inhibiting interactions between molecules in a
+biochemical system. We introduce a category-theoretic formalism for regulatory networks, using
+signed graphs to model the networks and signed functors to describe occurrences of one network
+in another, especially occurrences of network motifs. With this foundation, we establish functorial
+mappings between regulatory networks and other mathematical models in biochemistry. We
+construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory
+networks, enabling us to precisely define when a reaction network could be a physical mechanism
+underlying a regulatory network. Turning to quantitative models, we associate a regulatory
+network with a Lotka-Volterra system of differential equations, defining a functor from the
+category of signed graphs to a category of parameterized dynamical systems. We extend this
+result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not
+only inclusions and collapsings of regulatory networks, but also the process of building up complex
+regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured
+cospans to produce a lax double functor from the double category of open signed graphs to that
+of open parameterized dynamical systems. Throughout the paper, we ground the categorical
+formalism in examples inspired by systems biology.
+1
+Introduction
+The genes, proteins, and RNA molecules that comprise living cells interact in complex, varied ways
+to sustain the cell throughout its lifecycle and respond to changes in its environment. Intensive
+experimental study of these interactions is distilled in an idealized form as regulatory networks,
+a kind of directed graph in which vertices represent molecules and edges represent interactions
+between molecules (Figure 1.1). The edges are labeled with a positive or negative sign according to
+whether the interaction is activating or inhibiting. Regulatory networks are the subject of a large
+body of experimental and theoretical work, notably reviewed by Alon [Alo07; Alo19] and Tyson
+et al [TN10; TLK19] among others. Particular attention has been paid to network motifs [Alo07;
+TN10], the simple but functionally meaningful patterns that recur frequently in regulatory networks,
+and to various quantitative dynamics [TLK19] that can be assigned to the networks.
+Although regulatory networks are simple enough to define mathematically—we shall define them
+to be directed graphs, possibly with multiple edges and loops, whose edges are assigned a positive
+or negative sign—important scientific concepts involving them, such as occurrences of motifs in
+networks and biochemical mechanisms generating networks, are often treated imprecisely. Likewise
+for relationships between regulatory networks and other mathematical models in biochemistry,
+particularly dynamical models based on ordinary or stochastic differential equations. Hence a first
+aim of this paper is to put certain concepts and relations concerning regulatory networks on a firm
+mathematical footing. To do so, we will use methods from category theory.
+1
+arXiv:2301.01445v1  [q-bio.MN]  4 Jan 2023
+
+Ash1
+Cdk1/ClbS
+Sld2
+Figure 1.1: A small biochemical regulatory network: regulation of Sld2 by Cdk1 or ClbS with Ash1 as a
+predicted transcription factor. Adapted from Csikász-Nagy et al [Csi+09, Figure 3C].
+Category theory, in both the small and the large, is a natural tool for this study. In saying that
+a motif occurs in a network, one should allow for the possibility that the occurrence is indirect,
+involving a sequence of appropriately signed interactions. For example, positive autoregulation
+can occur directly but also indirectly through a double-negative feedback loop. Since a small
+category is exactly a graph in which consecutive edges can be composed, subject to certain laws,
+regulatory networks should be viewed not only as signed graphs (Section 2.1) but also as signed
+categories freely generated by those (Section 2.2). Sign-preserving functors, unlike sign-preserving
+graph homomorphisms, can express indirect occurrences and are in this sense a better notion of
+morphism for regulatory networks. Here we are doing category theory in the small, using categories
+as algebraic structures comparable to familiar ones like graphs, groups, and monoids.
+Having laid these foundations for regulatory networks, we turn to category theory in the large, a
+mathematical theory of structure well suited to describe the passages between regulatory networks
+and other mathematical models of biochemical systems. Formally speaking, these passages are
+functors into or out of the category of regulatory networks. Making a functor is significantly stronger
+than making an objects-only mapping, as is typically done in the literature, since if morphisms
+of signed graphs formalize relationships between different regulatory networks, then functorality
+requires that these relationships be transported to or from other models of interest. By contrast,
+an objects-only mapping is, abstractly speaking, entirely unconstrained and so is capable of acting
+highly irregularly across different models of a given class. Functorality thus serves as a kind of
+safeguard for model transformation: it does not, on its own, ensure that a transformation makes
+good scientific sense but it does impose nontrivial logical constraints and coherences.
+A first illustration of this principle is the connection between regulatory networks and biochemical
+reaction networks (Section 2.3). When modeling the complex biochemical systems that constitute a
+living cell, it is often practically necessary to abstract away certain details of the underlying chemical
+processes. Regulatory networks generally do not capture all the species or reactions involved in
+a given system; nor can they capture multispecies reactions faithfully because they describe only
+pairwise interactions. Given that regulatory networks are, to some degree, phenomenological models,
+it is natural to ask whether a given network could arise as a summary of a specific chemical process.
+The latter are described by biochemical reaction networks, graph-like structures allowing reactions
+or transitions with multiple inputs and outputs. Inspired by graphical syntax from systems biology
+[Voi00; Voi13], we formalize reaction networks as “Petri nets with links,” and we construct a functor
+from the category of Petri nets with signed links to the category of signed graphs. This functor
+enables us to propose a formal definition for when a reaction network could be a mechanism for a
+regulatory network, a concept that is rarely if ever treated in a precise way.
+This concludes the content of Section 2. In Section 3, we turn from qualitative to quantitative
+analysis, seeking a functorial assignment of continuous dynamics to regulatory networks. Although
+rarely made explicitly functorial, systematic ways to formulate a model belonging to a mathematically
+homogeneous class of models are ubiquitous in science. Voit calls these “canonical representations”
+2
+
+or “canonical models,”1 and identifies Lotka-Volterra models and BST models/S-systems as two
+prominent examples in biology [Voi13, §3]. Reflecting their phenomenological status, regulatory
+networks do not admit a single, obvious dynamical interpretation, and so a wide variety of dynamical
+models have been considered, spanning the discrete and continuous, deterministic and stochastic
+[TLK19]. We consider the Lotka-Volterra systems of ordinary differential equations. While not
+necessarily the most biologically plausible, it is among the simplest continuous models and hence a
+natural place to begin a functorial study.
+A Lotka-Volterra system of equations has the form
+˙xi = ρi xi +
+n
+�
+j=1
+βi,j xi xj,
+i = 1, . . . , n.
+or equivalently, has logarithmic derivatives that are affine functions of the state variables:
+d
+dt[log xi(t)] = ρi +
+n
+�
+j=1
+βi,j xj(t),
+i = 1, . . . , n.
+The coefficients ρi specify baseline rates of growth or decay, according to their sign, and the
+coefficients βi,j rates of activation or inhibition, according to their sign. We construct a functor
+that sends a signed graph (regulatory network) to a Lotka-Volterra model suitably constraining the
+signs of the rate coefficients (Section 3.2). As a prerequisite, we define a category of parameterized
+dynamical systems (Section 3.1), a construction of intrinsic interest that is by no means confined to
+Lotka-Volterra dynamics.
+In order to comprehend complex biological systems, we must decompose them into small, readily
+understandable pieces and then compose them back together to reproduce the behavior of the
+original system. This is the mantra of systems biology, which stresses that compositionality is no
+less important than reductionism in biology. With this motivation, a secondary aim of this paper
+is to extend the above constructions from closed systems to open ones, which can be composed
+together by gluing them along their interfaces. Mathematically, we pass from categories to double
+categories [Gra19], two-dimensional categorical structures in which the usual morphisms of systems
+compose along one direction (by convention, the “vertical” one) and open systems compose along
+the other direction (the “horizontal” one). Among other results, we show that the Lotka-Volterra
+dynamics functor extends to a lax double functor from the double category of open signed graphs
+to a double category of open parameterized dynamical systems (Section 3.3).
+The mathematics developed here is motivated by biochemistry but need not be restricted to it.
+Famously, Lotka-Volterra systems originated in ecology to model predator-prey dynamics [Lot25].
+Regulatory networks and Lotka-Volterra systems can be used as generic models of entities that
+“regulate” each other in some manner, be it at the scale of individual cells or animal ecosystems.
+Regulatory networks are highly reminiscent of the causal loop diagrams in system dynamics [Ste00,
+Chapter 5], where the latter explicitly label feedback loops and their polarities.
+The language of category theory is indispensable to this work but the level of knowledge assumed
+by the reader is not constant. We assume throughout that the reader is familiar with the basic
+notions of category theory, such as categories, functors, and natural transformations. Our main
+reference for facts about category theory is Riehl’s text [Rie16], although there are many others.
+In the definitions and theorems, we have tried to minimize the technical level and explicate the
+statements in concrete terms. In the proofs, we have aimed for efficiency and freely use concepts
+1This usage of “canonical” should not be confused with the unrelated, in fact incompatible, meaning of “canonical”
+in pure mathematics, especially category theory.
+3
+
+and results from the literature that do not appear in the main text. The reader can omit the proofs
+without disrupting the continuity of the paper.
+2
+Qualitative analysis: motifs and mechanisms
+2.1
+Regulatory networks as signed graphs
+To begin, we clarify the notion of graph to be used throughout in this paper. The following definition
+is standard among category theorists. In other fields, it might be called a “directed multigraph,”
+but we will call it simply a “graph.”
+Definition 2.1 (Graphs). The schema for graphs is the category Sch(Graph) freely generated
+by two parallel morphisms:
+V
+E
+src
+tgt
+.
+A graph is a functor X : Sch(Graph) → Set. A graph homomorphism from a graph X to another
+graph Y is a natural transformation φ : X → Y . Graphs and graph homomorphisms form the
+category Graph.
+To restate the definition in explicit terms, a graph X consists of
+• a set X(V ) of vertices;
+• a set X(E) of edges; and
+• functions X(src), X(tgt) : X(E) → X(V ), assigning to each edge its source and target.
+A graph homomorphism φ : X → Y consists of a function φV : X(V ) → Y (V ), the vertex map,
+and another function φE : X(E) → Y (E), the edge map. These maps must preserve sources and
+targets, meaning that the following squares commute:
+X(E)
+X(V )
+Y (E)
+Y (V )
+X(src)
+φE
+Y (src)
+φV
+X(E)
+X(V )
+Y (E)
+Y (V )
+X(tgt)
+φE
+Y (tgt)
+φV .
+We now turn to the main notion of this section, signed graph. Write Sgn for the set of (nonzero)
+signs, whose two elements may be denoted {1, −1} or {+, −}. The set of signs is an abelian group,
+isomorphic to the cyclic group Z2, under the usual multiplication.
+Definition 2.2 (Signed graphs). The category of signed graphs is the slice category
+SgnGraph := Graph/Sgn,
+where, by abuse of notation, Sgn is regarded as a graph with one vertex and two loops.
+Unpacking the definition, a signed graph is seen to be a graph X equipped with a function
+X(sgn) : X(E) → Sgn that assigns a sign to each edge. Given signed graphs X and Y , a morphism
+of signed graphs from X to Y is a graph homomorphism φ that preserves signs, meaning that
+the following triangle commutes:
+X(E)
+Y (E)
+Sgn
+X(sgn)
+Y (sgn)
+φE
+.
+4
+
+Signed graphs are a mathematical description of the regulatory networks studied in systems
+biology [Alo07; TN10].
+For the purposes of this paper, we will simply define a regulatory
+network to be a signed graph. The vertices of the graph represent the components of the network,
+which could be proteins, genes, or RNA molecules. Signed edges represent interactions between
+components, where the source has the effect of either activating/promoting the target (positive sign)
+or inhibiting/repressing it (negative sign). As is customary, we denote activation interactions by
+arrows with pointed heads (−→) and inhibition interactions by arrows with flat heads (−−⊣). For
+instance, the two drawings
+x
+y
++
+−
+↭
+x
+y
+represent the same network, a negative feedback loop in which x activates y, which in turn inhibits
+x [TN10, Scheme 1, Motif B].
+In the literature [TN10], regulatory networks are often modeled as sign-valued matrices. This
+approach is a special case of ours in that an n-by-n matrix valued in {+1, −1, 0} can be interpreted
+as a simple signed graph on n vertices, with signed edges defined by the nonzero matrix elements.
+Unlike the matricial formalism, our formalism allows multiple edges between the same pair of edges,
+which can model multiple interactions based on different mechanisms. Allowing multiple edges and
+self-loops also ensures that graphs and signed graphs form well behaved categories, as the following
+proposition shows.
+Proposition 2.3. The category of signed graphs is complete (has all limits) and cocomplete (has
+all colimits).
+Proof. Because Graph is a copresheaf category, it is complete and cocomplete [Rie16, Proposition
+3.3.9]. The slice category SgnGraph = Graph/Sgn is hence also complete and cocomplete [Rie16,
+Proposition 3.5.5]; alternatively, this follows because slices of copresheaf categories are again
+(equivalent to) copresheaf categories [Str00, Remark p. 303].
+Colimits of signed graphs can be used to construct a category, or rather a double category, of
+open signed graphs. Composition of open signed graphs formalizes the process of building large
+regulatory networks from smaller pieces, including network motifs.
+Proposition 2.4 (Open signed graphs). There is a symmetric monoidal double category of open
+signed graphs, Open(SgnGraph), having
+• as objects, sets A, B, C, . . . ;
+• as vertical arrows, functions f : A → B;
+• as horizontal arrows, open signed graphs, which consist of a signed graph X together with
+a cospan of sets A0
+ℓ0
+−→ X(V )
+ℓ1
+←− A1;
+• as cells, morphisms of open signed graphs (X, ℓ0, ℓ1) → (Y, m0, m1), which consist of a
+map of signed graphs φ : X → Y along with functions fi : Ai → Bi, i = 0, 1, making the
+following diagram commute:
+A0
+X(V )
+A1
+B0
+Y (V )
+B1
+ℓ0
+f0
+ℓ1
+φV
+m0
+f1
+m1
+.
+5
+
+Vertical composition is by composition in Set and in SgnGraph. Horizontal composition and monoidal
+products are by pushouts and coproducts in SgnGraph, respectively, viewing the sets in the feet of the
+cospans as discrete signed graphs.
+Proof. To construct this symmetric monoidal double category, we use the method of structured
+cospans [FS07] in its double-categorical form [BC20]. The categories of sets and of signed graphs
+are related by an adjoint pair of functors
+Set
+SgnGraph
+Disc
+evV
+⊣
+.
+Here evV : SgnGraph → Set is the evaluation at V functor, sending a signed graph X to its set of
+vertices X(V ) and a morphism of signed graphs φ to its vertex map φV , and Disc : Set → SgnGraph
+is the discrete signed graph functor, sending a set A to the signed graph with vertex set A and no
+edges. We obtain a symmetric monoidal double category of open signed graphs as the L-structured
+cospans for the functor L := Disc : Set → SgnGraph [BC20, Theorems 2.3 and 3.9].
+To show that this symmetric monoidal double category is the same one in the proposition
+statement, suppose that L ⊣ R : A → X is an adjoint pair of functors, where in our application
+L = Disc and R = evV . By the defining bijection of an adjunction, L-structured cospans, i.e.,
+objects A0 and A1 in A together with a cospan L(A0) → X ← L(A1) in X, correspond exactly
+to “R-decorated cospans,” i.e., an object X in X together with a cospan A0 → R(X) ← A1 in A.
+Furthermore, by the naturality of this bijection [Rie16, Lemma 4.1.3], morphisms of L-structured
+and R-decorated cospans
+L(A0)
+X
+L(A1)
+L(B0)
+Y
+L(B1)
+L(f0)
+φ
+L(f1)
+↭
+A0
+R(X)
+A1
+B0
+R(Y )
+B1
+f0
+R(φ)
+f1
+related by the adjunction are equivalent in that one diagram commutes if and only if the other does.
+We will tacitly reuse this reasoning in future constructions, such as Proposition 2.8 below.
+A morphism of signed graphs can do two things. Most obviously, it can pick out a signed graph
+as a subobject of another one, via a sign-preserving subgraph embedding. A signed graph morphism
+can also collapse multiple vertices onto a single vertex, and multiple edges onto a single edge with
+the same sign, in the fairly restrictive sense permitted by a graph homomorphism. To illustrate,
+consider the following morphism inspired by Alon’s review [Alo07, Figure 5].
+argR
+argCBH
+argD
+argE
+argF
+argI
+−→
+argR
+arg∗
+(2.1)
+The network in the domain is a “single-input module” in the arginine biosynthesis system, in which
+the regulator argR represses five different enzymes (argCHB, argD, etc.) involved in producing
+arginine. The morphism above forgets the distinction between these enzymes, collapsing them
+into a catch-all entity labeled “arg∗”. These two functions—embedding and collapsing—are all
+that a signed graph morphism can do, because any such morphism factors essentially uniquely as
+6
+
+an epimorphism (morphism with surjective vertex and edge maps) followed by a monomorphism
+(morphism with injective vertex and edge maps), using the epi-mono factorization available in any
+copresheaf category, or more generally in any topos [MM94, §IV.6].
+2.2
+Refining regulatory networks using signed categories and functors
+While morphisms of signed graph have their uses, they do not capture the important idea of refining
+regulatory networks, in which an interaction in one network is realized as a composite of several
+interactions in another. To express refinement, we must generalize our notion of morphism from
+graph homomorphisms to functors. This, in turn, requires the concept of a signed category.
+Definition 2.5 (Signed categories). The category of signed categories is the slice category
+SgnCat := Cat/Sgn,
+where Cat is the category of small categories and the group of signs, Sgn, is regarded as a category
+with one object and two morphisms.
+Unpacking the definition, a signed category is a category C in which every morphism f is
+assigned a sign sgn(f) ∈ {1, −1} in a functorial way, meaning that
+sgn(x0
+f1
+−→ x1
+f2
+−→ · · ·
+fn
+−→ xn) =
+n
+�
+i=1
+sgn(fi)
+for every n ≥ 0 and every sequence of composable morphisms f1, . . . , fn. In particular (n = 0), the
+identity morphisms have positive sign. A morphism of signed categories, or signed functor,
+is a functor F : C → D between signed categories that preserves the signs, meaning that
+sgnD(F(f)) = sgnC(f)
+for every morphism f in C.
+Since our aim is to have a more flexible notion of morphism between signed graphs, we will
+mostly restrict ourselves to those signed categories that are freely generated by a signed graph. The
+free signed category or signed path category functor
+Path : SgnGraph → SgnCat
+sends a signed graph X to the signed category Path(X) having
+• as objects, the vertices of X;
+• as morphisms from x to y, the paths in X from x to y, whose sign is defined to be the product
+of the signs of the edges comprising the path.
+Composition of paths is by concatenation, which clearly preserves the sign. The identity morphism
+at x is the empty path at x, which has positive sign. By convention, if X and Y are signed graphs,
+we say that a signed functor from X to Y is a signed functor F : Path(X) → Path(Y ) between
+the corresponding signed path categories. Since the morphisms of Path(X) are freely generated
+by the edges in X, a signed functor from X to Y is uniquely determined by a morphism of signed
+graphs from X to the underlying signed graph of Path(Y ). This means that each edge in X is
+sent to an appropriately signed path of edges in Y , which can be regarded as a refinement of the
+relationship that the edge represents.
+7
+
+Motif
+Generic instance
+Positive autoregulation
+L+ :=
+�
+•
+�
+Negative autoregulation
+L− :=
+�
+•
+�
+Coherent feedforward loop
+I++ :=
+�
+•
+•
+�
+Incoherent feedforward loop
+I± :=
+�
+•
+•
+�
+Positive feedback loop
+L++ :=
+�
+•
+•
+�
+Negative feedback loop
+L± :=
+�
+•
+•
+�
+Double-negative feedback loop
+L−− :=
+�•
+•
+�
+Table 2.1: Common motifs in biochemical regulation networks [Alo07; TN10]
+We now have a precise language with which to classify network motifs and their occurrences.
+As a first example, Alon identifies four types of incoherent feedforward loop (FFL) involving three
+components,
+x
+x
+x
+x
+y
+y
+y
+y
+z
+z
+z
+z
+,
+those of type 1, 2, 3, and 4, respectively [Alo07, Figure 2a]. Besides having three components, what
+these motifs have in common is that there exists a signed functor into each of them from the signed
+graph I± :=
+�
+•
+•
+�
+having two parallel arrows of opposite sign. The network I± is thus the
+“generic” incoherent feedforward loop, in the sense that signed functors out of it refine the pattern
+in specific ways. A similar situation holds for other common network motifs (Table 2.1), which
+motivates the following definition.
+Definition 2.6 (Motif instance). Given a signed graph A, regarded as a motif, an instance or
+occurrence of the motif A in a network X is a monic signed functor A ↣ X.
+Note that a signed functor is a monomorphism exactly when the functor is an embedding of
+categories, i.e., an injective-on-objects, faithful functor. Requiring the functor in the definition to
+be monic excludes “degenerate instances” of motifs where vertices or edges are identified.
+Now, should the incoherent FFL be regarded as a network motif, or is it the more specific types,
+such as the incoherent FFL of type 1, that are motifs? From our point of view, they are all equally
+motifs but they have different degrees of specificity, and the functorial language clarifies how motifs
+are iteratively refined. Specifically, an instance of an incoherent FFL of type 1 in a network X
+also gives an instance of an incoherent FFL in X (of unspecified type), simply by composing the
+monomorphisms involved:
+I± ∼=
+�
+x
+z
+�
+↣
+�
+x
+y
+z
+�
+↣
+X.
+8
+
+Similarly, in the notation of Table 2.1, any instance of double-negative feedback (L−−) also gives an
+instance of positive autoregulation (L+) [CP09], via the monomorphism L+ ↣ L−− that sends the
+positive loop to the double-negative 2-cycle.
+For any choice of motif A, the mapping that sends a regulatory network X to the set of
+occurrences of A in X is a functor
+HomSgnCatm(Path(A), Path(−)) : SgnGraphm → Set,
+where SgnGraphm and SgnCatm denote the wide subcategories of monomorphisms in SgnGraph and
+SgnCat, respectively. This functor is almost, but not quite, representable, due to the distinction
+between signed graphs and signed categories. More importantly, the existence of this functor means
+that a monomorphism between regulatory networks induces a map between instances of A, for any
+motif A.
+We now extend the construction of open signed graphs to open signed categories.
+Proposition 2.7. The category of signed categories is complete and cocomplete.
+Proof. Because the category Cat is complete and cocomplete [Rie16, Proposition 3.5.6], its slice
+SgnCat = Cat/Sgn is also [Rie16, Proposition 3.5.5].
+Proposition 2.8 (Open signed categories). There is a symmetric monoidal double category of open
+signed categories, Open(SgnCat), having
+• as objects, sets A, B, C, . . . ;
+• as vertical arrows, functions f : A → B;
+• as horizontal arrows, open signed categories, which consist of a signed category C together
+with a cospan of sets A0
+ℓ0
+−→ Ob(C)
+ℓ1
+←− A1;
+• as cells, morphisms of open signed categories (C, ℓ0, ℓ1) → (D, m0, m1), which consist of
+a signed functor F : C → D along with functions fi : Ai → Bi, i = 0, 1, making the diagram
+commute:
+A0
+Ob(C)
+A1
+B0
+Ob(D)
+B1
+ℓ0
+f0
+ℓ1
+Ob(F)
+m0
+f1
+m1
+.
+Vertical composition is by composition in Set and in SgnCat. Horizontal composition and monoidal
+products are by pushouts and coproducts in SgnCat, respectively, viewing the sets in the feet of
+cospans as discrete signed categories.
+Moreover, the signed path category functor extends to a symmetric monoidal double functor
+Path : Open(SgnGraph) → Open(SgnCat).
+Proof. We take Open(SgnCat) to be the symmetric monoidal double category of L′-structured
+cospans for the functor L′ := Disc : Set → SgnCat involved the composite adjunction
+Set
+SgnCat
+=
+Set
+SgnGraph
+SgnCat
+Disc
+Ob
+Disc
+evV
+U
+Path
+⊣
+⊣
+⊣
+.
+On the right hand side, the first adjunction was already used in the proof of Proposition 2.4, and
+the second adjunction is the free-forgetful adjunction between signed graphs and signed categories.
+9
+
+To prove the last statement, we notice that all functors involved in the commutative square
+Set
+SgnGraph
+Set
+SgnCat
+L=Disc
+L′=Disc
+Path
+are left adjoints, hence preserve colimits. We can therefore appeal to [BC20, Theorem 4.3] to obtain
+a symmetric monoidal double functor
+Open(SgnGraph) ∼= LCsp(SgnGraph) → L′Csp(SgnCat) ∼= Open(SgnCat).
+2.3
+Mechanistic models as Petri nets with links
+However challenging they may be to identify through experiments and data analysis, regulatory
+networks still only summarize how the components of a complex biochemical system interact.
+Regulatory networks typically include only a subset of the system’s components, and they do not
+model individual reactions and processes, only pairwise promoting or inhibiting interactions between
+components. In this sense, regulatory networks are not fully mechanistic models, even if they have
+a stronger causal interpretation than, say, a correlation matrix.
+By contrast, mechanistic models in biochemistry model individual reactions, which requires a
+different formalism. Pictures like the following, adapted from Voit’s review [Voi13, Figure 4], are
+common in systems biology.
+A
+B
+D
+C
+−
+(2.2)
+This diagram possesses two distinctive features. First, directed hyperedges represent reactions
+having a number of inputs or outputs different than one. There are, for example, hyperedges from
+B and C to D, from nothing to A (an inflow), and from D to nothing (an outflow). If, in lieu of
+hyperedges, we introduce a second type of vertex, we obtain a structure similar to a Petri net
+A
+B
+C
+D
+−
+(2.3)
+but with the second distinctive feature of having signed links from the first type of vertices (species)
+to the second type (transitions), whose signs indicate promotion or inhibition.
+In this section, we explain how a Petri net with signed links can provide a mechanism for a
+regulatory network. This involves constructing a functor from Petri nets with signed links to signed
+graphs, approximating the former as the latter. As a prerequisite, we need a rigorous definition of a
+Petri net with links, which seems to be absent from the literature.
+10
+
+Definition 2.9 (Petri net with links). The schema for Petri nets with links is the category
+Sch(LPetri) freely generated by these objects and morphisms:
+I
+S
+O
+T
+L
+srcL
+tgtL
+srcI
+tgtI
+tgtO
+srcO
+.
+A Petri nets with links is a functor P : Sch(LPetri) → Set, and a morphism of these is a natural
+transformation. A morphism φ : P → Q preserves arities if the naturality squares associated
+with the morphisms I → T and O → T are also pullback squares:
+P(I)
+P(T)
+Q(I)
+Q(T)
+P(tgtI)
+φI
+φT
+P(tgtI)
+⌟
+P(O)
+P(T)
+Q(O)
+Q(T)
+P(srcO)
+φO
+φT
+P(srcO)
+⌟
+.
+Petri nets with links and their morphisms form the category LPetri.
+To explicate the definition, a Petri net with links P consists of
+• a set P(S) of species;
+• a set P(T) of transitions;
+• a set P(I) of input arcs going from species to transitions, via maps P(srcI) : P(I) → P(S)
+and P(tgtI) : P(I) → P(T);
+• a set P(O) of output arcs going from transitions to species, via maps P(srcO) : P(O) → P(T)
+and P(tgtO) : P(O) → P(S); and finally
+• a set P(L) of links going from species to transitions, via maps P(srcL) : P(L) → P(S) and
+P(tgtL) : P(L) → P(T).
+The property of preserving arities, called “etale” by Kock [Koc22, §3.4], means that a morphism
+φ : P → Q of Petri nets with links preserves the input and output arities of all transitions. Namely,
+for each transition t in the net P the map φI : P(I) → Q(I) restricts to a bijection between the
+input arcs to t and to φT (t), and similarly the map φO : P(O) → Q(O) restricts to a bijection
+between the output arcs from t and from φT (t). This property seems appropriate for many purposes,
+including in biochemistry, but for mathematical convenience we will not always assume it.
+Remark 2.10 (Related literature). Our definition of a Petri net with links, while apparently novel,
+is the obvious joint generalization of two existing concepts. Kock has described Petri nets as
+copresheaves on a category with objects S, T, I, O [Koc22], calling them whole-grain Petri nets to
+distinguish them from classical Petri nets, whose semantics are subtly different [Bae+21]. Meanwhile,
+the concept of a link is essential to stock and flow diagrams, originating in the field of system
+dynamics [For61; Ste00] and recently given a rigorous categorical account [Bae+22]. We also note
+that the Petri nets with catalysts proposed by Baez, Foley, and Moeller [BFM19] differ significantly
+from Petri nets with links: the former fix a subset of the species to be catalysts throughout the net,
+whereas the latter uses links to make catalyzation specific to individual reactions.
+11
+
+Remark 2.11 (Petri nets as typed graphs). Like bare Petri nets, Petri nets with links can be described
+as graphs with two types of vertices. To see this, take the graph
+TLPetri :=
+�
+�
+�
+�
+�
+�
+�
+S
+T
+I
+L
+O
+�
+�
+�
+�
+�
+�
+�
+with vertices S and T and edges I, O, and L. The category of Petri nets with links and natural
+transformations is isomorphic to the slice category Graph/TLPetri. Moreover, the schema category
+Sch(LPetri) is isomorphic to the category of elements of the functor TLPetri : Sch(Graph) → Set,
+exemplifying a general fact about slices of copresheaf categories [Str00, Remark p. 303].
+Petri nets with signed links are defined analogously to signed graphs (Definition 2.2).
+Definition 2.12 (Petri nets with signed links). The category of Petri nets with signed links
+is the slice category
+SgnPetri := LPetri/PSgn,
+where PSgn is the Petri net with links having one species, one transition, one input arc, one output
+arc, and two links, namely the elements of Sgn.
+Incidentally, the morphism P → PSgn defining a Petri net with signed links does not preserve
+arities unless every transition in P has exactly one input and one output.
+We now turn to the main construction of this section, a functor that “approximates” a Petri net
+with signed links as a signed graph. On the example in Equations (2.2) and (2.3), this functor gives
+the signed graph
+A
+B
+D
+C
+.
+(2.4)
+In general, the resulting signed graph has, as vertices, the Petri net’s species and has signed edges
+for each of the four cases:
+(a) for every input-output pair to a transition, a positive edge from input to output;
+(b) for every input to a transition, a negative self loop, representing consumption by the reaction;
+(c) for every signed link, an edge of opposite sign going from the linked species to each input to
+the linked transition;
+(d) for every signed link, an edge of equal sign going from the linked species to each output from
+the linked transition.
+All four cases are visible in the example of Equation (2.4). Set-theoretically, each of these cases is
+the result of a conjunctive query, or equivalently of a representable functor
+HomLPetri(P, −) : LPetri → Set
+associated with a particular Petri net with links P, the generic instance for that query. The four
+generic instances we need are shown in Figure 2.1. Their sum is a disjoint union of conjunctive
+queries, or duc-query for short.
+12
+
+(a) Input-output pair to tran-
+sition
+(b) Input to transition
+(c) Input to transition with
+incident link
+(d) Output from transition
+with incident link
+Figure 2.1: Four different Petri nets with links. For each of these instances P, evaluating the representable
+functor HomLPetri(P, −) : LPetri → Set gives the edges for one case in the case analysis that defines the functor
+from Petri nets with signed links to signed graphs (Proposition 2.13).
+To make the construction just sketched on objects fully precise and functorial, we use Spivak’s
+theory of functorial data migration based on duc-queries [Spi21]. In order to apply it, we fully
+schematize the definitions of signed graphs and Petri nets with signed links.
+The schema for signed graphs is the category Sch(SgnGraph) freely generated by these objects
+and morphisms:
+V
+E
+A
+neg
+src
+tgt
+sgn
+.
+A signed graph as in Definition 2.2 is equivalent to a functor X : Sch(SgnGraph) → Set such that
+X(A) = Sgn, the set of signs, and X(neg) : Sgn → Sgn is negation (i.e., multiplication by −1). Note
+that negation is not needed to define the data of a signed graph but is relevant to the data migration.
+A morphism of signed graphs X → Y is a natural transformation φ : X → Y whose component at
+A is the identity function, φA = 1Sgn. We thus obtain a category isomorphic to SgnGraph.
+Similarly, the schema for Petri nets with signed links is the category Sch(SgnPetri) freely
+generated by:
+S
+I
+O
+L
+A
+T
+neg
+srcL
+tgtL
+srcI
+tgtI
+tgtO
+srcO
+sgn
+one
+.
+A Petri net with signed links, as in Definition 2.12, is equivalent to a functor P : Sch(SgnPetri) → Set
+such that P(A) = Sgn, the map P(neg) : Sgn → Sgn is negation, and P(one) : P(T) → Sgn is the
+constant map at +1. Again, these maps are needed for data migration, not for the data itself. A
+morphism of Petri nets with signed links is a natural transformation φ : P → Q such φA = 1Sgn,
+yielding a category isomorphic to SgnPetri.
+Proposition 2.13 (Regulatory net induced by Petri net). A functor
+Net : SgnPetri → SgnGraph
+13
+
+is specified by the following functor from Sch(SgnGraph) to the category of duc-queries on Sch(SgnPetri):
+Sch(SgnGraph) → ⨿
+��
+SetSch(SgnPetri)�op�
+V �→ S
+E �→ I ×T O + I + I ×T L + O ×T L
+A �→ A
+src �→ [srcI ◦πI, srcI, srcL ◦πL, srcL ◦πL]
+tgt �→ [tgtO ◦πO, srcI, srcI ◦πI, tgtO ◦πO]
+sgn �→ [one ◦πT , neg ◦ one ◦ tgtI, neg ◦ sgn ◦πL, sgn ◦πL]
+neg �→ neg .
+(2.5)
+Proof. We will define the functor Net : SgnPetri → SgnGraph as the restriction of a functor
+SetD → SetC between the categories of copresheaves on D := Sch(SgnPetri) and C := Sch(SgnGraph).
+In fact, the functor SetD → SetC will be of the special kind known as a parametric right adjoint
+[Str00, Definition p. 311].
+According to the theory of data migration [Spi21, Corollary 2.3.6], giving a parametric right
+adjoint SetD → SetC is equivalent to giving a functor from C to ⨿((SetD)op), the free coproduct
+completion of the free limit completion of D. Our functor C → ⨿((SetD)op) is defined by Equa-
+tion (2.5). The assignment of E ∈ C can also be described as the sum of the four representables
+associated with the Petri nets with links in Figure 2.1.
+Finally, the assignments A �→ A and neg �→ neg ensure that if P is a copresheaf on D with
+P(A) = Sgn and P(neg) is negation, then applying this parametric right adjoint functor to P yields
+a copresheaf X on C where again X(A) = Sgn and X(neg) is negation. Thus, this functor between
+copresheaf categories restricts to a functor SgnPetri → SgnGraph as claimed.
+With this construction, we can give a formal account of what it means to have a mechanistic
+model for a regulatory network.
+Definition 2.14 (Mechanism). A mechanistic model for a regulatory network X is a Petri
+net with signed links P together with an occurrence of X in Net(P), i.e., a monic signed functor
+X ↣ Net(P).
+For example, the Petri net with signed links in Equation (2.3) is a mechanistic model for a
+regulatory network in which A and D participate in a positive feedback loop:
+A
+D .
+3
+Quantitative analysis: parameters and dynamics
+3.1
+Parameterized dynamical systems
+Pioneering the idea of functorial semantics for scientific models, Baez and Pollard extended the
+mass-action model of reaction networks to a functor from the category of Petri nets with rates
+into a category of dynamical systems [BP17]. In this picture, the reaction rate coefficients are
+known constants associated with the reaction network. In practice, however, rate coefficients are
+often unknown and must be extracted from existing literature or estimated from experimental data.
+We therefore change our perspective slightly and consider dynamical systems not in isolation but
+as parameterized families. This shift also turns out to have formal advantages: the category of
+14
+
+parameterized dynamical systems is better behaved than the category of dynamical systems, which
+has too few morphisms.
+To begin, we recall the dynamics functor, nearly identical to Baez-Pollard’s [BP17, Lemma 15]:
+Lemma 3.1 (Dynamics). There is a functor Dynam : FinSet → VectR that sends
+• a finite set S to the vector space of algebraic vector fields v : RS → RS, where algebraic
+means that the components of the vector field are polynomials in the state variables;
+• a function f : S → S′ between finite sets to the linear transformation
+(v : RS → RS)
+�→
+(f∗ ◦ v ◦ f∗ : RS′ → RS′),
+where the linear map f∗ : RS′ → RS is the pullback along f
+f∗(x′)(i) := x′(f(i)),
+x′ ∈ RS′, i ∈ S,
+and the linear map f∗ : RS → RS′ is the pushforward along f
+f∗(x)(i′) :=
+�
+i∈f−1(i′)
+x(i),
+x ∈ RS, i′ ∈ S′.
+Proof. The functor Dynam : FinSet → VectR can be constructed as the composite
+FinSet
+⟨D,F⟩
+−−−−→ Vectop
+R × VectR
+Poly(−,−)
+−−−−−−→ VectR,
+where F : FinSet → VectR is the free vector space functor (restricted to finite sets); D : FinSetop →
+VectR is the dual vector space functor (restricted to F), whose underlying set-valued functor is
+VectR(F(−), R) ∼= Set(−, R) : FinSetop → Set;
+and Poly(−, −) is the VectR-enriched hom-functor that sends a pair of vector spaces to the vector
+space of polynomial maps between them.2
+The dynamics functor is the same one studied by Baez and Pollard except that we take the
+vector space, rather than merely the set, of vector fields. That is because we are interested in
+linearly parameterized dynamical systems. In calling the functor “dynamics,” we implictly identify a
+vector field with the differentiable dynamical system that it generates. This common practice is not
+entirely innocent since even when a system of differential equations depends linearly on parameters,
+its solutions rarely do. We also note that the restriction to algebraic vector fields, as opposed to
+smooth or even just continuous ones, is inessential but suffices for us and agrees with Baez-Pollard.
+The dynamics functor is the main building block in constructing a category of parameterized
+dynamical systems.
+Definition 3.2 (Linear parameterizations). The category of linearly parameterized dynami-
+cal systems is the comma category
+Para(Dynam) := F/ Dynam,
+where F : FinSet → VectR, X �→ RX is the free vector space functor restricted to finite sets.
+2For a coordinate-free description of polynomial maps between vector spaces, see [Car71, §1.6].
+15
+
+So, by definition, a linearly parameterized dynamical system consists of a finite set P, the
+parameter variables, and a finite set S, the state variables, together with a linear map
+v : RP → Dynam(S)
+sending each choice of parameters θ ∈ RP to an algebraic vector field v(θ). In more conventional
+notation, we can write v(x; θ) := v(θ)(x) for x ∈ RS and θ ∈ RP . A morphism (P, S, v) → (P ′, S′, v′)
+of linearly parameterized dynamical systems is a pair of functions q : P → P ′ and f : S → S′
+making the square
+RP
+Dynam(S)
+RP ′
+Dynam(S′)
+v
+f∗◦(−)◦f∗
+v′
+q∗
+(3.1)
+commute, or equivalently making the equation
+f∗(v(f∗(x′); θ)) = v′(x′; q∗(θ))
+hold for all x′ ∈ RS′ and θ ∈ RP .
+While certainly not all dynamical models depend linearly on their parameters, a great many
+of them do, including several important canonical models in biology and chemistry. The law of
+mass action defines dynamical systems that depend linearly on the rate coefficients. The generalized
+Lotka-Volterra equations, studied in the next section, are linear in the rate and affinity parameters.
+Of course, the mass-action and Lotka-Volterra equations are nonlinear ODEs; linearity of a vector
+field in state or in parameters are separate matters. Nevertheless, even for nonlinear models such
+as Lotka-Volterra, linearity in parameters is a useful assumption that aides in the identifiability
+analysis of the model [SRS14, §5].
+To express important physical constraints and to define a semantics for signed graphs, we will
+restrict the dynamical system and its parameters to be nonnegative. This is straightforward enough
+but requires a bit of additional formalism.
+Write R+ := {x ∈ R : x ≥ 0} for the semiring of nonnegative real numbers. A module over R+
+is called a conical space, and the category of conical spaces and conic-linear (R+-linear) maps is
+denoted Con := ModR+. A conical space is a structure in which one can take linear combinations
+with nonnegative real coefficients, just as a real vector space (R-module) is a structure in which
+one can take linear combinations with arbitrary real coefficients. Any convex cone in a real vector
+space is a conical space. Our main example is the nonnegative orthant of RS for some set S: the
+function space RS
++ := {x : S → R+}, with conical combinations taken pointwise. A real vector space
+can itself be regarded as a conical space; more precisely, the inclusion of semirings R+ �→ R induces
+a forgetful functor VectR → Con by restriction of scalars.
+Recall that a dynamical system is nonnegative if whenever the initial condition is in the
+nonnegative orthant, its trajectory always remains in the nonnegative orthant. A dynamical system
+of form ˙x = v(x) is nonnegative if and only if vi(x) ≥ 0 whenever x ≥ 0 componentwise and
+xi = 0 [HCH10, Proposition 2.1], in which case the vector field v is called essentially nonnegative
+[HCH10, Definition 2.1]. Using this criterion, it is easy to see that a reaction network with mass-
+action kinetics is nonnegative assuming the rate constants are nonnegative, as is a Lotka-Volterra
+system for any choice of parameters. Hence both systems satisfy the obvious physical constraint
+that no species should have negative concentration or population.
+Lemma 3.3 (Nonnegative dynamics). There is a functor Dynam+ : FinSet → Con that sends a
+finite set S to the conical space of essentially nonnegative, algebraic vector fields v : RS → RS and
+sends a function f : S → S′ to the transformation v �→ f∗ ◦ v ◦ f∗.
+16
+
+Proof. The proof is similar to that of Lemma 3.1. It is clear that the essentially nonnegative functions
+are stable under pointwise conical combinations, hence form a conical space. (They are, of course, not
+stable under arbitrary linear combinations.) We just need to check that if v : RS → RS is essentially
+nonnegative, then so is the transformed vector field f∗ ◦ v ◦ f∗ : RS′ → RS′. Fix x′ ∈ RS′
++ and i′ ∈ S′,
+and suppose that x′(i′) = 0. For every i ∈ f−1(i′), we have f∗(x′)(i) = x′(f(i)) = x′(i′) = 0 and so
+v(f∗(x′))(i) ≥ 0, whence the inequality of essential nonnegativity follows:
+(f∗ ◦ v ◦ f∗)(x′)(i′) =
+�
+i∈f−1(i′)
+v(f∗(x′))(i) ≥ 0.
+We now define the conical analogue of linearly parameterized dynamical systems.
+Definition 3.4 (Conical parameterizations). The category of conically parameterized non-
+negative dynamical systems is the comma category
+Para(Dynam+) := F+/ Dynam+ .
+where F+ : FinSet → Con, X �→ RX
++ is the free conical space functor restricted to finite sets.
+So, a conically parameterized nonnegative dynamical system consists of finite sets P and S
+together with a conic-linear map
+v : RP
++ → Dynam+(S).
+Proposition 3.5 (Colimits of parameterized dynamical systems). The categories of linearly and
+conically parameterized dynamical systems are finitely cocomplete. Moreover, these finite colimits
+are computed by colimits in FinSet of the parameter variables and of the state variables.
+Proof. The category FinSet has finite colimits and the functors F : FinSet → VectR and F+ :
+FinSet → Con preserve finite colimits, since they are composites of the inclusion FinSet �→ Set with
+the left adjoints
+Set
+VectR
+F
+U
+⊣
+and
+Set
+Con
+F+
+U+
+⊣
+to the underlying set functors on vector spaces and conical spaces. By Lemma 3.6 below, the comma
+categories Para(Dynam) = F/ Dynam and Para(Dynam+) = F+/ Dynam+ have finite colimits,
+which are preserved by the projection functors onto FinSet.
+To illustrate, we describe the initial object and binary coproducts in Para(Dynam). The initial
+linearly parameterized dynamical system has no parameter variables, no state variables, and the
+unique (trivial) vector field on the zero vector space. The coproduct of two linearly parameterized
+dynamical systems (P1, S1, v1) and (P2, S2, v2) has parameter variables P1 + P2, state variables
+S1 + S2, and parameterized vector field
+RP1+P2 ∼= RP1 ⊕ RP2
+v1⊕v2
+−−−−→ Dynam(S1) ⊕ Dynam(S2)
+[Dynam(ι1),Dynam(ι2))]
+−−−−−−−−−−−−−−−→ Dynam(S1 + S2),
+where ι1 : S1 → S1 +S2 and ι2 : S2 → S1 +S2 are the canonical inclusions. In conventional notation,
+the coproduct system has parameterized vector field
+v
+��
+x1
+x2
+�
+;
+�
+θ1
+θ2
+��
+=
+�
+v1(x1; θ1)
+v2(x2; θ2)
+�
+.
+The proof of Proposition 3.5, as well as of Theorems 3.7 and 3.8 below, depends on the following
+technical lemma about comma categories, which the reader can omit without loss of continuity.
+17
+
+Lemma 3.6 (Colimits in comma categories). Let C0
+F0
+−→ C
+F1
+←− C1 be a cospan of categories such
+that C0 and C1 have colimits of shape J and F0 preserves J-shaped colimits. Then the comma category
+F0/F1 also has J-shaped colimits, and the projection functors πi : F0/F1 → Ci, i = 0, 1, preserve
+those colimits.
+Furthermore, a functor G : X → F0/F1 into the comma category preserves J-shaped colimits
+whenever the associated functors Gi := πi ◦ G : X → Ci, i = 0, 1, do so.
+Proof. Colimits in the comma category F0/F1 are constructed in [RB88, §5.2]. To make the rest of
+the proof self-contained, we recall the construction here.
+By the universal property of the comma category, a diagram D : J → F0/F1 is equivalent to
+diagrams Di := πi ◦ D : J → Ci, i = 0, 1, along with a natural transformation ⃗D : F0 ◦ D0 ⇒ F1 ◦ D1.
+Let (ci, λi) be a colimit cocone for the diagram Di in Ci, having legs Di(j)
+λi
+j
+−→ ci for each j ∈ J.
+The family of morphisms
+F0(D0(j))
+⃗Dj
+−−→ F1(D1(j))
+F1(λ1
+j)
+−−−−→ F1(c1),
+j ∈ J,
+is then a cocone under F0 ◦ D0. Since F0 preserves J-shaped limits, (F0(c0), F0 ∗ λ0) is a colimit
+cocone for F0 ◦ D0, so by its universal property, there exists a unique morphism f : F0(c0) → F1(c1)
+making the squares commute:
+F0(D0(j))
+F1(D1(j))
+F0(c0)
+F1(c1)
+⃗Dj
+f
+F0(λ0
+j)
+F1(λ1
+j) ,
+j ∈ J.
+Setting λ := (λ0
+j, λ1
+j)j∈J, the cocone ((c0, c1, f), λ) can be shown to be a colimit of the diagram D.
+To prove the last statement about colimit preservation, let D : J → X be a diagram with colimit
+cocone (x, λ), having legs Dj
+λj
+−→ y for j ∈ J. We must show that its image cocone (G(x), G ∗ λ) is
+a colimit of the diagram G ◦ D in F0/F1. By the universal property of the comma category, the
+functor G : X → F0/F1 is equivalent to the functors Gi : X → Ci, i = 0, 1, along with a natural
+transformation ⃗G : F0 ◦ G0 ⇒ F1 ◦ G1. The image cocone (G(x), G ∗ λ) then amounts to cocones
+(G0(x), G0 ∗ λ) and (G1(x), G1 ∗ λ), which by hypothesis are colimits of the diagrams G0 ◦ D and
+G1 ◦ D in C0 and C1, together with a family of commutative squares in C:
+F0(G0(Dj))
+F1(G1(Dj))
+F0(G0(x))
+F1(G1(x))
+⃗GDj
+F0(G0(λj))
+⃗Gx
+F1(G1(λj)) ,
+j ∈ J.
+But a morphism ⃗Gx making all these squares commute is already uniquely determined by the
+universal property of the colimit cocone (F0(G0(x)), F0 ∗ G0 ∗ λ) for the diagram F0 ◦ G0 ◦ D, using
+the hypothesis that F0 preserves J-shaped colimits. Indeed, this is precisely how one constructs the
+colimit of the diagram G ◦ D in F0/F1, as shown above. It follows that (G(x), G ∗ λ) is a colimit
+cocone for G ◦ D.
+18
+
+3.2
+The Lotka-Volterra dynamical model
+A Lotka-Volterra system with n species has, using matrix notation, the vector field
+v(x; ρ, β) := x ⊙ (ρ + βx) = diag(x)(ρ + βx)
+with state vector x ∈ Rn and arbitrary real-valued parameters ρ ∈ Rn and β ∈ Rn×n [SMH18, §2.2].
+In coordinates, the vector field is
+vi(x; ρ, β) = xi
+�
+�ρi +
+n
+�
+j=1
+βi,jxj
+�
+� = ρixi +
+n
+�
+j=1
+βi,jxixj,
+i = 1, . . . , n.
+The parameter ρi sets the baseline rate of growth (when positive) or decay (when negative) for
+species i, whereas βi,j defines a promoting (when positive) or inhibiting (when negative) effect of
+species j on species i. In typical applications the signs of the parameters are fixed and known in
+advance of any data. For example, in the famous predator-prey Lotka-Volterra system
+˙x = ax − bxy
+˙y = dxy − cy,
+with prey x and predators y, the parameters ρ =
+�
+a
+−c
+�
+and β =
+�
+0
+−b
+d
+0
+�
+are specified by nonnegative
+real numbers a, b, c, d ≥ 0.
+In this section, we define quantitative semantics for graphs and signed graphs using the Lotka-
+Volterra dynamical model. To illustrate the main ideas, we first construct a functor from finite
+graphs (Definition 2.1) to linearly parameterized dynamical systems (Definition 3.2), giving a
+semantics for unlabeled graphs. It is more useful to have a semantics for regulatory networks, which
+we have defined to be signed graphs. We therefore construct a second functor from finite signed
+graphs (Definition 2.2) to conically parameterized nonnegative dynamical systems (Definition 3.4).
+Recall that a graph is finite if its vertex and edge sets are both finite. Let FinGraph denote the
+full subcategory of Graph spanned by finite graphs.
+Theorem 3.7 (Lotka-Volterra model for finite graphs). There is a functor
+LV : FinGraph → Para(Dynam)
+that sends
+• a finite graph X to the linearly parameterized dynamical system with parameter variables
+P := X(V ) + X(E), state variables S := X(V ), and algebraic vector field3
+v(x; ρ, β)(i) := ρ(i) x(i) +
+�
+(e:i′→i)∈X
+β(e) x(i′) x(i),
+x ∈ RX(V ), i ∈ X(V ),
+parameterized by vectors ρ ∈ RX(V ) and β ∈ RX(E);
+• a graph homomorphism φ : X → Y to a morphism of systems with parameter variable map
+φV + φE : X(V ) + X(E) → Y (V ) + Y (E) and state variable map φV : X(V ) → Y (V ).
+Moreover, the functor LV preserves finite colimits.
+3For a fixed graph X and vertex i ∈ X(V ), the notation (e : i′ → i) ∈ X means any edge e ∈ X(tgt)−1(i) incoming
+to i, whose source i′ = X(src)(e) varies with e.
+19
+
+Proof. By the universal property of the comma category Para(Dynam) = F/ Dynam, to give a
+functor LV : FinGraph → Para(Dynam) is to give a pair of functors LV0, LV1 : FinGraph → FinSet
+along with a natural transformation
+⃗
+LV : (F ◦ LV0) ⇒ (Dynam ◦ LV1) : FinGraph → VectR.
+We set LV0(X) := X(V ) + X(E) and LV1(X) := X(V ). Using the universal property of the
+coproduct in VectR, the components
+⃗
+LVX : RX(V ) ⊕ RX(E) ∼= RX(V )+X(E) → Dynam(X(V )).
+of the transformation ⃗
+LV themselves decompose into two parts, call them
+v0
+X := ⃗
+LV
+0
+X : RX(V ) → Dynam(X(V ))
+and
+v1
+X := ⃗
+LV
+1
+X : RX(E) → Dynam(X(V )).
+We define these to be
+v0
+X(x; ρ)(i) := ρ(i) x(i)
+and
+v1
+X(x; β)(i) :=
+�
+e∈X(tgt)−1(i)
+β(e) x(X(src)(e)) x(i).
+Putting the pieces back together reproduces the first statement of the theorem. We just need to
+check that the transformation ⃗
+LV is, in fact, natural.
+Given a graph homomorphism φ : X → Y , the naturality square for the transformation ⃗
+LV is
+RX(V )+X(E)
+Dynam(X(V ))
+RY (V )+Y (E)
+Dynam(Y (V ))
+⃗
+LVX
+(φV +φE)∗
+(φV )∗◦(−)◦(φV )∗
+⃗
+LVY
+,
+(3.2)
+which decomposes into two squares,
+RX(V )
+Dynam(X(V ))
+RY (V )
+Dynam(Y (V ))
+v0
+X
+(φV )∗
+(φV )∗◦(−)◦(φV )∗
+v0
+Y
+and
+RX(E)
+Dynam(X(V ))
+RY (E)
+Dynam(Y (V ))
+v1
+X
+(φE)∗
+(φV )∗◦(−)◦(φV )∗
+v1
+Y
+.
+Let us check that both squares commute. For the first, we have
+(φV )∗(v0
+X(φ∗
+V (y); ρ))(j) =
+�
+i∈φ−1
+V (j)
+v0
+X(y ◦ φV ; ρ)(i) =
+�
+i∈φ−1
+V (j)
+ρ(i) y(φV (i))
+=
+�
+�
+�
+�
+i∈φ−1
+V (j)
+ρ(i)
+�
+�
+� y(j) = v0
+Y (y; (φV )∗(ρ))(j)
+20
+
+for all y ∈ RY (V ), ρ ∈ RX(V ), and j ∈ Y (V ). For the second, we have
+(φV )∗(v1
+X(φ∗
+V (y); β))(j) =
+�
+i∈φ−1
+V (j)
+v1
+X(y ◦ φV ; β)(i)
+=
+�
+i∈φ−1
+V (j)
+�
+e∈X(tgt)−1(i)
+β(e) y(φV (X(src)(e))) y(j)
+=
+�
+f∈Y (tgt)−1(j)
+�
+e∈φ−1
+E (f)
+β(e) y(Y (src)(φE(e))) y(j)
+=
+�
+f∈Y (tgt)−1(j)
+�
+�
+�
+�
+e∈φ−1
+E (f)
+β(e)
+�
+�
+� y(Y (src)(f))y(j)
+= v1
+Y (y; (φE)∗(β))(j),
+for all y ∈ RY (V ), β ∈ RX(E), and j ∈ Y (V ). When exchanging the order of the summations we
+have used the facts that the graph homomorphism φ : X → Y preserves sources and targets, the
+latter in its contravariant form
+X(E)
+X(V )
+Y (E)
+Y (V )
+X(tgt)
+φE
+Y (tgt)
+φV
+⇝
+P(Y (V ))
+P(X(V ))
+P(Y (E))
+P(X(E))
+X(tgt)−1
+φ−1
+E
+Y (tgt)−1
+φ−1
+V
+,
+where P(S) denotes the power set of a set S.
+Finally, we must verify that the functor LV preserves finite colimits. By Lemma 3.6, that
+happens provided both functors LV0, LV1 : FinGraph → FinSet preserve finite colimits. The functor
+LV1 = evV is an evaluation functor on a copresheaf category, hence preserves colimits [Rie16,
+Proposition 3.3.9].
+Since coproducts commute with colimits, the pointwise coproduct of two
+evaluation functors
+LV0 =
+�FinGraph
+⟨evV ,evE⟩
+−−−−−−→ FinSet × FinSet +
+−→ FinSet
+�
+also preserves colimits. This completes the proof.
+A quantitative semantics for signed graphs can be defined similarly, subject to a caveat about
+the vertex parameters. Our notion of signed graph, designed to capture regulatory networks as
+studied in the biochemistry literature, attaches signs only to edges. We are thus led to assume that,
+in the Lotka-Volterra dynamical model, all species have baseline rates of decay rather than growth.
+This assumption is generally valid for protein regulatory networks, but not for gene regulatory
+networks in which mediating proteins are ignored [TN10], nor for predator-prey models in ecology.
+More flexible approaches are certainly possible. It would be straightforward to attach signs to
+vertices as well as edges and use them in the quantitative semantics. Alternatively, at the expense
+of a more cumbersome formalism, one could define dynamical systems with mixed linear-conical
+parameterizations, allowing the vertex parameters to be arbitrary reals while the edge parameters
+are constrained to be nonnegative.4 For simplicity and uniformity of presentation, we do not describe
+these extensions further.
+Let FinSgnGraph denote the full subcategory of SgnGraph spanned by finite signed graphs.
+4Similar mixed parameterizations are a practical necessity for parametric statistical models, studied in detail in
+one author’s PhD thesis [Pat20].
+21
+
+Theorem 3.8 (Lotka-Volterra model for finite signed graphs). There is a functor
+LV : FinSgnGraph → Para(Dynam+)
+that sends a finite signed graph X to the conically parameterized nonnegative dynamical system with
+parameter variables P := X(V ) + X(E), state variables S := X(V ), and essentially nonnegative,
+algebraic vector field
+v(x; ρ, β)(i) := −ρ(i) x(i) +
+�
+(e:i′→i)∈X
+X(sgn)(e) β(e) x(i′) x(i),
+x ∈ RX(V ), i ∈ X(V ),
+parameterized by ρ ∈ RX(V )
++
+and β ∈ RX(E)
++
+. Moreover, the functor LV preserves finite colimits.
+Proof. Similarly to the previous proof, the functor LV : FinSgnGraph → Para(Dynam+) is defined
+by functors LV0, LV1 : FinSgnGraph → FinSet along with a natural transformation
+⃗
+LV : F+ ◦ LV0 ⇒ Dynam+ ◦ LV1 : FinSgnGraph → Con,
+now having components ⃗
+LVX given by the copairing of
+v0
+X := ⃗
+LV
+0
+X : RX(V )
++
+→ Dynam+(X(V ))
+and
+v1
+X := ⃗
+LV
+1
+X : RX(E)
++
+→ Dynam+(X(V )),
+where we define
+v0
+X(x; ρ)(i) := −ρ(i) x(i)
+and
+v1
+X(x; β)(i) :=
+�
+e∈X(tgt)−1(i)
+X(sgn)(e) β(e) x(X(src)(e)) x(i).
+The proof of naturality is essentially the same as before, using the crucial additional fact that
+morphisms of signed graphs preserve signs. The proof that the functor LV preserves finite colimits
+is unchanged.
+To exemplify the theorem, let us see how the Lotka-Volterra dynamics functor acts on a
+monomorphism and on an epimorphism of signed graphs.
+In order to compare the dynamics of two species A and B involved in a negative feedback loop
+versus A and B in isolation, we take the inclusion of signed graphs
+A
+B
+A
+B
+ι
+Labeling the edges in the feedback loop as AB and BA, the morphism LV(ι) sends the conically
+parameterized dynamical system
+�
+vA(x; ρ) = −ρA xA
+vB(x; ρ) = −ρB xB
+,
+ρ ∈ R{A,B}
++
+,
+to the parameterized dynamical system
+�
+vA(x; ρ, β) = −ρA xA − βBA xB xA
+vB(x; ρ, β) = −ρB xB + βAB xA xB
+,
+ρ ∈ R{A,B}
++
+, β ∈ R{AB,BA}
++
+,
+by setting the latter’s interaction coefficients to zero: βAB = βBA = 0.
+This formalizes the
+commonsense fact that the first system is a degenerate case of the second.
+22
+
+For a more interesting example, we return to the projection map between regulatory networks
+given by Equation (2.1) of Section 2.1, inspired by the arginine biosynthesis system. Call this
+projection map p, and abbreviate the regulator molecule as R and the enzymes as S := {C, D, E, F, I}.
+The morphism LV(p) sends the parameterized dynamical system
+�
+�
+�
+�
+�
+�
+�
+vR(x; ρ, β) = −ρR xR − βR x2
+R
+vC(x; ρ, β) = −ρC xC − βC xR xC
+vD(x; ρ, β) = −ρD xD − βD xR xD
+vE(x; ρ, β) = −ρE xE − βE xR xE
+vF (x; ρ, β) = −ρF xF − βF xR xF
+vI(x; ρ, β) = −ρI xI − βI xR xI
+with state variables {R} + S and parameters ρ, β ∈ R{R}+S
++
+to the parameterized dynamical system
+�
+vR(x; ρ, β) = −ρR xR − βR x2
+R
+v∗(x; ρ, β) = −ρ∗ x∗ − β∗ xR x∗
+with state variables {R, ∗} and parameters ρ, β ∈ R{R,∗}
++
+, in two different but equivalent ways. The
+first way sets the latter system’s coefficients equal to sums of the former’s coefficients, namely
+ρ∗ =
+�
+i∈S
+ρi
+and
+β∗ =
+�
+i∈S
+βi.
+The second way substitutes x∗ for each xi, i ∈ S, in the first system and then takes the vector field
+v∗ to be the sum of the vi’s, i ∈ S, with these substitutions. The equivalence of these operations
+is precisely the condition for LV(p) to be a morphism of parameterized dynamical systems, cf.
+Equations (3.1) and (3.2).
+3.3
+Composing Lotka-Volterra models
+To complete this part of the story, we extend the Lotka-Volterra dynamics functors between
+graphs and parameterized dynamical systems, constructed in Theorems 3.7 and 3.8, to double
+functors between open graphs and open parameterized dynamical systems. We begin by making
+parameterized dynamical systems into open systems.
+Proposition 3.9 (Open parameterized dynamical systems). There is a symmetric monoidal double
+category of open linearly parameterized dynamical systems, Open(Para(Dynam)), having
+• as objects, finite sets A, A′, . . . ;
+• as vertical arrows, functions f : A → A′;
+• as horizontal arrows, open linearly parameterized dynamical systems, which consist of
+a linearly parameterized dynamical system (P, S, v : RP → Dynam(S)) along with a cospan
+A0
+ℓ0
+−→ S
+ℓ1
+←− A1 whose apex is the set S of state variables;
+• as cells, morphisms of such open systems (P, S, v, ℓ0, ℓ1) → (P ′, S′, v′, ℓ′
+0, ℓ′
+1), which
+consist of a morphism (q, f) : (P, S, v) → (P ′, S′, v′) between linearly parameterized dynamical
+systems along with functions f0 : A0 → A′
+0 and f1 : A1 → A′
+1 making the diagram commute:
+A0
+S
+A1
+A′
+0
+S′
+A′
+1
+ℓ0
+ℓ1
+f0
+f
+ℓ′
+0
+f1
+ℓ′
+1
+.
+23
+
+Vertical composition is by composition in FinSet and in Para(Dynam). Horizontal composition and
+monoidal products are by pushouts and coproducts in Para(Dynam), respectively, interpreting the
+finite sets in the feet of the cospans as linearly parameterized dynamical systems with no parameter
+variables and identically zero vector fields.
+Similarly, there is a symmetric monoidal double category Open(Para(Dynam+)) of open conically
+parameterized nonnegative dynamical systems.
+Proof. We perform the construction for linearly parameterized dynamical systems. The construction
+for conically parameterized nonnegative dynamical systems is perfectly analogous, replacing R with
+R+ and vector spaces with conical spaces.
+The projection functor πS : Para(Dynam) → FinSet, (P, S, v) �→ S that sends a linearly parame-
+terized dynamical systems to its set of state variables has a left adjoint Z : FinSet → Para(Dynam)
+that sends a finite set S to the system (∅, S, 0) with empty set of parameter variables. By linearity, its
+parameterized vector field 0 ∼= R∅ → Dynam(S) is necessarily the zero vector field. This indeed gives
+an adjunction Z ⊣ πS, because to any function f : S → S′ and linearly parameterized dynamical
+system (P ′, S′, v′) there corresponds a unique morphism (0P ′, f) : Z(S) → (P ′, S′, v′), where the
+required square
+0
+Dynam(S)
+RP ′
+Dynam(S′)
+f∗◦(−)◦f∗
+v′
+commutes trivially, since the zero vector space is initial in VectR.
+Since Para(Dynam) has finite colimits (Proposition 3.5), we can construct a symmetric monoidal
+double category of Z-structured cospans [BC20, Theorem 3.9]. As we have argued before, it will be
+isomorphic to Open(Para(Dynam)).
+With this definition, we can construct double functors between open graphs and open param-
+eterized dynamical systems, but the vertex parameters under Lotka-Volterra dynamics cause a
+twist in the story compared to Baez and Pollard’s compositionality result for mass-action kinetics
+[BP17, Theorem 18]. When composing open dynamical systems in the image of the Lotka-Volterra
+functor, one takes a coproduct of the parameter variables, i.e., a direct sum of the parameter spaces,
+belonging to identified vertices. However, if one composes the open graphs first, then the identified
+vertices receive a single copy of the parameters from the Lotka-Volterra functor. Thus this functor
+does not preserve composition of open systems, not even up to isomorphism. Nevertheless, there is
+a natural (noninvertible) comparison between them: given a pair of parameters in the direct sum,
+we can reduce them to a single parameter simply by summing them. In mathematical terms, we get
+a lax double functor: a double functor that strictly preserves vertical composition, as usual, but
+preserves horizontal composition only up to specified comparison maps.5
+Theorem 3.10 (Open Lotka-Volterra models). There is a symmetric monoidal lax double functor
+LV : Open(FinGraph) → Open(Para(Dynam))
+that acts
+• on objects and vertical arrows, as the identity;
+5The precise definition of a lax double functor can be found in the textbook [Gra19, §3.5], among other sources.
+24
+
+• on horizontal arrows and cells, by the functor LV : FinGraph → Para(Dynam) on graphs and
+graph homomorphisms and as the identity on the associated cospans and cospan morphisms:
+�
+X, A0
+ℓ0
+−→ X(V )
+ℓ1
+←− A1
+�
+�→
+�
+LV(X), A0
+ℓ0
+−→ X(V )
+ℓ1
+←− A1
+�
+.
+The comparison cells are defined using the morphisms of linearly parameterized dynamical systems
+αS : Z(S) → LV(Disc S), where
+αS := (0S, 1S) : (∅, S, 0) → (S, S, ⃗
+LV(Disc S)),
+S ∈ FinSet.
+• Given composable open graphs (X, A → X(V ) ← B) and (Y, B → Y (V ) ← C), the comparison
+cell for horizontal composition is given by the morphism of systems
+LV(X) +Z(B) LV(Y )
+id +αB id
+−−−−−−→ LV(X) +LV(Disc B) LV(Y )
+∼
+=
+−→ LV(X +Disc B Y ).
+• Given a finite set A, the comparison cell for the horizontal unit is given by the morphism of
+systems αA : Z(A) → LV(Disc A).
+Similarly, there is a symmetric monoidal lax double functor
+LV : Open(FinSgnGraph) → Open(Para(Dynam+)).
+Proof. To construct the lax double functor, we use a lax version of [BC20, Theorem 4.3]. The
+family of morphisms αS : Z(S) → LV(Disc S), S ∈ FinSet, in the theorem statement assemble into
+a natural transformation
+FinSet
+FinGraph
+FinSet
+Para(Dynam)
+L=Disc
+LV
+L′=Z
+α
+.
+The functors involved in this cell all preserve finite colimits: the top and bottom ones because they
+are left adjoints and the right one by Theorem 3.7. The hypotheses of [BC20, Theorem 4.3] are
+therefore satisfied, except that α is not a natural isomorphism but merely a natural transformation.
+By inspection of the proof, the result still holds except that the resulting double functor is lax rather
+than pseudo. We obtain a lax double functor
+Open(FinGraph) ∼= LCsp(FinGraph) → L′Csp(Para(Dynam)) ∼= Open(Para(Dynam)).
+To show that this double functor is the same one in the theorem statement, we once again
+use the adjunctions to pass between L-structured and R-decorated cospans (recalling terminology
+introduced in the proof of Proposition 2.4). Notice that the natural transformation α has as its
+mate [CGR14, §1] the identity transformation ¯α = 1evV :
+FinSet
+FinGraph
+FinSet
+Para(Dynam)
+R=evV
+LV
+R′=πS
+¯α
+.
+25
+
+Thus the action of the double functor F := LV on L-structured cospans simplifies to the identity
+when translated to R-decorated cospans.
+L(A0)
+X
+L(A1)
+L′(A0)
+F(L(A0))
+F(X)
+F(L(A1))
+L′(A1)
+↭
+A0
+R(X)
+A1
+A0
+R(X)
+R′(F(X))
+R(X)
+A1
+ℓ0
+ℓ1
+αA0
+F(ℓ0)
+F(ℓ1)
+αA1
+¯ℓ0
+¯ℓ1
+¯ℓ0
+¯αX
+¯αX
+¯ℓ1
+A similar statement holds for the action of the double functor on morphisms of L-structured and
+R-decorated cospans.
+4
+Conclusion
+Summary.
+Regulatory networks are a minimalistic but widely used tool to describe the interactions
+between molecules in biochemical systems. We have made the first functorial study of regulatory
+networks, formalized as signed graphs, and their connections with other mathematical models in
+biochemistry. Among the latter, we have studied reaction networks, formalized as Petri nets with
+signed links, and parameterized dynamical systems, focusing on Lotka-Volterra dynamics. This
+project fits into a broader program by applied category theorists and other scientists aiming to
+systematize, in a completely precise way, the language and methods of describing, composing, and
+transforming scientific models.
+The major categories of this paper, and the functors between them, are summarized in the
+following diagram, where “LV” is the Lotka-Volterra dynamics functor (§3.2).
+SgnCat (§2.2)
+SgnGraph (§2.1)
+SgnPetri (§2.3)
+FinSgnGraph
+Para(Dynam+) (§3.1)
+Path
+U
+LV
+Net
+⊣
+Most of the main results extend from closed systems to open systems, which compose by gluing
+along their boundaries. Of the diagram above, we have extended the following parts to double
+categories of open systems and double functors between them.
+Open(SgnCat)
+Open(SgnGraph)
+Open(FinSgnGraph)
+Open(Para(Dynam+)) (§3.3)
+LV
+Path
+26
+
+Future work.
+Of many possible directions for future work, we mention a few. As noted in the
+introduction, Lotka-Volterra dynamics are only one of numerous dynamics that could be considered
+as a canonical model for regulatory networks, and they are not even among the most commonly
+studied in the biochemistry literature [TLK19]. It would be desirable to have dynamics functors for
+regulatory networks that draw on more flexible or more biologically plausible classes of dynamical
+systems. In another direction, the two halves of this paper—qualitative and quantitative—are not as
+tightly as integrated as one might hope. How does the presence of a motif in a regulatory network,
+such as an incoherent feedforward loop perhaps even of a specific type, manifest in the continuous
+dynamics of that network? Put in category-theoretic terms, the Lotka-Volterra dynamics functor is
+defined on signed graphs, so how does it relate to the freely generated signed categories in which
+motifs are expressed? These intriguing questions are suggestive of “feedback loop analysis” in the
+field of system dynamics [Ric95], to which stronger connections should be made.
+Acknowledgments.
+The authors thank the American Mathematical Society (AMS) for hosting
+the 2022 Mathematical Research Community (MRC) on Applied Category Theory, where this
+research project began. The AMS MRC was supported by NSF grant 1916439. We thank John Baez,
+our group’s mentor at the MRC, for suggesting this project and for much helpful advice along the
+way. Authors Fairbanks, Patterson, and Shapiro acknowledge subsequent support from the DARPA
+ASKEM and Young Faculty Award programs through grants HR00112220038 and W911NF2110323.
+Author Ocal acknowledges subsequent support from an AMS-Simons Travel Grant and from the
+Hausdorff Research Institute for Mathematics funded by the German Research Foundation (DFG)
+under Germany’s Excellence Strategy - EXC-2047/1 - 390685813.
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+29
+

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+arXiv:2301.11392v1  [cond-mat.mes-hall]  26 Jan 2023
+Noise and Thermal Depinning of Wigner Crystals
+C. Reichhardt and C. J. O. Reichhardt
+Theoretical Division and Center for Nonlinear Studies, Los Alamos National
+Laboratory, Los Alamos, New Mexico 87545, USA
+30 January 2023
+Abstract.
+We examine changes in the depinning threshold and conduction noise
+fluctuations for driven Wigner crystals in the presence of quenched disorder. At low
+temperatures there is a well defined depinning threshold and a strong peak in the noise
+power with 1/f noise characteristics. At higher temperatures, the depinning threshold
+shifts to lower drives and the noise, which is also reduced in power, becomes more white
+in character. At lower temperatures, a washboard frequency appears when the system
+depins elastically or forms a moving smectic state; however, this washboard signal is
+strongly reduced for higher temperatures and completely disappears above the melting
+temperature of a system without quenched disorder. Our results are in good agreement
+with recent transport and noise studies for systems where electron crystal depinning is
+believed to arise, and also show how noise can be used to distinguish between crystal,
+glass, and liquid phases.
+1. Introduction
+When a collection of interacting particles is driven over quenched disorder, the system
+can exhibit a pinned phase, a depinning threshold, and a sliding phase [1, 2].
+The
+existence of these phases can be deduced from changes in transport measures such
+as the velocity-force and differential resistance curves [1, 2, 3, 4, 5, 6, 7, 8].
+If the
+particles maintain the same neighbors during the depinning and sliding process, the
+depinning is considered elastic and is associated with specific scaling features in the
+velocity-force curves [1, 2, 9, 10], while if there is tearing or mixing of the particles,
+the behavior is plastic and can produce multiple steps or jumps in the transport curves
+[1, 2, 9, 11].
+In the sliding phase, there can also be dynamical transitions between
+different types of plastic flow or fluidlike flow, as well as dynamical ordering transitions
+where the driven particles move so rapidly over the substrate that the effectiveness of
+the pinning is reduced and a disordered system can organize into a moving crystal or
+smectic state [2, 12, 13, 14, 15, 16, 17, 18, 19, 20]. When thermal effects are included,
+additional behaviors can occur both during depinning and in the sliding states.
+In
+general, sharp depinning thresholds become rounded due to thermal creep; however, a
+peak in the differential velocity can still arise near the T = 0 depinning threshold due
+to a transition from creep to sliding dynamics [1, 2, 13]. If the temperature is higher
+
+Noise and Thermal Depinning of Wigner Crystals
+2
+than the melting temperature of the quenched disorder-free system, a system containing
+quenched disorder will always be in a disordered state, and can form a glass phase with
+thermal creep or a fluctuating liquid state at high drives [2, 13, 21, 22].
+Another method to examine the driven dynamics is to measure changes in the noise
+for systems in which time series of the velocity or density fluctuations can be obtained
+as a function of drive [2, 4, 17, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].
+One of
+the most common characterization techniques is to determine the power spectrum of
+the fluctuations and to measure the noise power over some frequency range. For elastic
+depinning or ordered moving states where the particles maintain a fixed set of neighbors
+and travel at speed v, there is typically a narrow band noise signature containing peaks
+at specific frequencies ω = 2πv/a that are associated with the average spacing a between
+the particles [2, 4]. Additional peaks appear at higher harmonics of these characteristic
+frequencies. Narrow band noise signatures have been observed for sliding charge density
+waves [4], superconducting vortex lattices [17, 26, 27, 28, 34], moving charge crystals [33],
+and skyrmion systems [30, 35]. In some cases it is possible to observe multiple frequencies
+when the system is broken up into several large sections that locally behave elastically
+but globally provide multiple degrees of freedom, permitting switching behavior to occur
+[33, 36]. If the depinning is strongly plastic, the narrow band noise signal is lost and is
+typically replaced by 1/f α noise with 0.75 < α < 2.0 [2, 17, 25, 27, 28], while for a fluid
+the noise power is often low and the fluctuations have white noise characteristics with
+α = 0. In other cases, the fluctuations are Lorentizan and the noise has a 1/f shape at
+low frequencies but becomes white above a characteristic frequency ωc that is associated
+with the average time between collisions of particles with pinning sites [2]. Broad band
+1/f noise has been observed near the depinning transition for superconducting vortices
+[2, 17, 25, 28], magnetic skyrmions [30, 32], and charge crystals [33, 37]. The noise
+measurements can also be used to identify a transition between different dynamical
+states such as plastic flow to dynamically ordered flow. In this case, 1/f noise occurs in
+the plastic flow phase just above depinning, but at higher drives there is a crossover to
+narrow band noise as dynamical ordering emerges [2, 17, 28, 30]. These different noise
+features can be modified significantly when thermal effects become important [2].
+Another example of an assembly of particle-like objects that can be coupled to
+quenched disorder and driven is electron crystals or Wigner crystals [38, 39, 40, 41, 42],
+where transport measures provide evidence for a conduction threshold that is consistent
+with the existence of a depinning transition [39, 40, 43, 44, 45, 46, 36, 47, 48]. Recently, a
+growing number of materials have been identified that can support Wigner crystals, such
+as moir`e superlattices [49, 50], transition metal dichalcogenide monolayers [51, 52, 53],
+and systems where Wigner crystals are stable at zero magnetic field [54].
+It would
+be interesting to examine conduction and noise measures as a function of drive and
+temperature in these new systems. Previous experiments that showed evidence of a
+conduction threshold also revealed a large increase in the conduction noise just above
+depinning [46], and previous numerical studies of driven Wigner crystals also showed
+both a conduction threshold and 1/f noise features near depinning followed by a
+
+Noise and Thermal Depinning of Wigner Crystals
+3
+crossover to narrow band noise at higher drives [55]. Brussarski et al. [56] examined the
+transport and noise of Wigner crystals near depinning as function of temperature, and
+found that at low temperature, there is a sharp depinning threshold that is correlated
+with a large peak in the noise power. Additionally, the noise near depinning is of 1/f 0.75
+form. As the temperature is increased, the depinning threshold shifts to lower values
+and the peak noise power is also reduced. This suggests that at higher temperature, the
+system forms a Wigner liquid in which the correlated motion associated with glassy or
+plastic flow phases and large noise power is lost. Noise studies have also been performed
+near the metal-insulator transition, which could be associated with a change from a
+Wigner glass to a Wigner liquid, and a drop in the noise power is observed at higher
+temperatures where a fluid phase may be present [57, 58]. Particle-based simulations
+across a Wigner glass to Wigner fluid crossover show high power 1/f α noise in the
+Wigner glass state and lower noise power with a white spectrum at higher temperatures
+in the fluid state [59]. Thermal effects and thermal melting in Wigner crystals have
+also been extensively studied [60, 61, 62, 63, 64], so it should be feasible to perform
+experimental noise and transport measures across a thermal melting transition while
+the system is being driven.
+In this work, we consider thermally induced transport and noise measurements for
+a two-dimensional (2D) electron system driven over quenched disorder. Previous work
+on this system focused on the T = 0 case, and showed that for plastic depinning, there is
+strong 1/f noise with a peak in the noise power near the depinning transition, followed
+by a drop in the noise power and a transition to white or narrow band noise at high
+driving where a moving smectic or moving crystal phase emerges [55, 65]. Here we find
+that as we increase the temperature, the depinning threshold decreases and the noise
+power drops, in agreement with experiments. Additionally, we find that the narrow
+band noise visible for T = 0 at high drives is strongly reduced at higher temperatures
+and vanishes above the temperature Tm at which the system melts in the absence of
+quenched disorder. This suggests that narrow band noise signals may only be accessible
+at temperatures well below melting.
+We map out the dynamic phase diagram as a
+function of drive versus temperature and show that at Tm there is a divergence in the
+drive at which a transition to ordered or partially ordered flow occurs, similar to the
+dynamic phase diagram proposed by Koshelev and Vinokur for driven superconducting
+vortex systems [13, 22]. For the case of elastic depinning, we find a thermally induced
+creep regime in which the lattice moves by one lattice constant at a time, and show
+that a narrow band signal can still arise even in the creep regime. The spectral peaks
+become sharper and shift to higher frequencies with increasing drive, but the narrow
+band signature is lost with increasing temperature even before the system reaches the
+clean melting temperature Tm.
+
+Noise and Thermal Depinning of Wigner Crystals
+4
+2. Simulation and System
+We model a 2D classical Wigner crystal with charge density n = Ne/L2, where Ne is the
+number of electrons and L is the system size. We employ periodic boundary conditions
+in the x and y directions, and the sample contains Np randomly placed pinning sites
+modeled as short range attractive wells with a density of np = Np/L2. Throughout
+this work we fix n = 0.208 and np = 0.25. At T = 0 and in the absence of quenched
+disorder, the charges form a triangular lattice that has a well defined melting transition
+temperature Tm [66]. Additionally, when T = 0 there is a well defined quenched disorder
+strength above which the system disorders [66].
+We represent the charges using a
+previously studied model [55, 67, 65, 66, 68, 69, 70, 71], where the equation of motion
+for charge i is
+αdvi =
+N
+�
+j
+∇U(rij) + Fp + FD + FT
+i .
+(1)
+Here αd is a damping term and Ui = q2/r is the long range Coulomb repulsion between
+charges of magnitude q. As in previous work [55, 71], we employ a Lekner summation to
+evaluate the long range interactions. The second term on the right hand side represents
+pinning sites modeled as finite range parabolic traps that impart a maximum pinning
+force of Fp at radius rp. The thermal fluctuations are applied with the term FT , which
+has the following properties: ⟨F T⟩ = 0 and ⟨F T(ti)F T(t′
+j)⟩ = 2kBTδijδ(t − t′). The
+initial positions of the charges are obtained through simulated annealing at zero drive.
+Once the system has been initialized, we apply a driving force FD = FDˆx representing
+an applied voltage. The drive can be set to a constant value, in which case we wait
+for the system to reach a steady state before measuring the average velocity per charge
+⟨V ⟩ =
+�Ne
+i
+vi · ˆx or obtaining a time series of the velocity to examine the temporal
+fluctuations. By considering a range of drives and measuring the average velocity at
+each drive, we can create a current-voltage curve. If there is a magnetic field present,
+the changes experience an additional force qB × vi that can create a Hall angle for the
+electron motion. This effect is generally small and we neglect it in the present work,
+but we have studied it in detail elsewhere [71].
+3. Elastic and Plastic Regimes
+In Fig. 1 we plot the fraction P6 of six-fold coordinated charges versus temperature T/Tm
+for a system with no quenched disorder. The melting temperature Tm is defined to be the
+temperature at which a proliferation of topological defects or non-sixfold coordinated
+charges occurs. For T/Tm < 1.0, P6 is close to 1.0, as expected for a triangular lattice,
+while for T/Tm > 1.0, a large number of fivefold and sevenfold coordinated charges
+appear, causing P6 to drop.
+Once we have defined Tm by measuring a clean system, we introduce quenched
+disorder in order to study the conduction noise and transport response above and below
+Tm for varied disorder strengths Fp. We apply a constant drive with FD = 0.01 to
+
+Noise and Thermal Depinning of Wigner Crystals
+5
+0
+0.5
+1
+1.5
+T/Tm
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P6
+Figure 1.
+The average fraction of sixfold-coordinated charges P6 versus temperature
+T/Tm in a system with no quenched disorder. Tm is defined to be the temperature at
+which a proliferation of non-sixfold coordinated charges occurs in a clean system.
+samples with different Fp and measure the time average velocity per charge ⟨V ⟩ over
+4 × 106 simulation time steps. When Fp = 0, the charge velocity V0 is identical to the
+driving force, V0 = FD = 0.01, so a measurement of ⟨V ⟩/V0 = 1 indicates that the
+flow of the charges has reached the pin-free limit. In Fig. 2(a), where we plot ⟨V/V0⟩
+versus Fp, at T/Tm = 0 there is a large drop in ⟨V ⟩/V0 near Fp = 0.75. Figure 2(b)
+shows the corresponding values of P6 versus Fp, where for T/Tm = 0 there is a well
+defined transition from an ordered Wigner crystal to a disordered Wigner glass, and the
+proliferation of defects correlates with the velocity drop in Fig. 2(a). At T/Tm = 0.3,
+the overall velocity is higher than for the T/Tm = 0 sample due to the lowering of
+the effectiveness of the pinning by the thermal fluctuations. Additionally, the pinning
+strength required to disorder the system is shifted upward to a value close to Fp = 0.1,
+which is again due to the partial reduction of the pinning effectiveness by the thermal
+fluctuations. A similar trend occurs for T/Tm = 0.6, where the velocity is higher. For
+T/Tm = 1.03, the system is disordered for all values of Fp and the velocity is even higher
+but has a gradual drop with increasing Fp, and the same trend occurs for T/Tm = 1.36. A
+more detailed study of the general phase diagram for the disordered and ordered phases
+as a function of pinning strength versus temperature appears in Ref. [66]. The results in
+
+Noise and Thermal Depinning of Wigner Crystals
+6
+0
+0.2
+0.4
+0.6
+0.8
+1
+<V>/V0
+0
+0.05
+0.1
+0.15
+Fp
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P6
+Figure 2. (a) The average charge velocity ⟨V ⟩/V0 vs pinning strength Fp at FD = 0.01,
+where V0 = FD = 0.01 is the average velocity in a disorder-free system. T/Tm = 0
+(dark blue), 0.3 (light blue), 0.6 (green), 1.03 (yellow) and 1.38 (red).
+(b) The
+corresponding P6 vs Fp showing that for T/Tm < 1.0, there is a well defined pinning-
+induced order to disorder transition.
+Fig. 1 and Fig. 2 indicate that the system exhibits three distinct regimes. These are an
+ordered or crystal regime containing sixfold-coordinated charges, which occurs at low
+temperatures or low pinning strengths; a disordered or plastic regime where the system
+has low mobility and is strongly affected by the pinning; and a high temperature fluid
+phase where the effectiveness of the pinning is reduced and the system is in a strongly
+fluctuating state. In terms of transport, in the presence of pinning the ordered state
+exhibits elastic depinning in which the charges maintain their same neighbors. The glass
+state undergoes plastic depinning, and the fluid state does not have a pinned phase but
+can still have a regime in which the charges are trapped for a time before thermally
+
+Noise and Thermal Depinning of Wigner Crystals
+7
+hopping out of the pinning sites.
+4. Transport and Noise in the Plastic Regime
+We next examine the noise and transport in the three regimes identified above. We
+consider samples with Fp = 0.5, a pinning strength at which the charges are disordered
+for T/Tm = 0, so the system is in a strongly disordered glass phase. The T/Tm = 0
+plastic depinning that occurs in this regime was studied in detail in [66], where a pinned
+phase, a filamentary flow phase, a disordered flow phase, and a dynamically ordered
+moving smectic phase appear in sequence as a function of increasing drive.
+In Fig. 3(a) we show a snapshot of the charge locations, pinning site locations, and
+trajectories in the plastic flow regime for FD = 0.15 at T/Tm = 0.15, where a portion
+of the charges are moving in a series of well defined channels, with occasional jumps
+between the channels when certain channels open or close again. In general, for strong
+pinning, at low temperature the system exhibits channel flow just above depinning,
+similar to that studied in other systems at zero temperature. Figure 3(b) shows the
+same system at T/Tm = 0.606 where there is a combination of channel flow and random
+thermal hopping, indicating that as the temperature increases, there is a transition
+from one-dimensional (1D) channels to two-dimensional (2D) flow.
+In Fig. 3(c), at
+T/Tm = 1.03 the motion is 2D and fluidlike. For higher temperatures, the images look
+similar to what is shown in Fig. 3(c).
+In Fig. 4(a) we plot ⟨V ⟩ versus FD for the system in Fig. 3 at T/Tm = 0, 0.606,
+0.9, 1.21, 1.81, and 2.42. For the lower temperatures, there is a well-defined depinning
+threshold followed by a nonlinear regime, while when T/Tm > 1.0, the threshold is
+replaced by a creep regime and the nonlinear regime at higher drives persists. At the
+highest drives, all of the curves approach the pin-free limit. In Fig. 4(b) we show the
+corresponding d⟨V ⟩/dFD versus FD curves, where for T/Tm ≥ 1.21 there is a peak
+in d⟨V ⟩/dFD due to the S shape of the velocity-force curves.
+Similar peaks in the
+differential conductivity were observed for driven superconducting vortices in the plastic
+flow regime [12, 17, 21, 22]. For T/Tm > 1.21, the peaks are lost and a creep regime
+appears. The dashed line is the differential conductivity for the pin free system, and
+all of the curves approach this value at high drives. We note that for T/Tm = 0, at
+low drives there are a number of jumps in the conduction as well as a few regimes of
+negative differential conduction. This arises due to a filamentary flow channel effect
+that is described in more detail in [65]. For T/Tm > 0.5, the jumps associated with
+the filamentary flow phase are lost and a single large peak in d⟨V ⟩/dFD appears in the
+plastic flow regime where there is a combination of moving and pinned charges.
+In Fig. 5 we plot P6 versus FD for the system in Fig. 4 for T/Tm = 0, 0.303, 0.606,
+0.757, 0.91, 1.21, and 2.42. For T/Tm < 1.0 there is an initial dip in P6 at the onset of
+plastic flow, and at high drives where d⟨V ⟩/dFD starts to flatten, P6 approaches values
+of 0.9 or higher as the system forms a moving smectic phase. In the moving smectic
+state, the charges move in well defined channels and a small number of dislocations are
+
+Noise and Thermal Depinning of Wigner Crystals
+8
+x
+(a)
+y
+x
+(b)
+y
+x
+(c)
+y
+Figure 3.
+Charge locations (red circles), trajectories (blue lines), and pinning
+site locations (black circles) for the system in Fig. 2 at FD = 0.15 and Fp = 0.5. (a)
+Filamentary flow at T/Tm = 0.15. (b) Disordered flow with channels at T/Tm = 0.606.
+(c) T/Tm = 1.03.
+present that have their Burgers vectors aligned with the driving direction [2, 55, 65].
+As T/Tm increases further, the drive at which the smectic state emerges shifts to higher
+values of FD, and for T/Tm > 1.0, the system no longer forms a moving smectic but
+instead becomes a moving fluid.
+From the features in the transport curves and P6 plotted in Figs. 4 and 5, we
+construct a dynamic phase diagram of the evolution of the different phases as a function
+of FD versus T/Tm in Fig. 6. At low drives we find a pinned or creep regime denoted
+C, where d⟨V ⟩/dFD < 0.5. The dynamically ordered moving smectic phase MS appears
+
+Noise and Thermal Depinning of Wigner Crystals
+9
+0
+0.2
+0.4
+0.6
+0.8
+1
+<V>
+0
+0.25
+0.5
+0.75
+1
+FD
+0
+0.5
+1
+1.5
+d<V>/dFD
+(a)
+(b)
+Figure 4.
+(a) Velocity ⟨V ⟩ vs drive FD for the system in Fig. 3 with Fp = 0.5 at
+T/Tm = 0.0 (purple), 0.606 (blue), 0.91 (dark green), 1.21 (light green), 1.81 (orange),
+and 2.42 (red). (b) The corresponding d⟨V ⟩/dFD vs FD curves. The dashed lines
+indicate the pin-free limit.
+when P6 > 0.9. The disordered regime is where the system is structurally disordered but
+moving, and it can be either a moving glass MG for T/Tm < 1.0, or a moving liquid ML
+for T/Tm > 1.0. The overall features of the phase diagram are similar to those observed
+in driven superconducting vortex systems with quenched disorder, as first proposed by
+Koshelev and Vinokur [13], where the transition between the MG and MS states shifts to
+higher drives as T/Tm is approached. In Ref. [13], the transition line from the disordered
+to moving ordered phase was argued to be proportional to A/(Tm −T), where A is some
+prefactor and the moving ordered phase can be described in terms of having an effective
+temperature that is decreasing toward zero. This picture assumes the formation of a
+
+Noise and Thermal Depinning of Wigner Crystals
+10
+0
+0.5
+1
+1.5
+2
+FD
+0.4
+0.6
+0.8
+1
+P6
+Figure 5.
+P6 vs FD for the system in Fig. 4 with Fp = 0.5 at T/Tm = 0 (purple),
+0.303 (dark blue), 0.606 (light blue), 0.757 (green), 0.91 (yellow), 1.21 (orange), and
+2.42 (red). The system reaches an ordered state at high drives for T/Tm < 1.0.
+moving crystal at high drive, and is somewhat modified in our system since the moving
+state we observe is a smectic in which the dynamic fluctuations are anisotropic [16]. We
+find that a better fit to the transition line in our case is (Tm − T)−0.7, which is likely
+due to the anisotropic nature of the moving smectic.
+Now that we have established the dynamic phase diagram as a function of drive
+versus temperature, we can ask how the velocity fluctuation power spectra change as a
+function of FD and T. The power spectrum as a function of ω = 2πf can be calculated
+using the time series v(t) of the velocity fluctuations,
+S(ω) =
+����
+�
+v(t)e−iωt
+����
+2
+(2)
+At T = 0 the noise has a 1/f α signature with α ≈ 0.8, in agreement with recent
+experiments [56]. The noise power is reduced at high drives and shows a crossover to
+a narrow band signature when the system forms a moving smectic phase; however, the
+experiments do not detect a narrow band noise signature at higher drives, suggesting
+that thermal effects could be coming into play.
+In Fig. 7 we plot power spectra of the velocity time series for the system in Fig. 6
+at a drive of FD = 0.15 for T/Tm = 0, 0.303, 0.606, and 1.03. For T/Tm = 0, the
+
+Noise and Thermal Depinning of Wigner Crystals
+11
+0
+0.5
+1
+1.5
+T/Tm
+0
+1
+2
+3
+4
+FD
+MS
+ML
+MG
+C
+Figure 6.
+Dynamic phase diagram as a function of FD vs T/Tm for the system in
+Figs. 4 and 5 with Fp = 0.5. There is a pinned or creep phase C (red), a disordered
+moving phase (blue) that is a moving glass, MG, at lower temperatures and a moving
+liquid, ML, at higher temperatures, and a moving smectic MS (green).
+low frequency noise has a 1/f α form, where the dashed line is a fit with α = −0.8,
+while at higher frequencies the noise tail has α = −2.0. At T/Tm = 0.5, the lower
+frequency noise power is reduced and α drops closer to α = 0, the value expected for
+white noise; however, the high frequency noise still has a 1/f 2 form. For higher T/Tm,
+the low frequency noise power is further reduced while the higher frequency noise power
+is enhanced, and the spectrum becomes much whiter overall.
+To better characterize the system, we measure the noise power S0, which is the
+value of the spectral power integrated in a small window around a specific frequency
+ω = 20. In Fig. 8 we show S0 versus FD for T/Tm = 0, 0.303, 0.606, and 1.03 on a log-
+linear plot. For T = 0 there is a large peak in S0 over the range 0.01 < FD < 0.5, which
+corresponds to the appearance of 1/f 0.85 noise. The noise is white for 0.5 < FD < 0.9,
+and for FD > 0.9 a narrow band noise signal appears. For T/Tm = 0.303 and 0.606,
+there is still a peak in the noise near FD = 0.2, but as the temperature increases, the
+peak power diminishes and the peak location shifts to lower drives. This is correlated
+with a whitening of the low frequency noise, as shown in Fig. 7. For T/Tm = 1.03, the
+
+Noise and Thermal Depinning of Wigner Crystals
+12
+10
+100
+1000
+ω
+10
+-9
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+S(ω)
+Figure 7.
+Power spectra S(ω) vs ω for the system in Fig. 6 with FD = 0.15 for
+T/Tm = 0 (dark blue), 0.303 (light blue), 0.606 (yellow), and 1.03 (red). The spectral
+signature changes from 1/f to white at low frequencies as the temperature increases,
+while the amount of noise power at higher frequencies increases with increasing T .
+sharp noise power peak is lost. At large FD, we find that the noise power increases with
+increasing temperature due to the transition from flow through narrow 1D channels
+in the smectic state to a 2D Brownian like motion in the liquid state.
+The overall
+behavior of the noise power that we find is in agreement with experimental observations
+[56], where there is a large peak in the noise power near the depinning threshold at
+low temperatures, while for higher temperatures the noise power peak is reduced and
+shifts to lower drives before disappearing at sufficiently high temperature.
+Another
+feature that is also observed in the experiments is that the noise power increases with
+temperature at large drives.
+We next consider thermal effects in the high drive limit where the system forms a
+moving smectic at T = 0. In Fig. 9(a) we show S(ω) vs ω for the system from Fig. 6 at
+FD = 1.5 and T/Tm = 0, where there are a series of peaks associated with a narrow band
+noise signature. At T/Tm = 0.303 in Fig. 9(b), there are still strong peaks associated
+with the narrow band noise but the higher harmonic peaks are strongly reduced in
+power. In Fig. 9(c) at T/Tm = 0.606, the level of background noise has increased and
+the narrow band peaks are diminished in size, while at T/Tm = 1.03 in Fig. 9(d), the
+
+Noise and Thermal Depinning of Wigner Crystals
+13
+0
+0.5
+1
+1.5
+2
+FD
+10
+-10
+10
+-9
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+S0
+Figure 8.
+The noise power S0 at fixed ω = 20 vs FD for the system in Fig. 7 at
+FD = 0.15 for T/Tm = 0 (dark blue), 0.303 (light blue), 0.606 (yellow), and 1.03 (red).
+moving smectic phase is lost and the narrow band peaks disappear into the background
+noise. To better characterize the change in the narrow band noise signature, in Fig. 10
+we plot the noise power S0 at ω = 323, which is the location of the most pronounced
+narrow band noise peak in Fig. 9(a). For T/Tm < 0.5 there is a strong narrow band
+noise signal; however, at T/Tm = 0.75 the narrow band noise level is close to the value
+of the background noise. This suggests that thermal effects can strongly reduce the
+narrow band noise signal even at temperatures well below T/Tm = 1.0, which could
+explain why the narrow band noise signals are difficult to see in experiment. To better
+understand the origins of the changes in the noise signals, in Fig. 11(a) we plot the
+trajectories of the charges at T/Tm = 0.606 where narrow band noise is present. The
+system is still in a moving smectic state but the channels have been broadened by the
+thermal fluctuations, and there are several regions in which the channel structures are
+starting to break down. Figure 11(b) shows the trajectories for T/Tm = 1.07, where
+the 1D channel structure is lost, there is a significant amount of transverse diffusion,
+and the narrow band noise peaks disappear. This result indicates that the narrow band
+noise occurs only when the motion of the charges is mostly 1D in character.
+
+Noise and Thermal Depinning of Wigner Crystals
+14
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+2.0×10
+-5
+2.5×10
+-5
+S(ω)
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+2.0×10
+-5
+2.5×10
+-5
+S(ω)
+0
+1000
+2000
+3000
+ω
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+2.0×10
+-5
+2.5×10
+-5
+S(ω)
+0
+1000 2000 3000 4000 5000
+ω
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+S(ω)
+(a)
+(b)
+(c)
+(d)
+Figure 9.
+S(ω) vs ω for the system in Fig. 6 at FD = 1.5 where the system is in the
+moving smectic phase. T/Tm = (a) 0, (b) 0.303, (c) 0.606, and (d) 1.03.
+5. Thermal Depinning Noise in the Elastic Regime
+We next consider the thermal depinning and noise in the elastic regime where the charges
+maintain their same neighbors. From Fig. 2 we select a value of Fp = 0.05, well below
+the T/Tm = 0 disordering threshold of Fp = 0.075. In Fig. 12(a) we show ⟨V ⟩ versus
+FD at Fp = 0.05 for T/Tm = 0, 0.0378, 0.0756, 0.17, 0.303, 0.606, and 1.03, and we plot
+the corresponding d⟨V ⟩/dFD curves in Fig. 12(b). As T/Tm increases, the depinning
+threshold shifts to lower FD, and in Fig. 12(b), the peak in d⟨V ⟩/dFD that appears for
+T = 0 is lost for T/Tm > 0.303. We note that the system remains in an ordered state
+up to T/Tm = 1.0 for all values of FD. The d⟨V ⟩/dFD curves also show a multiple peak
+feature at high temperatures, with one peak at the finite temperature threshold and a
+second peak near the T = 0 depinning threshold. In between these two peaks, the flow
+is creep-like in nature.
+In Fig. 13 we plot S(ω) versus ω for the system in Fig. 12 at T/Tm = 0.1515 for
+different values of FD = 0.025, 0.03, 0.04, 0.046, 0.06, and 0.01. At FD = 0.025, the
+motion occurs mostly in the form of avalanches, and no clear narrow band signatures
+are present but the low frequency noise has high power. For FD = 0.03, the system
+starts to develop a narrow band noise signature that sharpens with increasing drive,
+and for FD ≥ 0.06, which is above the zero temperature depinning threshold, the low
+frequency noise is strongly suppressed and the narrow band noise peaks become much
+
+Noise and Thermal Depinning of Wigner Crystals
+15
+0
+0.5
+1
+T/Tm
+1×10
+-5
+S0
+Figure 10.
+The value of the noise power S0 at ω = 323, the frequency of the largest
+narrow band noise peak in Fig. 9, vs T/Tm for the system in Fig. 8 with FD = 0.15.
+Here the narrow band noise peaks are lost near T/Tm = 0.75.
+x
+(a)
+y
+x
+(b)
+y
+Figure 11.
+Charge locations (red circles), trajectories (blue lines), and pinning site
+locations (black circles) for the system in Figs. 9 and 10 at FD = 1.5 for T/Tm = (a)
+0.606 and (b) 1.03.
+
+Noise and Thermal Depinning of Wigner Crystals
+16
+0
+0.01
+0.02
+<V>
+0
+0.01
+0.02
+FD
+0
+2
+4
+6
+d<V>/dFD
+(a)
+(b)
+Figure 12. (a) ⟨V ⟩ vs FD for a system that exhibits elastic depinning at T/Tm = 0,
+where Fp = 0.05. The different curves are for temperatures of T/Tm = 0, 0.0378,
+0.0756, 0.17, 0.303, 0.606, and 1.03, from right to left. The dashed line is the expected
+curve in the pin free limit. (b) The corresponding d⟨V ⟩/dFD vs FD curves.
+sharper. This result shows that in the elastic flow regime, the narrow band noise signal
+is more robust than in the plastic phase, and it appears once the system has depinned.
+In Fig. 14 we plot S(ω) vs ω for the system in Fig. 13 at T/Tm = 0 and T/Tm = 0.303
+at a drive of FD = 0.02. At T/Tm = 0, there is a strong narrow band noise feature.
+Interestingly, at T/Tm = 0.303, although the level of background noise has increased,
+the primary narrow band noise peak is enhanced in power. The increase in the narrow
+band peak occurs when thermal effects weaken the effectiveness of the pinning and allow
+the charges to become better ordered. This effect is diminished in the case of strong
+pinning.
+
+Noise and Thermal Depinning of Wigner Crystals
+17
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+-10
+10
+-9
+10
+-8
+10
+-7
+S(ω)
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+ω
+10
+-11
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+ω
+10
+-11
+10
+-10
+10
+-9
+10
+-8
+10
+-7
+S(ω)
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 13.
+S(ω) vs ω for the system in Fig. 12 with Fp = 0.05 at T/Tm = 0.1515
+for FD = (a) 0.025, (b) 0.03, (c) 0.04, (d) 0.046, (e) 0.06, and (f) 0.01.
+To better characterize the narrow band noise behavior for the system in Figs. 12 and
+13, in Fig. 15 we plot the noise power S0 versus T/Tm for FD = 0.02 where the system is
+always in a moving state at the narrow band peak of ω = 80 and the background noise
+signal at ω = 300, along with the difference between these two noise powers. Unlike the
+case for strong pinning, the power of the narrow band noise signal generally increases
+with increasing T/Tm; however, the background noise power also increases, and the
+amount of power in the two signals becomes equal near T/Tm = 1.0. The narrow band
+noise peak has the greatest amount of additional power compared to the background
+noise near T/Tm = 0.3. This is again due to thermal effects washing out any additional
+avalanche-like motion and permitting the charge lattice to become better ordered.
+6. Discussion
+Narrow band noise has been observed experimentally in superconducting vortex [26, 28],
+magnetic skyrmion [32], and charge density wave [4] systems, but has not been seen for
+Wigner crystals. There have been reports of periodic noise in charge ordering systems
+such as stripe or bubble forming states [33, 37]; however, this noise generally appears at
+low frequencies and is probably not associated with the lattice-scale narrow band noise,
+
+Noise and Thermal Depinning of Wigner Crystals
+18
+0
+200
+400
+600
+ω
+0
+2×10
+-8
+4×10
+-8
+6×10
+-8
+S(ω)
+Figure 14.
+The power spectra S(ω) vs ω for the system in Fig. 13 with Fp = 0.05
+at FD = 0.02 in the moving phase for T/Tm = 0 (blue) and T/Tm = 0.303 (green).
+At T/Tm = 0.303, although the overall background noise power is higher, there is an
+enhancement of the narrow band noise signal.
+but instead arises due to the motion of some other periodically moving macroscopic scale
+structure. In the experiments of Brussarski et al. [56], the peak noise power decreased
+with increasing temperature, similar to what we observe, but no narrow band noise signal
+was observed. This could be the result of several possible factors. If the drive applied to
+the system is not uniform, there could still be strong plastic flow at low drives; however,
+at high drives the system may not form a uniformly ordered moving state but could
+instead break into several locally ordered regions that are moving at different speeds.
+Related to this, if the quenched disorder has a wide range of strength so that some of
+the charges are moving while a small number remain pinned, a disordered flow regime
+would emerge in which narrow band noise is absent. A narrow band noise signal could
+also be masked by strong background noise. In this case, the signal could be boosted
+by applying an additional ac drive. If the frequency of this ac drive is swept, phase
+locking or Shapiro steps would appear when the frequency comes into resonance with
+the narrow band signal [26]. Another possible issue is that the narrow band frequency
+could be too high to detect with the available experimental setup; however, for a system
+in the elastic depinning limit, fairly low frequency periodic signals could be generated in
+the creep regime. The lack of experimentally observed narrow band noise may suggest
+
+Noise and Thermal Depinning of Wigner Crystals
+19
+0
+0.5
+1
+T/Tm
+0
+2×10
+-8
+4×10
+-8
+6×10
+-8
+8×10
+-8
+S0
+Figure 15.
+The noise power S0 vs T/Tm for the system in Figs. 12 and 13 with
+Fp = 0.05 at FD = 0.02 for the narrow band frequency of ω = 80 (green circles), the
+background noise at ω = 300 (red squares), and the difference (blue triangles).
+that elastic depinning of the Wigner crystal is not occurring and that the systems are
+generally in the disordered or plastic flow regimes where the only available narrow band
+noise signals are of the moving smectic type. In principle, we think that the best place
+to look for a narrow band noise signature is in a sample with relatively weak pinning just
+above the depinning threshold. In our work, we focused on samples that were entirely
+within the elastic or plastic regimes; however, close to the transition between the elastic
+and plastic regimes, the plastic flow noise may be enhanced.
+7. Summary
+We have investigated the thermally induced depinning and noise fluctuations for driven
+Wigner crystal systems with quenched disorder. We identify an elastic regime in which
+the charges maintain the same neighbors at depinning as well as a plastic regime in which
+the system is broken up into moving and non-moving regions. In the plastic depinning
+regime, the velocity noise has a 1/f shape and there is a peak in the noise power above
+the depinning threshold at lower temperatures, while for large temperatures, the noise
+power peak is reduced and the spectrum becomes white, in agreement with experiments.
+For high drives at low temperatures in the plastic regime, the system forms a moving
+
+Noise and Thermal Depinning of Wigner Crystals
+20
+smectic with a narrow band noise signal. We find that this narrow band signal persists
+up to T/Tm = 0.75, where Tm is the temperature at which the charge lattice melts in the
+absence of quenched disorder. In the elastic regime, the system remains ordered up to
+temperatures approaching T/Tm = 1.0, although thermal effects reduce the depinning
+threshold. In the elastic regime, 1/f noise appears only in the creep regime where there
+are avalanches or jumps of motion, while in the sliding regime, pronounced narrow band
+noise appears that reaches its lowest power at the disorder-free melting temperature.
+Our results show that measurements of the velocity noise spectra and noise power can
+be used in connection with transport curves to distinguish different phases of driven
+Wigner crystals.
+Acknowledgments
+We gratefully acknowledge the support of the U.S. Department of Energy through
+the LANL/LDRD program for this work.
+This work was supported by the US
+Department of Energy through the Los Alamos National Laboratory.
+Los Alamos
+National Laboratory is operated by Triad National Security, LLC, for the National
+Nuclear Security Administration of the U. S. Department of Energy (Contract No.
+892333218NCA000001).
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+Information Flow Tracking Methods for Protecting
+Cyber-Physical Systems against Hardware Trojans
+- a Survey -
+Sofia Maragkou
+Institute of Computer Technology, TU Wien
+Vienna University of Technology
+Gusshausstr. 27–29 / 384, 1040 Wien, Austria
+sofia.maragkou@tuwien.ac.at
+Axel Jantsch
+Institute of Computer Technology, TU Wien
+Vienna University of Technology
+Gusshausstr. 27–29 / 384, 1040 Wien, Austria
+axel.jantsch@tuwien.ac.at
+Abstract—Cyber-physical systems (CPS) provide profitable
+surfaces for hardware attacks such as hardware Trojans. Hard-
+ware Trojans can implement stealthy attacks such as leaking
+critical information, taking control of devices or harm humans.
+In this article we review information flow tracking (IFT) methods
+for protecting CPS against hardware Trojans, and discuss their
+current limitations. IFT methods are a promising approach for
+the detection of hardware Trojans in complex systems because the
+detection mechanism does not necessarily rely on potential Trojan
+behavior. However, in order to maximize the benefits research
+should focus more on black-box design models and consider real-
+world attack scenarios.
+Index Terms—hardware Trojans, detection, hardware security,
+real hardware attacks, information flow tracking, cyber-physical
+production systems, cyber-physical systems
+I. INTRODUCTION
+Hardware security began facing desultory challenges much
+later than software [1]. In 1996 a timing attack was published
+[2] based on which sensitive information could be leaked from
+a cryptographic component. After this point, hardware security
+research became more systematic. From 2005 on [1, 3] the
+field of hardware security has gained ground in the academic
+and the industrial world because it breaks the chain of trust
+known so far.
+This chain of trust, from the hardware security perspective,
+begins at the integrated circuit (IC) supply chain, where
+security vulnerabilities are formed by the needs of the market
+for fast and cheap technology. The involvement of external en-
+tities in the design process and the internationally outsourced
+fabrication can create security breaches that can be even
+relevant for national security. Design houses, in order to stay
+competitive, purchase third-party intellectual property (3PIP)
+cores from vendors and outsource the fabrication process
+without always verifying the returned product with respect
+to hardware security breaches. The reason for that is that the
+verification of the purchased cores is an expensive process that
+requires resources and time. Those intellectual property (IP)
+cores or chips are integrated and distributed to the customers.
+Consequently, hardware security has to deal with attacks like
+IP piracy, reverse engineering, counterfeit chips and hardware
+Trojans.
+In a real world scenario, when an IP core is being purchased,
+the design house requests some design specification and the
+3PIP core vendor replies with the IP core and the specifications
+of the IP core. Throughout this information exchange, the only
+trusted part is the specification requested by the design house.
+The core in return, is considered untrusted and it is treated
+as black box. Information flow tracking (IFT) methods are
+a promising research direction for the detection of hardware
+Trojans because the verification can be based on the security
+specification of the application and not only on potentially
+malicious designs. Thus, the verification methods can be
+adapted based on the application. In addition, those methods
+can be flexible regarding new attacks, and can be expandable
+in case of the alteration of the security specifications.
+A. Known Real World Attacks
+The real world hardware attacks are much more complicated
+than the attacks developed by the research community, since
+real world attacks interact with different layers of the comput-
+ing system and communicate with external systems over long
+distance. Compared to software, real world hardware attacks
+are less frequent. The information that is publicly available
+about real world attacks is limited and specific details are
+rarely known to the public.
+The real world attack that received most attention is the
+2007 attack on a Syrian military radar [4, 5]. Even though
+the details were not officially revealed, all the indications
+suggest that the radar at a nuclear installation in Syria has
+been tampered. The attack took place in September 2007 and
+the nuclear installation was completely destroyed by Israeli
+bombing jets. The Israeli jets, took off from southern Israel,
+crossed the Mediterranean Sea and the Syrian-Turkish borders
+and returned four hours later. The state of the art radars did not
+detect the jets, which raised suspicions for malicious alteration
+of their functionality. Adee [4] suspects a kill-switch or a
+backdoor in the off-the-shelf microprocessor that could block
+a bombing radar by an apparently remote command (trigger)
+arXiv:2301.02620v1  [cs.CR]  27 Nov 2022
+
+without shutting down the whole system. The difference of
+a kill switch and a backdoor is that the kill switch will shut
+off a specific chip when triggered, but a backdoor requires
+an intruder to implement the same effect. The hypothesis of
+the kill switch is more likely and, in order to be implemented,
+requires the injection of extra logic. The HW and SW overhead
+for such an attack is very small which makes it hard to
+detect during testing, and the threat models discussed are
+the malicious designer and the malicious manufacturer. The
+microprocessor used remains unknown. This is not the only
+occasion where microprocessors including a kill switch have
+supposedly been used. According to anonymous sources from
+U.S defense department, it is known that a European chip
+maker is building microprocessors with a kill switch, and the
+French defense uses this technology for military applications.
+Undocumented microchips were found in the servers assem-
+bled by Supermicro [6, 7], that implemented a doorway to the
+network of the original system, which incorporated memory,
+networking capacity and processing power. The attack aimed
+at leaking sensitive information over a long term.
+Stuxnet attack provides an example of the real world attack
+capabilities in the industrial environment [8]. Stuxnet is a
+worm that was introduced in the Microsoft Windows operating
+systems and it was targeting specific industrial control systems
+of Siemens which were used in Iran to run centrifuges. Until
+the target was found the worm was updating itself. The worm
+was compromising the targeted system by exploiting ’zero-
+day’ vulnerabilities. After monitoring the operation, the worm
+was taking the control of the control system and it ran the
+centrifuges to the point of failure, returning false feedback to
+cover the failure until the damage was irreversible.
+Hybrid attacks are very common in real world scenar-
+ios. The hybrid attacks can include hardware, software and
+firmware parts. Such an attack can be malicious software that
+exploits vulnerabilities of the hardware, damaging physical
+resources such as Stuxnet [8].
+B. Cyber-Physical Production Systems
+Cyber-physical systems (CPS) are sophisticated systems that
+combine physical and cyber units. They are used in many
+different applications and they are the fundamental units of
+the internet of things (IoT) . Their functionality is based on
+the information exchange and the interaction with each other.
+According to the [9], the nature of the CPS makes them
+particularly sensitive to attacks, due to their heterogeneous
+nature, their reliance on data and their large scale.
+When those systems are integrated in the production envi-
+ronment then we refer to them as cyber-physical production
+systems (CPPS). Often, CPPS expose a profitable surface to
+adversaries for hardware Trojan introduction, because they are
+complex, sophisticated structures that manage sensitive infor-
+mation with extend communication among them, which facili-
+tates malicious functionality to stay hidden. Consequently, we
+consider securing the CPPS an emerging, critical issue.
+According to [10], the pyramid of the automation hierar-
+chy known until recently, is decentralized in the concept of
+Industry 4.0. The information processing has been distributed
+in many control units which exchanging information with the
+goal to optimize the production process. The control units have
+moved closer to the technical processes for efficiency, creating
+an interactive communication net among heterogeneous sys-
+tems. This creates the challenge to secure those components.
+Assume that a hardware Trojan is included in one of
+the control units. In Industry 4.0 machines use machine
+to machine (M2M) communication for sensitive information
+exchange. That means that the authentication keys are stored
+and processed in the machines. If the hardware Trojan leaks
+an authentication key to the adversary, she can take the control
+of the unit and possibly the control of the factory.
+In such a demanding environment the CPPS should stay
+consistent to the security requirements. Availability, integrity
+and confidentiality are only the basic guidelines of the prop-
+erties that should be taken into consideration. The proof that
+the units of those systems comply to those properties and to
+more detailed ones can be achieved with IFT methods as we
+discuss in the next sections.
+C. Scope
+The scope of this report is to survey how IFT methodologies
+can secure CPS against hardware Trojan attacks and how those
+methods need to be further developed in order to be applicable
+in real world scenarios.
+The remainder of this survey is organised as follows:
+Section II provides basic information about hardware Trojans.
+Section III refers to basic information for IFT methods and
+presents state of the art methodologies against information
+leakage. Finally, in section V we compare the IFT methods
+and we discuss future steps for research.
+II. HARDWARE TROJANS
+Hardware Trojans are circuits with hidden, unspecified,
+malicious functionality that can be included in any phase
+of the IC supply chain. In the environment of Industry 4.0,
+stealthy attacks like hardware Trojans can implement any kind
+of effect, including information leakage. In this report we are
+interested in this kind of malicious activity.
+Figure 1 shows a time bomb hardware Trojan from [11].
+This hardware Trojan is activated when the counter reaches
+the value 2k − 1. When the trigger is activated, the output
+value at ER* becomes different from the initial signal ER.
+The circuitry with the counter is the trigger and the circuitry
+that changes the value of the signal ER is the payload. This
+is a simplified example. More sophisticated mechanisms have
+been proposed from the research community like the Trojans
+mentioned above.
+According to the taxonomy of R. Karri, J. Rajendran, K.
+Rosenfeld, M. Tehranipoor [12], a hardware Trojan can be
+described by the insertion phase, the abstraction level, the
+activation mechanism (trigger), the effects (payload) and the
+location in the design.
+
+…
+0
+1
+K-1
+CLK
+ER
+ER*
+Fig. 1. Time bomb hardware Trojan based on [11]
+1) Insertion phase: The earlier a hardware Trojan is in-
+troduced in the design the broader the range of its impact is
+and the lower the cost of the attack is. For instance, assume
+that a third party vendor infects an IP core with a hardware
+Trojan. This IP core can be integrated in more than one
+design, increasing the number of infected systems. On the
+other hand, the scenario of the malicious manufacturer is
+design-specific. The attacker, in order to introduce a Trojan,
+should be aware of the design details which can be acquired by
+reverse engineering, a technique that needs special knowledge
+and is expensive in time and resources. Consequently, the
+phase of the hardware Trojan introduction, in combination
+with the value of the protected assets should be taken into
+consideration, during the development of countermeasures.
+2) Abstraction level: Depending on the abstraction level of
+the design, a hardware Trojan can be injected at system level,
+at the development environment, at register-transfer level as
+soft IP core, at gate level as firm IP core, at transistor level as
+hard IP core or at the physical level.
+3) Triggers: There are hardware Trojans exploiting don’t
+care conditions for their trigger mechanisms [13], or data
+patterns in specific memory addresses [14], or even dedicated
+input images [15]. Some attacks have even more sophisticated
+triggers which are activated during the design flow, leaving no
+trigger signal to the possible detection algorithm [16, 17].
+4) Payload: The most common attacks realized by hard-
+ware Trojans are sensitive information leakage and denial
+of service (DoS) attacks. Other attacks can be functional
+alteration, downgrade performance, data corruption, circuit
+aging, chip destruction, etc.
+5) Attack targets: The most common targets for hardware
+Trojan attacks are memory elements [18–21] and crypto-
+graphic components [13, 22, 23]. However, there are many
+proposals for attacking cores such as UARTs [24] or AXI4-
+bus interconnects [25], FPGA LUTs [16], CPUs [26–28], etc.
+6) Resources required: For the majority of the Trojans we
+study, the attacker needs knowledge of the design and access
+to it (e.g. bitstream [29], netlists [30] or access to the design
+tools [16, 17, 31]).
+III. INFORMATION FLOW TRACKING
+The basic idea behind IFT methods is that they track the
+influence of information of a system during computation. In
+order to achieve that, they assign tags (usually binary values)
+for each of the data element of the design and they update
+the value of the tag based on the applied method and the
+applied security properties. The verification is achieved by the
+observation of the value of the tags.
+IFT methods can be used with different verification tech-
+niques as it is described in the taxonomy in [32]. More
+specifically they can verify security properties through static
+methods like simulation, formal verification, emulation, and
+virtual prototyping or through dynamic methods like runtime
+monitoring techniques.
+There are many IFT methods used with different verification
+techniques and at different abstraction levels and tackling dif-
+ferent problems, since not all those methods address hardware
+Trojans.
+Here in this paper we chose to present different IFT ap-
+proaches and discuss their limitations and requirements. We
+present IFT static methods that tackle information leakage.
+Information leakage is the most common hardware Trojan
+effect and in the case of CPS it can cause economic loss or
+even set a human life in danger.
+As we discussed earlier, the runtime monitoring methods
+can be expensive in resources, and the recovery from those
+attacks can be costly too. Based on that, we chose to focus
+on the static IFT verification methods. Static IFT methods are
+applied in design-time, identifying the malicious behavior soon
+enough to minimize the recovery cost. Moreover, they do not
+add overhead in the original designs resources.
+IV. IFT METHODS AGAINST HARDWARE TROJANS
+Many methodologies are using theorem proving to verify
+the information flow in the designs [33–36]. In those methods
+the security properties are expressed as theorems and theorem
+proving tools such as Coq are used to verify them. In the
+proof-carrying hardware IP (PCHIP) framework [33] the IP
+vendors are required to deliver the HDL code of the design
+with formal proofs that the code is according to some security
+properties predefined among the two parties. For instance, such
+a property could describe that an instruction is allowed to
+access memory locations, which are defined in its op-code.
+With the provided security tags to the signals PCHIP can
+track the information flow in the design. The disadvantage
+of theorem proving methods is the manual conversion of
+the HDL core to the theorem proving language and proof
+checking environment (e.g. Coq and CoqIDE). Even though
+a conversion from HDL to Coq has been proposed [33, 34],
+theorem proving is far from an automated technique.
+The approach proposed in [37] addresses black box models.
+It is based on information flow security (IFS) verification
+which detects violations of security properties. An asset is
+modeled as stuck-at-0 and stuck-at-1 faults and, by leverag-
+ing the automatic test pattern generation (ATPG), faults are
+searched for. When a fault is detected, it means that there
+is an information flow from the asset to observation points.
+Finally the trigger mechanisms is extracted. This methodology
+is based on the fact that the trigger mechanism is injected in
+the original circuit.
+
+The tool Register Transfer Level Information Flow Tracking
+(RTLIFT)[38], can be applied directly to HDL code. Secu-
+rity tags (or labels) are assigned to every signal. Register
+transfer level information flow tracking (RTLIFT) uses IFT
+logic to securely propagate the tags throughout the design.
+The functionality of the additional IFT logic depends on the
+precision required. For instance, the output of an operation can
+be tainted when any of the inputs is tainted. If an untainted
+input influences the output to be untainted even though the
+other input is tainted, a false positive may occur. To avoid
+inaccuracies, the modules implementing the flow tracking
+logic take such cases into consideration. Based on the required
+trade off between complexity and precision, different precision
+levels can be achieved. Given the Verilog code, the control
+and the data flow precision flags (which define the required
+precision level), the tool generates a functionally equivalent
+Verilog code including IFT logic (IFT-Verilog code). The IFT-
+Verilog code is tested against the security properties requested
+for the design through simulation or formal verification. If the
+design passes this process, the extra logic is removed and the
+design is sent for fabrication. If it fails, the design has to be
+altered and to go through this process again.
+The methodology described in [39], gate-level information-
+flow tracking (GLIFT), can detect hardware Trojans injected
+by malicious third-party vendors, that alter the functionality
+of the original circuit or leak sensitive information. According
+to GLIFT, each data bit is assigned to a security label. This
+is implemented with additional tracking logic. It is up to the
+designers to define the security properties and use the GLIFT
+to verify the cores. For example, assume that the goal is
+to track the flow of a cryptographic key in order to ensure
+that it does not leak. The security labels of the keys will
+take the value ’confidential’ and the security property that
+verifies that there is no leakage should ensure that no bit with
+’confidential’ label ends up in an output or memory with the
+label ’untrusted’. Thus, this technique can identify violations
+of confidentiality and integrity and, hence, expose a hardware
+Trojan.
+Both methods discussed above [38, 39] face the problem of
+false positives results, which have to be resolved manually.
+The method proposed by Wang et al. [40], called HLIFT,
+detects hardware Trojans based on the trigger behavior at
+register transfer level (RTL) with the use of control and
+data flow graphs (CDFG). The method can identify hardware
+Trojans that leak information through specific outputs pins
+or side channel, without functional modification and through
+unspecified output pins. This approach is based on a feature
+matching methodology that captures specific Trojan features.
+The features are based on three kind of Trojan triggers: always-
+on, immediate-on, sequential-on. This methodology can be
+divided in the predefinition flow and the application flow.
+During the predefinition flow, statement CDFGs are build
+based on already known infected RTL designs. Statement
+CDFGs are abstract, high-level and compact RTL netlists. That
+way unnecessary information is removed which decreases the
+complexity. IFT is applied on the CDFGs and a list of Trojan
+IFT features is created. At the application flow, the statement-
+level CDFG is extracted from the unknown RTL design, and
+it is compared for matches with the list of the extracted Trojan
+features.
+The methodology proposed in [41] uses virtual prototyping
+(SystemC TLM 2.0) to identify information leakage or un-
+trusted access. At the behavioral level there is a lack of design
+details. Thus, the security properties applied are very strict.
+This can lead to false positives. This approach identifies the
+vulnerable paths and reports them to the user for inspection.
+Consequently, the inspection process is done manually, adding
+time overhead.
+The approach in [42], creates IFT models and optimizes
+them according to specific security properties. The security
+properties are compiled to security constraints and assertions,
+which are combined with the trimmed IFT model. Finally, the
+combination of the IFT model with the security constraints and
+assertions is verified through simulation, emulation or formal
+verification.
+In contrast to the methods presented above, the method in
+[43] does not use any of the mentioned verification methods.
+The HDL code is converted to an abstract syntax tree (AST)
+to identify, track and localize anomaly behavior. The AST is
+converted to directed data-flow graph (DFG). This process
+automatically recognizes interaction between IP cores. By
+identifying the sink and the source signals, the tool detects
+vulnerabilities and finally locates the threats.
+V. DISCUSSION AND CONCLUSIONS
+The development of hardware Trojans is flourishing as they
+attract interest from the academia and industry. As counter-
+measures, IFT methodologies are very promising, because
+they can be flexible, adaptable and expandable based on the
+application.
+However, the IFT verification methodologies proposed so
+far, cannot be applied in real world scenarios. To the best
+of our knowledge, usually the purchased IP cores are not in
+a white box form (usually the cores are purchased locked
+in order to avoid IP piracy), or the specifications of the
+cores provided are considered untrusted. Thus, the IP cores
+purchased are treated as black boxes. That means that the
+internals of the purchased modules are unknown and can
+be leveraged from other layers of the systems (firmware or
+software) for potential attacks.
+Thus, there is a need to explore more IFT methods for black
+box designs without the usage of known hardware Trojan
+behaviors. The reason we suggest, that the known Trojan
+behaviors should not be taken into consideration is because the
+attackers want their Trojans to stay hidden, pushing the limits
+of the current known Trojan behaviors, in order to make them
+more stealthy. A case in point is the development of trigger
+mechanisms. In recent years there is the tendency to include
+the trigger mechanisms in the design flow, so that the detection
+methods searching for trigger behaviors cannot detect them.
+On the other hand, methods that are based on security
+properties to identify unwanted or unspecified behavior in the
+
+TABLE I
+STATIC IFT METHODS - WB=WHITE BOX, BB=BLACK BOX,
+TP=THEOREM PROVING, MC=MODEL CHECKING, GL= GATE LEVEL,
+SL=SEQUENCIAL LOGIC
+Method
+Abstraction
+BB/
+Verification
+Limitations
+level
+WB
+method
+[33]
+RTL
+WB
+TP
+based on
+conservative
+rules [44]
+[35]
+GL
+WB
+TP
+manual proof
+or RTL
+construction
+[36]
+GL
+WB
+TP
+proof of genuine
+benchmark ,
+does not
+support SL
+[34]
+GL
+WB
+TP and MC
+high complexity,
+false positives
+[37]
+GL
+BB
+partial scan ATPG
+based on
+analysis
+trigger condition
+[38]
+RTL
+WB
+simulation or
+challenged in
+SAT solving
+complex
+structures
+[39]
+GL
+WB
+simulation
+creates
+false positives
+[40]
+RTL
+WB
+feature matching
+based on HT
+features
+[41]
+behavioral
+WB
+virtual prototypes
+lack of design
+details,
+manual inspection
+[42]
+RTL
+WB
+assertion based
+false positives
+or GL
+simulation
+emulation
+[43]
+RTL
+WB
+If-tracker
+false positives
+designs seem more flexible with respect to unknown attacks.
+However, the completeness of the security properties is an
+open problem. Another issue is the definition of the security
+properties by the engineers. Manual processes can result in
+vulnerabilities of the systems which can be leveraged by
+adversaries.
+Identifying a hardware Trojan in a real world example can
+be very challenging, especially since the trigger mechanism is
+not necessarily part of the original design. In some concepts
+a fault, a vulnerability, or a backdoor may be no different
+from a well covered Trojan. From the real world attacks we
+can conclude that the attack scenarios implemented are much
+more complete than the ones provided by academia. In the
+real world examples mentioned above we identify mechanisms
+that can communicate at great distance and can affect state of
+the art systems. The attacks were sophisticated enough with
+complicated mechanisms with more than negligible overhead.
+It will be useful for the research community to explore more
+complicated attacks, across the levels of a computing system
+in order to facilitate corresponding countermeasures.
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+

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+Deep Breath: A Machine Learning Browser
+Extension to Tackle Online Misinformation
+Marc Kydd
+School of Design and Informatics
+Division of Cyber Security
+Abertay University
+Dundee, United Kingdom
+m.kydd1800@abertay.ac.uk
+Lynsay A. Shepherd
+School of Design and Informatics
+Division of Cyber Security
+Abertay University
+Dundee, United Kingdom
+lynsay.shepherd@abertay.ac.uk
+Abstract—Over the past decade, the media landscape has seen
+a radical shift. As more of the public stay informed of current
+events via online sources, competition has grown as outlets vie
+for attention. This competition has prompted some online outlets
+to publish sensationalist and alarmist content to grab readers’
+attention. Such practices may threaten democracy by distorting
+the truth and misleading readers about the nature of events.
+This paper proposes a novel system for detecting, processing,
+and warning users about misleading content online to combat the
+threats posed by misinformation. By training a machine learning
+model on an existing dataset of 32,000 clickbait news article
+headlines, the model predicts how sensationalist a headline is and
+then interfaces with a web browser extension which constructs
+a unique content warning notification based on existing design
+principles and incorporates the models’ prediction. This research
+makes a novel contribution to machine learning and human-
+centred security with promising findings for future research. By
+warning users when they may be viewing misinformation, it is
+possible to prevent spontaneous reactions, helping users to take
+a deep breath and approach online media with a clear mind.
+Index Terms—Misinformation, Machine Learning, Human-
+Centred Security, Cyber Security, Web Technologies.
+I. INTRODUCTION AND BACKGROUND
+The Internet has rapidly become the dominant means for
+users to stay connected, even more so in the wake of the
+COVID-19 pandemic; however, this has led to several prob-
+lems [1]. From connecting with friends and family, comment-
+ing on recent events, or just being entertained, the Internet
+plays an integral role in how we inform and stay informed
+[2]. However, in a landscape that rewards attention rather than
+quality, some have turned to more dubious practices such as
+sensationalism, misinformation, and shocking imagery.
+A. The Incentivisation of Misinformation & Clickbait
+As more and more content is published every second, focus
+shifts from producing quality content to hijacking attention.
+When users’ interest is constantly pulled from one article,
+video, or image to the next, publishers need to find new
+ways to garner clicks. Consequently, this can easily lead
+to sensationalism, clickbait, and in the worst case, outright
+misinformation without the user being aware.
+The constant news cycle of the Information Age presents
+difficulties for news outlets to write up each emerging story.
+Instead, some outlets adopt the practice of aggregating content,
+either lightly editorialising a piece of existing content or
+directing users to another source to read the information there.
+The method of news aggregation is controversial, with some
+feeling that it is blatant theft of content, whilst others see
+it as the only viable solution to surviving in a fast-paced
+information economy [3] [4].
+B. The Impact of Sharing Misinformation and Clickbait
+Regardless of the ethics of clickbait and similar practices,
+publishing misinformation can be harmful once it reaches a
+broader audience. Many internet users utilise social network-
+ing, which compounds misinformation’s effects by allowing
+for rapid dissemination of falsehoods and half-truths. Often
+this creates a chain-reaction scenario where one user shares
+a story with another user, who in turn shares it with another,
+and so on. The impact of this phenomenon does not require a
+user to have a vast following on social media platforms. Cˆot´e
+and Darling [5] found that once a Twitter account surpassed
+approximately 1000 followers, it was much more likely to
+attract other users from a wide variety of backgrounds, in-
+cluding members of the general public, outreach groups, and
+even influential decision makers. This dramatically increases
+the potential audience that a piece of content can reach - and be
+shared by. Such a situation can result in users being presented
+with the same information multiple times, albeit being posted
+by different users.
+Fazio, Rand and Pennycook analysed the effects of
+widespread sharing of information [6]. The authors exam-
+ined how repeated exposure to information (both true and
+false) affects individuals’ perception of the data presented.
+Participants were asked to rate a series of statements on a
+percentage scale based on their believability. Statements were
+presented in two ways: some were shown only once, while
+others were repeatedly shown and were interspersed with one-
+off statements. Participants’ perception of truth across the
+accuracy scale rose without any influencing factor, other than
+being repeatedly shown the same information.
+When rated by participants, information regarded as the
+truth was rated as such. Similarly, information strongly con-
+sidered false did not influence participants sufficiently to
+arXiv:2301.03301v1  [cs.HC]  9 Jan 2023
+
+cross over into the “considered as true” category. This was
+noticeable in the 1%-30% bin, where repeated exposure did not
+generate a pronounced change in participant opinion. One of
+the paper’s key findings relates to information considered to be
+ambiguous. Information initially considered unclear gradually
+became regarded as accurate and authentic the more the user
+was exposed to the statement. This was particularly evident
+in the 41-70% bin which saw, on average, an increase of 6
+points compared to their original rating.
+Existing research [3]–[6] suggests that, in the current hyper-
+social age, users risk having their perceptions warped by
+misleading statements, hyperbole, and repeated exposure to
+information. This repetition does not have to stem purely from
+repeatedly seeing the same news article displayed but also
+from the conversation surrounding the topic. Increasingly frag-
+mented and confrontational viewpoints presented online com-
+plicates how users determine what is true or false. However,
+simply stating that a user is ‘misinformed’ or ‘wrong’ does
+not make them more inclined to reassess their understanding.
+Instead, a more thoughtful approach is required, giving users
+a starting point from which to come to their own conclusions.
+As Bronstein et al. [7] suggests, interventions catering towards
+the promotion of ‘open-minded and analytical thinking’ could
+be of benefit in curbing the impact of misinformation.
+C. Informing Users of Risk Online
+As
+misinformation
+has
+become
+more
+apparent
+and
+widespread in recent years, research into warning users when
+they are being misled has received increased interest, both in
+academia and industry. Many social media sites have taken
+steps to curb the impact of misinformation on their respective
+platforms, typically by presenting users with a warning label.
+This approach offers a simple means of quickly informing
+users that the content they are viewing may be misleading;
+proving popular with various social media platforms adopting
+similar practices including Twitter [8] and Meta’s Facebook
+[9].
+Ross et al. [10] analysed the effectiveness of warning labels
+adopted by major social platforms. Focusing primarily on the
+methods used to dissuade users from sharing misinformation,
+the authors tested two different label styles, one replicating
+those used by Meta and another informed by contemporary
+research. Participants (N = 151) were shown content consisting
+of six true and six false stories. Group one were presented
+all stories without any warning label. Group two were shown
+half the stories with a warning and the other half without. The
+participants in each group were then asked to determine which
+stories were manipulated or fake and which were unaltered.
+The research found that neither of the warning messages
+changed user behaviour. Users did not appear to be more
+suspicious of labelled content and were just as likely to interact
+with the content as that which was unlabelled.
+D. Designing Effective Warning Labels
+Ross et al. [10] indicate that providing additional context to
+the user can be beneficial in curbing the impact of misinfor-
+mation. Careful consideration in the design process on what
+information is communicated to the user and how could be
+instrumental in limiting misinformation’s reach.
+Shepherd and Renaud [11] conducted a literature review
+on designing effective security warning labels in browsers.
+Assessing existing work in this area, the authors found that
+time and resources are not adequately allocated to designing
+warning labels leading to user frustration and confusion. This
+sentiment appeared to be validated by Ross et al. [10]. The au-
+thors found that the effectiveness of current solutions differed.
+Some warnings wrongly assumed that users had background
+knowledge on a topic which decreased their effectiveness,
+whilst others were too vague in their language that users did
+not understand the ramifications of their choices.
+To combat the aforementioned issues, Shepherd and Renaud
+[11] concluded with the proposal of a set of design guidelines
+for browser warnings. The authors note that warnings designed
+for privacy and those designed for security differed, suggesting
+that different priorities must be considered depending on the
+intended use- case. The proposed guidelines recommend using
+simple and concise language to alert users to a potential
+issue and using neutral colours to avoid an undue emotional
+response. Furthermore, the guidelines propose linking to ad-
+ditional resources should the initial description not prove
+sufficient. Although the research in question is primarily
+targeted toward warning labels for security purposes, there
+is still value in applying these recommendations to tackling
+misinformation.
+E. Designing Effective Browser Warnings and Labels
+Much work has been done previously in the field of usable
+security concerning the design of warning labels. Early at-
+tempts at warning systems typically used contextual measures
+such as a small on-page popup informing the user of potential
+risk. However, work by Wu, Miller and Garfinkel [12] illus-
+trated these popups are often ignored, misunderstood, or users
+do not even recognise they are there.
+Further research in this area took on a different approach,
+utilising interstitial warnings. This approach required users to
+interact with the warning label before proceeding, the under-
+lying theory of this approach being that making the warning
+the central focus of the users’ attention would increase the
+likelihood of users reading and making an informed decision
+on the contents of the warning.
+Until recently, most research on effective warning design
+has been limited to web-security topics such as expired cer-
+tificates or phishing links. Kaiser et al. [13] examined warning
+label design to inform users of potential disinformation online.
+Evaluating several different warning design styles, ranging
+from information-dense with minimal colouring, to warnings
+with a strong visual impact but minimal detail, the authors
+assessed how users responded to the designs in a realistic
+environment.
+In a survey conducted with 238 participants, the authors
+found that participants responded most favourably to designs
+which featured a reference to the perceived risk (“This page
+
+contains misinformation”) and the recommended next step
+(“Consider finding alternative sources.”). The authors note
+that none of the designs evaluated showed any significant
+difference in how likely users were to consult a second
+or alternative source afterwards. The authors propose that
+changes in behaviour were more likely to stem from the
+friction caused by having to manually click through a warning
+rather than the content of the warning itself.
+Multiple factors play a role in shaping how users respond
+to warning labels, including the language used within them.
+Findings from a research study conducted by Mozilla [14] to
+understand how to design better warning labels highlighted
+that employing opinionated design was more important than
+providing objective information. This means it is more im-
+portant to convey the idea of a threat rather than what the
+threat is - prior research suggested overly technical warnings
+lead to confusion among users. Mozilla implemented this
+by simplifying the warning heading to feature abstract but
+understandable language. Additionally, for scenarios where
+users want to know details of the underlying issue, the warning
+provides an accessible description of the risks associated with
+the security fault.
+The issue of labelling misleading content is a challenging
+one. What counts as misinformation must be determined,
+and designing warning systems that promote critical thinking
+rather than knee-jerk reactions is still an ongoing area of re-
+search. Although the means of warning users adopted by major
+social platforms may have limited efficacy [10], Shepherd and
+Renaud [11] indicate that warning labels can cater to users’
+assumed knowledge and understanding without provoking
+undue alarm or concern. Similarly, Kaiser et al. [13] suggest
+that such research can be used to combat misinformation.
+F. Detecting Clickbait with Machine Learning
+The vast array of content posted online every second makes
+it impossible for human moderators to assess and review all
+dubious uploads. The use of an automated system is merited,
+one capable of rapidly and reliably analysing content for
+potentially misleading information which can integrate inter-
+vention measures. Machine learning’s inherent capabilities for
+finding and predicting patterns in information are well-suited
+to tackling misinformation. Furthermore, machine learning
+has seen renewed interest over the past decade as computing
+power and data storage have matured to enable real-world
+applications across a host of use cases.
+Chen, Conroy and Rubin [15] explored if clickbait, and by
+extension, misinformation, could be detected using machine
+learning methods. Conducting a holistic view of research in
+the field, the authors note four unique means of detecting
+clickbait. Initially focusing on the textual content of clickbait
+articles, the authors found clickbait often displays lexical and
+semantic features unique to its form. The authors cited work
+by Lex, Juffinger and Granitzer [16] which analysed clickbait
+based upon factors such as word length, word choice, and
+terminology, and found that a machine learning model could
+be trained to detect clickbait with 77% accuracy regardless of
+the topic discussed.
+Appealing to users’ innate curiosity by using unresolved
+pronouns or alluding to content within the article was also
+consistent with clickbait. As a subsequent paper by Rubin et al.
+[17] noted, automated fact-checking and verification systems
+could help detect language patterns in text and warn users
+that the content they are about to read may be misleading.
+The authors also note that such a tool could prove helpful
+for journalists too, alerting them when they may be conflating
+claims or accidentally misleading.
+Clickbait is not strictly limited to the text of the article
+in question; the authors also found that surrounding factors
+such as imagery and how the user interacts with the article
+play a key role. Regarding the former, the authors [17] cite
+Ecker et al. [18] who found that clickbait articles were likely
+to feature images which were incongruent with the headline.
+In such articles, an image can be used to grab the would-be
+readers’ attention with an impactful but unrelated image or
+shape opinion before the article was read.
+The authors [17] also noted that previous research had found
+clickbait outlets typically aimed to attract user attention before
+funnelling them towards sponsored content or advertising.
+Additionally, the time difference between “time spent reading
+the article” and “time spent sharing and commenting about
+the article” could also be a signifier of clickbait. In this
+aspect, clickbait articles tend to use alarmist or sensationalist
+headlines to provoke knee-jerk responses (whether that be
+commenting or sharing) before the user has actually read the
+content within.
+G. Problem Space
+The rapid rise of the information age has led some to
+adopt unethical practices to drive engagement. Whether these
+practices are deployed purposefully or not, they pose a serious
+risk to society. Although existing work has explored the use of
+warning labels, depending on how these are designed, these
+may be ineffective. Given the amount of content published
+every second, it would be impossible to label the accuracy
+of content manually. Instead, machine learning offers a com-
+pelling alternative. Misinformation poses a severe threat; there-
+fore combining advances in machine learning and warning
+design means an effective solution can be proposed to keep
+users safe.
+II. METHODOLOGY
+The proposed method consists of two-components: the
+machine learning model, for analysing and classifying content,
+and the web extension for communicating potential risk to the
+user. A simplified pipeline can been seen in Figure 1.
+Using TensorFlow [19] and adopting the same dataset as
+used by Chakraborty et al. [20], a Sequential Model was
+trained on 32,000 news article headlines, labelled as either
+‘clickbait’ or ‘non-clickbait’. The model, consisting of four
+layers (excluding the input layer), tokenises input text into a
+64-dimensional dense vector before running it through a global
+
+average pooling filter. Output is then fed through a layer of
+ReLU nodes, followed by a final layer of Sigmoid nodes to
+arrive at a real number between zero (indicating neutral) and
+one (indicating strongly misleading).
+The model was then connected to a browser extension via a
+Native Manifest, which allowed the browser to send portions
+of an article (e.g., the headline) to the model to analyse and
+generate a rating. The rating is then returned to the browser,
+after which a relevant warning can be presented to the user.
+Warnings were designed to be informative and actionable for
+the user, presenting clear detailing about the perceived risk
+and recommended next steps.
+A. Developing the machine learning model
+1) Dataset: The same dataset used by Chakraborty et al.
+[20] was adopted for this project and consisted of 32,000
+news article headlines labelled as either ‘clickbait’ or ‘non-
+clickbait’. The dataset offered a robust and relevant base upon
+which to build. In particular, clickbait and misinformation rely
+on emotional language to provoke a response, suggesting the
+dataset would help develop a model well suited to detecting
+such language.
+2) Model: In practice, the backend of this project cen-
+tres around binary classification: Is this piece of text click-
+bait/misinformation or not? As such, using a Sequential model
+was deemed the most suitable due to its singular input-output
+structure, a structure well suited to classification tasks such as
+this.
+Fig. 1. Example simplified data pipeline.
+3) Preprocessing Data : The dataset was split into 26,666
+training samples and 5,334 testing samples which equates to
+83% of the dataset for training and the remaining 17% for test-
+ing. Before training, the dataset had to be formatted such that
+the model could determine distinctions between clickbait and
+non-clickbait. This was achieved by converting, the headlines
+in the dataset into a vocabulary of word embeddings (Figure
+2). To ensure uniformity across the dataset, all sequences
+were padded to 24 tokens long which was considered a safe
+maximum length for a headline. Headlines longer than 24
+words long were automatically truncated to the maximum
+length.
+Fig. 2. Example word embedding.
+4) Building the Model: The model is visualised in Figure
+3, and consists of the input layer, two hidden layers, and
+the output layer. The first layer takes the input (a tokenised
+sentence) and transforms it into a 64-dimensional dense vector.
+The usage of dense vectors allows for the semantic meaning
+Fig. 3. Visualisation of the Model.
+of the sentence to be compressed, ensuring better general-
+isation. Despite their ability to derive underlying meanings
+and connections for a given sentence, the aforementioned
+dense vectors can result in overfitting if they become too
+detailed. To address the issue, the second layer consists of
+a Global Average Pool. This layer takes the 64-dimensional
+vector and determines the mean of each input channel (the 24-
+dimensional token sequences) which allows the model to learn
+approximations of embeddings rather than their exact values
+(Figure 4).
+Fig. 4. Visualisation of 1-Dimensional global average pooling.
+At this stage, the input data is now formatted and approx-
+imated to limit overfitting, and layers can be constructed,
+which will inform the output of the model. The first activation
+function of the model uses a rectified linear activation function
+(or ‘ReLU’). ReLU ensures that the next layer of the network
+receives a positive value as ReLU outputs 0 for input values
+equal to or less than 0 or the original value for those greater
+than 0. Finally, the data is passed through a Sigmoid activation
+layer, ensuring the resultant output falls between 0 and 1, i.e.,
+“Is this piece of text clickbait or not?”.
+The model measures its performance based on the ac-
+curacy of its predictions. The task involves classification;
+
+9 makeup tips you won't believe!
+Model
+ Clickbait: 1If Disney Princesses Were From Florida
+[10122 
+752
+6586 4 1 737
+0 
+0 0 (
+0 (
+0 (
+000000000000(
+01input:
+[(None, 24)]
+InputLayer
+output:
+[(None, 24)]
+input:
+(None,24)
+Embedding
+output:
+(None, 24, 64)
+input:
+(None, 24, 64)
+GlobalAveragePooling1D
+output:
+(None, 64)
+input:
+(None, 64)
+Dense
+output:
+(None, 2)
+input:
+(None, 2)
+Dense
+output:
+(None, 1)4
+6
+4
+2
+4Fig. 5. Accuracy graph during model training (Higher is better).
+thus, a binary cross-entropy loss function is used. The func-
+tion calculates how far the models’ predictions stray from
+the dataset’s labels. A gradient descent with a momentum
+optimiser (also known as Stochastic Gradient Descent or
+SGD) further minimises the loss function. The optimiser helps
+improve the model’s training rate by minimising loss across
+training iterations. By doing so, it is intended that the model
+predictions will gradually trend towards the expected output.
+Fig. 6. Loss graph during model training (Lower is better).
+The model was trained for 80 epochs with a batch size
+of 128. Although batch sizes larger than 32 can lead to
+underfitting, it was intended that the larger epoch size would
+gradually result in greater accuracy and convergence, which
+can be seen in Figures 5 and 6. After the model was compiled,
+trained, and evaluated, it was exported for use by the web
+extension.
+B. Web extension and warning messages
+1) Creating the web extension: A web browser extension
+was developed to ensure the model could be deployed in a real-
+world context. The extension means the model can analyse
+news articles as the user views them, providing a warning if
+misleading content is found.
+Native Manifests [21] were used to allow a web browser to
+interface with a native application, passing data back and forth
+between the two. These manifests allowed the TensorFlow
+model to perform predictions locally on the machine and then
+send the resultant prediction to the browser for further analysis
+and output.
+Within the browser, the analysis begins when the headline
+of the page the user has visited is fetched. Initially, white
+space and control characters are trimmed. The headline is
+processed, and a value is returned indicating how sensationalist
+the headline is.
+On the user’s device, the program transfers over to the native
+application, which handles parsing the headline into a format
+suitable for the model before computing a rating which is
+returned to the browser. Data from the browser is JSON-
+encapsulated and is sent via standard input (stdin), which
+the program reads from. Following this, the script begins
+importing libraries for loading the model and formatting the
+incoming data accordingly. The model is loaded, and the
+tokeniser is instantiated to convert the incoming headline. At
+this stage, the initial setup is complete and the model is ready
+for use.
+The core of the script features a loop which waits for a
+message from the browser to be received, at which point the
+decoding process can take place. Now that the headline of the
+article is available, the script tokenises it and provides padding
+to ensure compatibility with the model. From here, a standard
+model prediction call can be made, encoded, and returned to
+the browser for display to the user.
+2) Presenting warnings to the user: When the native ap-
+plication produces a result, it is returned to the browser. The
+browser extension then generates a message sent to the news
+article’s page, with the native applications result stored in a
+variable. This message is received by the content script, which
+is injected into each page by the extension.
+The content script dynamically generates a warning label
+for the user. This is done by waiting to receive a message
+(the result) from the background script. This value is then
+multiplied by ten (to accommodate any floating-point issues
+that may arise (i.e. converting 0.8 to 8)). To prevent repeatedly
+warning the user about innocuous content, only headlines that
+score above five out of ten have a warning generated. Scrolling
+is disabled whilst the warning is on-screen, ensuring the user
+has to acknowledge it.
+With regards to this project, the existing literature points
+towards interstitial warnings being the most likely to promote
+change in user behaviour. Additionally, even if most users do
+not appear to actually read the content of a warning label, they
+do show a preference for such information being present.
+Informed by the papers discussed in Section I-D, the web
+extension was designed to ensure strong visual clarity to
+effectively convey risks associated with a piece of misleading
+content.
+The warning adopts a paywall-style design, mimicking an
+approach that users will likely already be familiar with from
+other news sites. This helps to ensure that the warning is
+not overlooked, which can happen with contextual warnings.
+To further ensure that the warning is brought to the user’s
+attention, an overlay is used to darken the article and scrolling
+
+model accuracy
+tain
+0.85
+val
+0.80
+0.75
+accuracy
+0.70
+0.65
+0.60
+0.55
+0.50
+10
+70
+0
+20
+30
+40
+50
+60
+80
+epochmodel loss
+0.70
+tain
+val
+0.65
+0.60
+los5
+0.55
+0.50
+0.45
+0.40
+10
+20
+30
+40
+50
+0
+70
+60
+80
+epochis prevented while the warning is on screen. The warning
+design comes in 5 variants - ranging from most minor to most
+severe, depending on the article’s rating. Exemplar designs can
+be seen in Figure 7
+Fig. 7. Sample of warning designs.
+A vital problem with previous warning designs is the poor
+communication of risk, whereby warnings may be obscured
+by jargon [13]. The design of the warnings seeks to minimise
+existing issues by conveying as much relevant information
+as possible in an easy-to-read format. Prominent symbols
+represent increasing risk levels based on the article’s rating.
+Articles lower on the risk scale are given a more general
+’magnifying glass’ symbol, promoting the notion of thinking
+more critically about the article’s merit. If an article includes
+more severe levels of misinformation, increasingly prominent
+’alert’-oriented symbols are deployed, such as warning signs,
+stop signs and symbols of authority such as police figures.
+Additionally, an oscillating gradient is placed behind the
+warning. Depending on the severity of the warning, the colour
+used will shift from yellow to orange to red. The subtle
+movement of the gradient is intended to draw the user’s eye
+to the warning, with the unique colour of each warning also
+helping the user understand the associated level of risk.
+Ultimately, the extension seeks to change user behaviour
+and provide education on meaningful steps users can take to
+protect themselves from misinformation in the future. As such,
+each warning label features unique phrasing that informs the
+user of not just what the perceived risk is but also advice on
+actionable next steps.
+Two buttons are presented to the user at the bottom of the
+warning, allowing them to dismiss the warning and continue,
+or navigate away from the page. To indicate the intended
+behaviour, the option to navigate away is displayed in promi-
+nent green with a ’Recommended’ label included in brackets.
+Conversely, the option to dismiss the warning is presented in
+red and is slightly faded out to deliberately be obscured against
+the background until the user hovers over the button.
+III. RESULTS AND DISCUSSION
+Fig. 8. Accuracy comparison between training and evaluation.
+During training, the model achieved an accuracy of approx-
+imately 85% and a loss of 0.39, and when pitted against
+the evaluation dataset, the model achieved an accuracy of
+approximately 45% with a loss of 1.15 (Figure 8, Figure 9).
+This decline in accuracy likely stems from inconsistencies in
+the existing evaluation dataset, e.g., improperly formatted data,
+such as some of the labels assigned to the headlines do not
+appear to be correct. This could be resolved via an additional
+data cleaning. Another point of note is that the evaluation
+dataset used only binary labels (Is this headline ‘clickbait’ or
+‘news’?), which may have also contributed to the discrepancy
+in accuracy as the model was producing a result between 0
+and 1 instead of a pure binary output.
+
+ThisContent PutsYouatRisk
+Misleadingarticlescanwarpyourperceptionofevents
+Think twice before continuing
+Deep Breathhas detectedthatthis article is misleading
+and should not be viewed.
+Backto safety (Recommended)
+Misleading Content Alert
+Sensationalistcontentcantrickyouinto consumingfalse
+information.Consultmultiplesources
+Websites can use sensationalist languageto mislead and
+deceive.Deep Breaths detection algorithmbelieves that
+this article is misleading.
+Dismiss and continue
+Back to safety (Recommended)
+SensationalistContentWarning
+Thispagemaycontainsensationalistormisleading
+content.Considerconsulting additionalsources.
+Websites can use sensationalist languagetograb your
+attention.DeepBreathsdetectionalgorithmbelievesthat
+this article may be misleading.
+Dismiss and continue
+Backto safety (Recommended)100
+90
+85%
+80
+70
+Accuracy
+60
+50
+45%
+40
+30
+20
+10
+0
+Training
+Evaluation
+(Higher is better.)Fig. 9. Loss comparison between training and evaluation.
+With regards to machine learning, the project confirms the
+findings of Lex, Juffinger and Granitzer (2010) that clickbait
+and misinformation can be detected based upon lexical seman-
+tics, namely word choice, word length, and word commonality,
+i.e., Word x appears frequently alongside word y.
+IV. CONCLUSION AND FUTURE WORK
+The model demonstrated in this paper has shown a re-
+liable degree of performance, however, it could be refined
+further to derive even better results. The models’ accuracy
+and loss were still increasing and decreasing, respectively,
+suggesting better performance could be obtained before the
+curves flattened out. Furthermore, the capability of the model
+could be extended further. The model has been trained only on
+clickbait-styled headlines, which was effective. However, more
+robust results may be achieved by training on the contents of
+clickbait articles which would allow the model to develop a
+deeper understanding of the article and make a more nuanced
+prediction. The model used in this paper is a Sequential model
+designed to take a single output and produce a single result.
+Although this is effective at classifying a single headline as
+used in this paper, greater functionality could be achieved
+by allowing multiple inputs and outputs. This could include
+assessing the article’s headline but also a selection of sentences
+from the article. The rating assigned to content is dynamic;
+however, the underlying warning remains static. By expanding
+the models’ capabilities, it may also be possible to provide
+personalised warnings relevant to the content. In practice, this
+could mean warning the user about specific aspects of the
+article, such as sensationalist authors, misleading sentences,
+and miscaptioned images. Although every effort was taken to
+ensure the model made balanced and accurate predictions, no
+system is infallible. Conducting user testing and introducing
+the option for users to report when the model makes a
+perceived miscalculation could help adjust for missteps.
+The work presented in this paper makes promising ad-
+vances toward tackling the issue of misinformation online
+by combining machine learning, human-computer interaction
+research, and web technologies. Findings validate and build
+upon prior research, and incorporating machine learning with
+usable security is still a relatively under-explored area of study.
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+

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+arXiv:2301.04861v1  [cs.IT]  12 Jan 2023
+Grant-Free Random Access of IoT devices in
+Massive MIMO with Partial CSI
+Gilles Callebaut1, Franc¸ois Rottenberg1, Liesbet Van der Perre1, and Erik G. Larsson2
+1Department of Electrical Engineering (ESAT-DRAMCO), KU Leuven, 9000 Ghent, Belgium
+2Department of Electrical Engineering (ISY), Link¨oping University, Link¨oping, Sweden
+Abstract—The number of wireless devices is drastically in-
+creasing, resulting in many devices contending for radio re-
+sources. In this work, we present an algorithm to detect ac-
+tive devices for unsourced random access, i.e., the devices are
+uncoordinated. The devices use a unique, but non-orthogonal
+preamble, known to the network, prior to sending the payload
+data. They do not employ any carrier sensing technique and
+blindly transmit the preamble and data. To detect the active
+users, we exploit partial channel state information (CSI), which
+could have been obtained through a previous channel estimate.
+For static devices, e.g., Internet of Things nodes, it is shown that
+CSI is less time-variant than assumed in many theoretical works.
+The presented iterative algorithm uses a maximum likelihood
+approach to estimate both the activity and a potential phase
+offset of each known device. The convergence of the proposed
+algorithm is evaluated. The performance in terms of probability
+of miss detection and false alarm is assessed for different qualities
+of partial CSI and different signal-to-noise ratio.
+Index Terms—activity detection, grant-free, massive MIMO,
+maximum likelihood, random access.
+I. INTRODUCTION
+Massive machine-typed communication (mMTC) is envi-
+sioned to enable low-power connectivity to a very large
+number of devices and open up new applications. While it
+has been put forward as a key building block in 5G and
+beyond, it has so far received less attention than enhanced
+Mobile Broadband (eMBB) and Ultra-Reliable Low Latency
+Communications (URLLC). The challenge in mMTC, and
+in general Internet of Things (IoT) systems, resides in the
+low-power operation, sporadic nature of the traffic and a
+large amount of uncoordinated devices. As these devices are
+often battery-powered, they are constrained in the signalling
+overhead they can handle [2]. Furthermore, they are often
+deployed in remote areas, where network coverage is low or
+non-existent [2, 3]. To address this, novel protocols need to
+be designed tailored to the low-power, sporadic and massive
+nature of mMTC traffic. One promising direction, and the
+focus of this work, is the use of massive MIMO, where a
+large number of antennas is present at the base station.
+Two multiple access approaches can be taken, grant-based
+and grant-free random access. The former requires the devices
+to obtain a grant from the network, where after it can use a
+The research reported herein was partly funded by the European Union’s
+Horizon 2020 research and innovation programme under grant agreement No
+101013425.
+This paper is presented at IEEE WCNC 2023. G. Callebaut, F. Rottenberg,
+L. Van der Perre, and E. G. Larsson, “Grant-Free random access of IoT devices
+in massive MIMO with partial CSI,” in 2023 IEEE Wireless Communications
+and Networking Conference (WCNC) (IEEE WCNC 2023), Glasgow, United
+Kingdom (Great Britain), Mar. 2023
+collision-free radio resource to transmit its data. The disadvan-
+tage of this approach for IoT and mMTC is that devices need
+to compete for grants, requiring, e.g., dedicated preambles or a
+lot of signalling for collision resolution. Due to the number of
+competing devices such schemes are not practical and scalable.
+Therefore, grant-free approaches are advocated over grant-
+based solutions [4–6]. In grant-free random access, devices
+do not contend for a grant, but just access the network when
+required. Often these devices are unaware of the other devices
+in the system and operate in an uncoordinated fashion, which
+is called unsourced grant-free random access [6]. Different
+approaches have been studied to detect the active devices in
+massive MIMO systems. Detecting active devices is facilitated
+when each device uses a unique and orthogonal preamble
+sequence. This entails that each preamble should have the
+same length as the number of devices to have no collisions,
+which is unpractical. Therefore a number of studies have
+considered a pool of orthogonal pilots [7] or the use of non-
+orthogonal pilots. Both of these strategies have been elaborated
+in [4], where they formulate the device-activity detection
+problem as a compressed sensing problem, where the sparsity
+of the active devices at each time slot is exploited. However,
+compressed sensing requires the preamble length to be larger
+than the active devices, increasing the energy expenditure of
+data transfer. To combat this, in [6, 8] a covariance-based
+method is suggested using two estimators for device activity
+recovery, i.e., maximum-likelihood (ML) and non-negative
+least squares (NNLS). This work is generalized by Ganesan,
+Bjornson, and Larsson [9, 10] to the cell-free case where mul-
+tiple access points (APs) are geographically distributed. They
+considered different large-scale fading coefficients per APs.
+They employed a simplification of their proposed algorithm
+by including only the AP with the strongest contribution. They
+concluded that co-located massive MIMO is highly sensitive to
+low signal-to-noise ratios (SNRs), while cell-free deployment
+is better suited against shadowing fading effects.
+Contributions. Inspired by these works, we propose a
+new algorithm exploiting the static nature of IoT devices. As
+demonstrated in [11] and elaborated in Secion II, the channel
+state information (CSI) can remain almost invariant over large
+period of times (hours) for scenarios with low mobility. This
+was not yet leveraged by practical algorithms in the literature
+to the best of our knowledge. Given that the base station
+knows a part of the CSI of each device the performance of
+the device activity scheme can be improved. To do so, we
+formulate the maximum-likelihood activity detection problem
+using partial CSI. Furthermore, a phase offset can be estimated
+which occurs due to carrier frequency offset (CFO) (when
+
+the CFO is static over the preamble duration). We validate
+the convergence of the iterative algorithm, study the impact
+of different initialization vectors for the device activity, the
+impact of the quality of the partial CSI and the SNR on the
+performance of the algorithm.
+The
+used
+mathematical
+notations
+are
+described
+in
+https://github.com/wavecore-research/math-notations.
+II. MOTIVATION – RECURRENCE IN CSI
+IoT technologies are often put in the field with a deploy-
+and-forget strategy, where the devices remain immobile after-
+wards. As such, we can expect that the channel conditions
+are less time-variant than typically assumed in literature [12,
+13]. An experimental campaign to investigate the long-term
+behavior of the channel is presented in [11]. The long-
+term behavior is measured by taking the channel correlation
+δi,j = |¯hH
+i ·¯hj|/∥¯hi∥∥¯hj∥ of the first channel estimate and the
+other channel N measurements, i.e., δ1,j for j ∈ {1, . . ., N}.
+The observed channel correlations are depicted in Fig. 1.
+It illustrates that most of the time the correlation coefficient
+is close to 1, indicating that the channel is highly correlated
+with the first estimate and thus can be considered static. It also
+shows that, while in some occasions the correlation drops,
+the channel quickly becomes again highly correlated with
+the first channel instance. More than 90% of the measured
+channels1 have a correlation coefficient higher than 0.9. This
+demonstrates the potential of re-using channel estimates in IoT
+contexts.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.6
+0.8
+1
+Sample index
+Channel Correlation δ1,j
+Node 1
+Node 2
+Figure 1: Channel correlation over a full day (9h24-17h48) with over 10 000
+channel instances, with an average correlation of 0.935/0.968 and a standard
+deviation of 0.06/0.03 for Node 1/Node 2.
+III. SYSTEM MODEL
+There is a total of K single-antenna devices and a total of M
+co-located base station (BS) antennas. The set of active users
+trying to access the network is denoted by Ka, with |Ka| ≤ K.
+To access the network, each active device k ∈ Ka sends a
+unique, non-orthogonal preamble of length T , known to the
+network. The pilot symbol of the preamble sent by device k at
+pilot symbol t is denoted by sk,t. The channel vector between
+the receive antenna m and user k is denoted by hk,m ∈ C, and
+1More specifically, 91% and 95% for Node 1 and Node 2, respectively.
+2We use here the term “re-using” to indicate that we no longer operate in
+a block fading model with independent channel realizations. Typically, these
+blocks are considered in the order of 50 ms.
+is considered fixed over the preamble duration. The received
+symbol at the m-th BS antenna at time t is
+yt,m =
+K−1
+�
+k=0
+hk,msk,tγk + wt,m,
+(1)
+where γk is an unknown complex scalar and wt,m ∈ C is
+independently and identically distributed (i.i.d.) zero mean
+circularly symmetric complex Gaussian (ZMCSCG) noise with
+variance σ2. The unknown complex scalar γk can be developed
+as
+γk = √ρkakejφk,
+(2)
+where ρk ∈ R+ is the transmit power of device k, ak ∈ {0, 1}
+is the device activity and φk models a potential phase offset.
+This offset φk can account for a CFO, where the phase is
+considered constant over the preamble duration. This occurs
+when no frequency drift has occurred during the preamble
+duration, which is a feasible assumption as the CFO is
+typically low, yielding negligible phase rotations during the
+preamble interval. By assuming that all M antennas are
+perfectly synchronized, this offset is only dependent on the
+device. In case the device is inactive, γk will be zero. We will
+introduce the term activity indicator to denote γk.
+Let us consider that the BS knows a part of the CSI, i.e.,
+gk,m in
+hk,m = gk,m + λkǫk,m,
+(3)
+where ǫk,m are i.i.d. ZMCSCG variables with unit variance,
+and λk ∈ R+ models the unknown part of the CSI. The large-
+scale fading coefficient of user k is βk = E(∥hk∥2 /M) =
+∥gk∥2 /M +λ2
+k. The factor λk models the quality of the known
+CSI. Hence, it quantifies the correlation of the actual channel
+hk,m to the known partial CSI gk,m. In the extreme case with
+λk = 0, the CSI is perfectly known and there is no uncertainty
+left, as was studied in [11]. This could be the case in a
+fully static environment and if the CSI estimates are noiseless.
+However, for a realistic IoT scenario, even for static devices,
+CSI is not perfect due to i) environment dynamics and ii) noisy
+estimates. The parameter λk then quantifies this imperfection.
+It is here assumed to be known3. Another extreme case,
+as considered in [8, 10], is obtained when gk,m = 0, ∀m,
+implying that only the large scale fading coefficient of user k
+is known, i.e., λk = √βk.
+IV. DEVICE ACTIVITY DETECTION
+This section describes the proposed activity detection al-
+gorithm. First, the log-likelihood of the received preamble is
+derived. Then, given its non-convex expression, an iterative
+approach is proposed to estimate the parameters γk ∀k. Finally,
+activity detection is performed.
+A. Log-Likelihood of the Received Symbols
+Combining (1) and (3), the symbol, received at BS antenna
+m and for pilot symbol t, is given by
+yt,m =
+K−1
+�
+k=0
+(gk,m + ǫk,mλk)st,kγk + wt,m.
+3It could be set to a certain value depending on the user activity profile
+and/or tracked for each user over time.
+
+Stacking the observations at antenna m gives
+ym =
+K−1
+�
+k=0
+gk,mskγk +
+K−1
+�
+k=0
+ǫk,mλkskγk + wm,
+where
+ym =
+
+
+
+y0,m
+...
+yT −1,m
+
+
+ , sk =
+
+
+
+s0,k
+...
+sT −1,k
+
+
+ , wm =
+
+
+
+wm,0
+...
+wT −1,m
+
+
+ .
+For a given value of γk, ym|γk has a circularly symmetric
+Gaussian distribution with mean �K−1
+k=0 gk,mskγk. After defin-
+ing the vector θm = �K−1
+k=0 ǫk,mλkskγk+wm, the covariance
+matrix is
+C = E
+�
+θmθH
+m
+�
+=
+K−1
+�
+k=0
+λ2
+k|γk|2sksH
+k + σ2IT ,
+(4)
+where we used the fact that ǫk,m were assumed to be i.i.d. and
+the additive noise is white. Note that this covariance matrix
+does not depend on the antenna index m and is thus valid for
+all ym. Defining the vector γ = (γ0, ..., γK−1)T , and Θm =
+ym − �K−1
+k=0 gk,mskγk the log-likelihood of the observation
+vector ym is
+log p(ym|γ) = − ln (|C|) − T ln(π) − ΘH
+mC−1Θm.
+Given the conditional independence of ǫk,m and wt,m over the
+antennas, the different ym are independent as well. Hence, the
+log-likelihood of the aggregated observations at all antennas
+y = (yT
+0 , ..., yT
+M−1)T becomes
+log p(y|γ) = −M ln (|C|) − MT ln(π) −
+M−1
+�
+m=0
+ΘH
+mC−1Θm.
+(5)
+B. Iterative Algorithm for Maximizing Likelihood
+The maximum likelihood estimator of γ is obtained by
+maximizing
+ˆγML = arg max
+γ
+log p(y|γ).
+This problem is not trivial to solve given the nonlinear and
+non-convex dependence of the log-likelihood, more specifi-
+cally the covariance matrix C, in γ. An idea to maximize
+the likelihood is to use an iterative approach, similarly as [8,
+10]: at each iteration, all γk are kept fixed but one, which is
+optimized and updated. This way, they get updated one by
+one until convergence is attained, i.e., a maximum number of
+iterations or a certain tolerance is reached. A block diagram
+of the algorithm is given in Figure 2 and the pseudocode is
+summarized in Algorithm 1.
+Let us consider that the complex-valued γk′ needs to be
+updated. Using the definition introduced in (2), we can rewrite
+γk′ with a phase-amplitude decomposition: γk′ = rk′eφk′,
+with rk′ = |γk′| = √ρk′ak′. The optimization with respect
+to γk′ is done in the following in several steps: i) optimizing
+the phase φk′ for a fixed value of rk′, ii) re-inserting this
+expression in the objective function to remove the dependence
+in φk′ and iii) optimizing the amplitude rk′.
+1) Phase optimization: To highlight the dependence of the
+objective function in γk′ for constant values of other γk, k ̸=
+k′, let us define the vector
+yk′,m = ym −
+K−1
+�
+k=0,k̸=k′
+gk,mskγk,
+(8)
+which can be seen as a cancellation of device interference to
+isolate the contribution from device k′. Hence, the objective
+function to maximize can be written as in (7)4.
+where we explicitly express the dependence in (rk′, φk′)
+while the other (rk, φk), k ̸= k′ do not appear since they are
+considered constant. Note that the matrix C, defined in (4),
+does not depend on φk′ but only rk′. In the extreme case of no
+prior CSI, i.e., gk′,m = 0 ∀m, the dependence of f(rk′, φk′)
+in φk′ disappears and there is an underdetermination and no
+estimate of the phase offset can be obtained. In other cases,
+we can find that, after some manipulations,
+ˆφk′ = ∠sH
+k′C−1
+M−1
+�
+m=0
+g∗
+k′,myk′,m.
+(9)
+This result has an intuitive understanding: the optimal phase
+φ′
+k tends to align the partial CSI with the observations due to
+device k′.
+2) Removing the phase dependence: Inserting this optimal
+value in the objective function f(rk′, φk′) makes the depen-
+dence in φk′ vanish and gives (6).
+3) Amplitude optimization: To alleviate the dependence on
+rk′ in (6), let us define
+C−k′ = C − λ2
+k′|γk′|2sk′sH
+k′
+(10)
+=
+�
+k\k′
+λ2
+k|γk|2sksH
+k + σ2IT ,
+which does not depend on rk′ and is full rank, thus invertible.
+Applying the Sherman-Morrison formulas [14] to C−1 and
+C−1sk′ gives
+C−1 = C−1
+−k′ − C−1
+−k′sk′sH
+k′C−1
+−k′r2
+k′λ2
+k′
+1 + sH
+k′C−1
+−k′sk′r2
+k′λ2
+k′
+(11)
+We insert these expressions in (6) and we omit terms that
+do not depend on rk′, which will vanish after taking the
+derivative. This gives
+˜f(rk′) = −M ln (|C|) + αr2
+k′ + βrk′
+1 + δr2
+k′
+,
+(12)
+where we defined the constants (independent of rk′) α, β and
+δ, as
+α =
+M−1
+�
+m=0
+|yH
+k′,mC−1
+−k′sk′|2λ2
+k′ − sH
+k′C−1
+−k′sk′
+M−1
+�
+m=0
+|gk′,m|2
+β = 2
+�����
+M−1
+�
+m=0
+yH
+k′,mC−1
+−k′sk′gk′,m
+�����
+δ = sH
+k′C−1
+−k′sk′λ2
+k′.
+(13)
+4For clarity, we omit in the following the constant term MT ln(π) which
+does not affect optimization as it does not depend on γ and vanishes after
+differentiation.
+
+f(rk′) = −M ln (|C|) −
+M−1
+�
+m=0
+yH
+k′,mC−1yk′,m − r2
+k′sH
+k′C−1sk′
+M−1
+�
+m=0
+|gk′,m|2 + 2
+�����
+M−1
+�
+m=0
+yH
+k′,mC−1sk′gk′,m
+����� rk′
+(6)
+f(rk′, φk′) = −M ln (|C|) −
+M−1
+�
+m=0
+�
+yk′,m − gk′,msk′rk′eφk′�H C−1 �
+yk′,m − gk′,msk′rk′eφk′�
+(7)
+Inputs
+σ2, λk, gm,k,
+ym ∀m, k
+Initialization
+k′ ← 0, ˆγ ← ˆγinit
+Amplitude Optimization
+(14), (16) or (17)
+ˆrk′ ← arg max ˜f(rk′)
+Phase Optimization (9)
+ˆφk′ ← ∠sH
+k′C−1 �M−1
+m=0 g∗
+k′,myk′,m
+Converged?
+Iterative maximum likelihood estimator
+Tolerance
+Max iteration
+Activity Detection
+ˆγk ≤ γth,k ∀k
+No
+Yes
+k′ ← k′ + 1 mod K
+Updated ˆγk′ ← ˆrk′eφk′
+ˆγ
+Figure 2: Block diagram of the iterative maximum likelihood estimator and activity detection.
+One can note that ˜f(rk′) in (12) can now be differentiated
+with respect to rk′, using the differential rule ∂(log |A|) =
+tr[A−1∂A]. Setting the derivative to zero gives, noting that
+the denominator is always strictly positive,
+0 = −r3
+k′2Mδ2 − r2
+k′βδ + rk′(−2Mδ + 2α) + β,
+(14)
+which is a polynomial of degree 3 in rk′. There are closed-
+form solutions for the roots of such polynomials. Following
+Descarte’s rule of signs, (14) has only one real and positive
+root, as required, in case the terms are non-zero. Below we
+discuss the special cases when the terms are not non-zero, i.e.,
+when there is no or complete CSI knowledge.
+The algorithm is summarized in the pseudocode Algo-
+rithm 1. At each iteration, the constants α, β and δ can
+be easily re-evaluated based on (13). They require the matrix
+inversion C−1
+−k′. To avoid re-computing a full inverse at each
+iteration, one can rely on the Sherman-Morrison formula and
+on the current knowledge of C−1, which is updated at the end
+of each iteration by inserting the obtained value of rk′ in (11).
+Using (10), we find
+C−1
+−k′ = C−1 + λ2
+k′|γk′|2C−1sk′sH
+k′C−1
+1 − λ2
+k′|γk′|2sH
+k′C−1sk′ .
+(15)
+Moreover, computations can be optimized as several quantities
+appear multiple times and can be computed only once. The
+complexity of the proposed algorithm is O(IMT 2), where I
+is the number of iterations.5
+We now investigate two particular cases, to gain further
+insights.
+Algorithm 1 Iterative maximum likelihood device activity
+detector
+Require: σ2, λk, ym, gk,m, ˆγinit ∀k, m
+k′ ← 0
+ˆγ ← ˆγinit
+C−1 ←
+��K−1
+k=0 λ2
+k|ˆγk|2sksH
+k + σ2IT
+�−1
+while Not converged do
+Compute yk′,m, C−1
+−k′, α, β and δ based on (8), (15) and
+(13)
+ˆrk′ ← arg max ˜f(rk′)
+⊲ Update amplitude based on (14),
+(16) or (17)
+ˆφk′ ← ∠sH
+k′C−1 �M−1
+m=0 g∗
+k′,myk′,m
+⊲ Update phase
+ˆγk′ ← ˆrk′e ˆφk′
+C−1 ← C−1
+−k′ −
+C−1
+−k′ sk′ sH
+k′ C−1
+−k′ r2
+k′ λ2
+k′
+1+sH
+k′ C−1
+−k′ sk′ r2
+k′ λ2
+k′
+k′ ← k′ + 1 mod K
+end while
+a) Particular case: device with no CSI: Now consider
+that, for a given k′, gk′,m = 0 ∀m. This could be because
+this device is new or moving a lot, such that its CSI is
+outdated. Only, its parameter λk′ is known, which is equal
+5Note that I will in practice depend on K as we will iterate N times over
+all users K, but this is not a requirement.
+
+to the large-scale fading coefficient √βk′. At iteration of user
+k′, evaluating (13) for gk′,m = 0 ∀m implies that β = 0.
+Hence, (14) simplifies to
+0 = 2rk′(−r2
+k′Mδ2 − Mδ + α),
+which has a trivial solution in rk′ = 0. One of the other roots
+is always negative. Keeping only the positive one, we find the
+amplitude update
+ˆrk′ =
+�
+α − Mδ
+Mδ2
+.
+(16)
+If this root is imaginary, we set ˆrk′ to zero. As discussed
+before introducing the phase update equation (9), in the case
+of no prior CSI, the phase ambiguity cannot be resolved.
+This particular case gives an update relatively similar to the
+maximum likelihood estimator derived in [8, (23)], where
+their ML expression estimates the error, while ours estimates
+directly the coefficient ˆrk′, given the same estimate of γk′ at
+each iteration.
+b) Particular case: device with complete prior CSI: Now,
+consider that, for a given k′, λk′ = 0, so that the CSI is
+perfectly known. Only the phase shift and the transmit power
+are unknown. At iteration of user k′, evaluating (13) for λk′ =
+0 implies that δ = 0. Hence, (14) simplifies to a linear equation
+0 = rk′2α + β, which gives the following amplitude update
+ˆrk′ = −β
+2α =
+|sH
+k′C−1 �M−1
+m=0 g∗
+k′,myk′,m|
+sH
+k′C−1sk′ �
+m′ |gk′,m′|2
+,
+(17)
+while the phase is updated according to (9).
+4) Initialization: To start the iterative algorithm, we con-
+sider different choices to initialize ˆγinit. A simple choice is
+to initialize to zero, i.e., ˆγ0
+init = 0. Another choice is to
+initialize solely based on the available prior CSI, considering
+that λk ≈ 0, ∀k. The estimator is similar to [11], except that,
+here, prior CSI is used instead of complete CSI.
+If λk ≈ 0, ∀k, the covariance matrix C, defined in (4),
+simplifies to C = σ−2IT , which is independent of γk. Hence,
+many terms of the log-likelihood in (5) become independent
+of γ. Maximizing (5) becomes equivalent to the following
+minimization
+max
+γ
+log p(y|γ) = min
+γ
+M−1
+�
+m=0
+�����ym −
+K−1
+�
+k=0
+gk,mskγk
+�����
+2
+= min
+γ ∥y − Γγ∥2 ,
+where we defined the vector and matrix notations
+y =
+�y0 . . . yM−1
+�T , Γ =
+�Γ0 . . . ΓM−1
+�T ,
+Γm =
+�s0 . . . sK−1
+�
+diag(g0,m . . . gK−1,m).
+This minimization problem is a quadratic function of γ, which
+is a least squares problem. The estimate has the following
+closed-form expression
+ˆγZF
+init =
+�
+ΓHΓ
+�−1
+ΓHy.
+(18)
+This last estimator can be seen as a zero-forcing (ZF) es-
+timator, which requires a matrix inversion. To avoid ill-
+conditioning, a first necessary condition is that K ≤ MT .
+This condition is not sufficient as the channel and preamble
+of two devices could be correlated, especially when K is on
+the order of MT . Moreover, if no prior information is available
+for a given user k′, i.e., gk′,m = 0, ∀m, the inverse will also
+be ill-conditioned. This implies that the k′-th column of Γ
+becomes null and thus Γ is rank deficient. Moreover, the prior
+CSI might be noisy, leading to unstable results.
+To make initialization more robust, we can use an least
+minimum mean square error (LMMSE) criterion. To do this,
+some prior knowledge must be assumed on the statistics of
+γ, more specifically, its first and second order moments.
+We here make the following assumptions: i) the activity of
+each device is independent of one another, ii) the average
+activity and average transmit power of each device is known
+and iii) no prior information is known on the phase offset
+so that φk is considered uniformly distributed between 0
+and 2π. Under these assumptions, we have E(γ) = 0 and
+D = E(γγH) = diag (E(a0)E(ρ0), ..., E(aK−1)E(ρK−1)).
+Hence, for the linear observation model y = Γγ + w, still
+considering that λk ≈ 0, ∀k, the LMMSE estimator of γ is
+then given by [15]
+ˆγLMMSE
+init
+=
+�
+ΓHΓ + σ2D−1�−1
+ΓHy.
+(19)
+Note that the matrix to be inverted is always well-conditioned.
+Finally, a matched filter (MF) estimator could be used to avoid
+the need for matrix inversion.
+ˆγMF
+init =
+�
+diag(ΓHΓ) + σ2D−1�−1
+ΓHy.
+(20)
+C. Activity Detection
+A non-negative activity threshold γth,k is applied for each
+device k. A device is considered active if |ˆγk| ≥ γth,k. The
+real-valued threshold is defined as,
+γth,k = v
+�
+SNRk
+−1,
+(21)
+where v is chosen to have a desired probability of false alarm
+and miss detection performance and with SNRk = Mβk/σ2 =
+E(∥hk∥2)/σ2 = (∥gk∥2 + Mλ2
+k)/σ2.
+Miss detection happens when a device was undetected,
+while it was actually transmitting. Equivalently, a false alarm
+occurs if a device is considered active by the algorithm but
+was not. We define the probability of miss detection as the
+average ratio of undetected devices to the number of active
+devices Pmd = 1 −
+����Ka ∩ ˆKa
+��� / |Ka|
+�
+, where Ka is the set
+of active devices and ��Ka = {k|ˆak = 1, ∀ ∈ [1, K]} denotes
+the estimated set of active devices. Note that on average
+|Ka| = Ka. Similarly, the probability of false alarm is the ratio
+of inactive devices considered active to the number of inactive
+devices and is given by Pfa =
+���� ˆKa \ Ka
+��� /(K − |Ka|)
+�
+.
+A trade-off can be made between the two probabilities
+by varying v in (21). A lower v yields a lower activity
+threshold, resulting in more devices considered active. This in
+turn lowers the probability of miss detection, while increasing
+the probability of generating a false alarm. In the simulations,
+the parameter v is swept across the range [−40, 40]dB.
+
+Table I: Simulation parameter set with default values.
+Parameter
+Symbol
+Default value
+Number of devices
+K
+500
+Number of total BS antennas
+M
+64
+Signal-to-noise ratio
+SNR
+20 dB
+Device activity probability
+ǫa
+0.1
+Pilot sequence
+sk
+∼ CN (0, 1)
+Pilot length
+τp
+10 symbols
+Phase offset
+φk
+∼ U[0,2π]
+Number of simulations
+Nsim
+>10 000
+Number of algorithm iterations
+Niter
+K · 4
+Initialization vector
+ˆ
+γinit
+ˆγLMMSE
+init
+(19)
+Unknown part of the CSI
+λ
+0.3
+V. NUMERICAL ASSESSMENT
+The default simulation configurations are summarized in
+Table I. The device activity profile is generated randomly and
+independently for each device with a probability ǫa = 0.1,
+meaning that on average ǫaK = 50 devices are active simul-
+taneously. Or equivalently, the devices have an average duty
+cycle of 10%, which is high for typical IoT applications [2].
+The channel between the BS and device k is modelled as
+in (3). The pilot sequence is randomly generated from a
+complex Gaussian distribution sk ∼ CN (0, 1), and is assumed
+to be known by the BS. Each device uses a pilot sequence
+of 10 symbols. A random phase offset φk
+∼ U[0,2π] is
+generated to simulate a carrier frequency offset (considered
+time-invariant over the preamble duration). The source code
+for all simulations can be accessed online6.
+A. Convergence of different initialization vectors
+The convergence of different initializations is evaluated with
+respect to the genie-aided approach. In the genie-aided case,
+the algorithm is initialized with the real activity indicators, i.e.,
+γ. The convergence is assessed via the likelihood (5) and the
+mean square error (MSE). The former should monotonically
+increase with each iteration, while the MSE can vary as it can
+not directly be minimized. The performance of the different
+initialization vectors for ˆγinit are depicted in Figure 3. The
+bottom figures zoom in on a smaller region to distinguish
+the performance of the initialization vectors when converging
+closer to the genie-aided case. While all initialization methods
+approximate the genie-aided case, the initialization vector has
+a non-negligible impact on the performance of the algorithm.
+An intuitive approach is to initialize with 0 because the activity
+probability is low and hence, on average, 90 % of the devices
+are expected to be inactive. As illustrated in Figure 3, ˆγinit = 0
+requires considerably more iterations to approach the other
+initialization methods.
+B. Impact of the quality of prior CSI
+Figure 4 illustrates the performance of the detector algo-
+rithms for different correlations between the actual channel
+and the known CSI, i.e., λ. With increased λ, and thus
+decreased channel knowledge, both the LMMSE estimator
+and the proposed algorithm have an increased probability of
+miss detection. The figure also demonstrates the gain of the
+6https://github.com/wavecore-research/grant-free-random-access-partial-csi
+0
+5
+10
+15
+20
+−8,000
+−6,000
+−4,000
+−2,000
+log p(y|γ) (5)
+LMMSE (19)
+ZF (18)
+MF (20)
+zeros
+genie (γ)
+0
+5
+10
+15
+20
+−8,000
+−6,000
+−4,000
+−2,000
+log p(y|γ) (5)
+LMMSE (19)
+ZF (18)
+MF (20)
+zeros
+genie (γ)
+0
+5
+10
+15
+20
+10−2
+10−1
+100
+101
+Number of iterations (Niter/K)
+MSE(ˆγ)
+0
+5
+10
+15
+20
+10−2
+10−1
+100
+101
+Number of iterations (Niter/K)
+MSE(ˆγ)
+Figure 3: The log-likelihood (5) and MSE of the estimated activity indicators
+for different initialization vectors. While with all initialization vectors the
+genie-aided case is approximated, different number of iterations are required.
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+10−4
+10−3
+10−2
+10−1
+λ
+Pmd
+Pfa
+0.1
+0.01
+0.001
+Figure 4: Performance of the proposed algorithm (
+) versus the LMMSE
+estimator (
+) for different values of channel knowledge. The probability
+of miss detection is shown for different values of Pfa (10 %, 1 %, 0.1 %).
+proposed algorithm with respect to the LMMSE estimator. The
+algorithm outperforms the LMMSE estimator for all λ and
+is most effective when the prior CSI has a strong correlation
+with the actual channel, and diminishes with decreased channel
+knowledge.
+C. Impact of the signal-to-noise ratio
+Figure 5 shows the false alarm and miss detection prob-
+ability of the LMMSE estimator and the iterative maximum
+likelihood device activity detector for different device SNRs.
+The full CSI case is included as a baseline for comparison,
+
+10−5
+10−4
+10−3
+10−2
+10−5
+10−4
+10−3
+10−2
+21x
+Pfa
+Pmd
+−6.67 dB
+6.67 dB
+20 dB
+Figure 5: The probability of false alarm and miss detection for different device
+SNRs for the LMMSE estimator when having full CSI (
+) and partial CSI
+(
+), and the proposed iterative algorithm with partial CSI (
+). No
+miss detection or false alarm occurred in the full CSI case for SNR values of
+−6.67 dB and 20 dB.
+where the full CSI is known instead of only a portion (λ).
+Fig. 5 demonstrates the large performance gain of the proposed
+algorithm with respect to the LMMSE estimator. The graph
+demonstrates that the iterative algorithm lowers the probability
+of miss detection by a factor of 21 for the same probability
+of false alarm.7 The performance is only marginally increased
+for very low SNRs (below zero).
+VI. CONCLUSION
+We formulated an iterative maximum-likelihood (ML) al-
+gorithm to detect active devices using prior channel state
+information (CSI) when performing grant-free random access.
+Previous experimental work has demonstrated that, in many
+massive machine-typed communication (mMTC) applications,
+the CSI is less time-variant than assumed in theoretical models.
+Given the static nature of Internet of Things (IoT) devices, we
+have exploited this feature in the activity detection estimator.
+During grant-free access, the devices transmit a unique, but
+non-orthogonal preamble, which is used for activity detection.
+Next to this, the algorithm is also able to detect a device-
+specific phase offset, which could be caused by carrier fre-
+quency offset (CFO). The algorithm is numerically evaluated
+and compared to the conventional least minimum mean square
+error (LMMSE) estimator with full channel knowledge and
+partial CSI. The presented results indicate that the iterative al-
+gorithm converges and outperforms the conventional LMMSE
+estimator. For a signal-to-noise ratio (SNR) of 6.67 dB, the
+probability of not detecting an active device is 21 times lower
+for the proposed iterative ML estimator than the LMMSE
+estimator for the same probability of wrongly considering an
+inactive device as an active device.
+This work can be extended to the cell-free or distributed case
+with geographically distributed access points. In that case, the
+partial CSI becomes access point-dependent.
+7Notably, the LMMSE estimator does not employ an iterative approach.
+Therefore, our proposed algorithm will be compared in future work with
+other iterative approaches.
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+

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+(Preprint) AAS 23-294
+DEEP MONOCULAR HAZARD DETECTION FOR
+SAFE SMALL BODY LANDING
+Travis Driver⋆*, Kento Tomita⋆†, Koki Ho‡, and Panagiotis Tsiotras§
+Hazard detection and avoidance is a key technology for future robotic small body
+sample return and lander missions. Current state-of-the-practice methods rely on
+high-fidelity, a priori terrain maps, which require extensive human-in-the-loop
+verification and expensive reconnaissance campaigns to resolve mapping uncer-
+tainties. We propose a novel safety mapping paradigm that leverages deep se-
+mantic segmentation techniques to predict landing safety directly from a single
+monocular image, thus reducing reliance on high-fidelity, a priori data products.
+We demonstrate precise and accurate safety mapping performance on real in-situ
+imagery of prospective sample sites from the OSIRIS-REx mission.
+INTRODUCTION
+Hazard detection and avoidance (HD&A) is a key technology for future robotic small body sam-
+ple return and lander missions. Current approaches rely on high-fidelity digital elevation maps
+(DEMs) derived from digital terrain models (DTMs), local topography and albedo maps, generated
+on the ground.1 However, DTM construction involves extensive human-in-the-loop verification,
+carefully designed image acquisition plans, and expensive reconnaissance campaigns to resolve
+mapping uncertainties.2,3 We, instead, propose a novel safety mapping paradigm that leverages
+Bayesian deep learning techniques to accurately predict landing safety maps directly from monocu-
+lar images in order to reduce reliance on expensive high-fidelity, a priori data products (i.e., DTMs).
+Safety mapping methodologies that leverage deep learning have demonstrated potential to im-
+prove the accuracy of onboard hazard detection. Previous works4,5 have leveraged deep semantic
+segmentation to classify safe and unsafe landing locations from digital elevation maps (DEMs)
+derived from simulated LiDAR scans. However, generating reliable DEMs from LiDAR scans is
+non-trivial and requires accurate state estimates and range measurements. Moreover, LiDARs typ-
+ically feature a relatively small effective operating range6 and increased size, weight, and power
+(SWaP) requirements relative to passive sensors such as monocular cameras. Thus, we propose to
+derive landing safety maps directly from monocular images without assuming any a priori data or
+relying on the fidelity of the current state estimate. Deep semantic segmentation has been previously
+employed for surface characterization of small bodies from simulated monocular images, primar-
+ily focusing on boulder detection.7,8 Conversely, we apply our models to real images and directly
+predict safety maps that conform to realistic landing parameters and constraints.
+⋆These authors contributed equally to this work.
+*PhD Student, Institute for Robotics and Intelligent Machines, School of Aerospace Engineering, Georgia Institute of
+Technology, Atlanta, GA 30332, USA.
+†PhD Student, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.
+‡Associate Professor, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.
+§David & Andrew Lewis Chair, Professor, Institute for Robotics and Intelligent Machines, School of Aerospace Engineer-
+ing, Georgia Institute of Technology, Atlanta, GA 30332, USA.
+1
+arXiv:2301.13254v1  [cs.CV]  30 Jan 2023
+
+The contributions of this paper are as follows: first, we develop a novel safety mapping paradigm
+that leverages Bayesian deep learning techniques to predict landing safety directly from monoc-
+ular images; second, we construct a dataset of real monocular images and corresponding land-
+ing safety maps that conform to realistic landing parameters for training and testing our mod-
+els; third, we demonstrate precise and accurate safety mapping performance on real imagery of
+prospective sample sites from the recent OSIRIS-REx mission to Asteroid 101955 Bennu. Our
+code, data, and trained models will be made available to the public at https://github.com/
+travisdriver/deep_monocular_hd.
+RELATED WORK
+Current hazard detection methodologies for small body missions rely on high-fidelity digital el-
+evation maps (DEMs) derived from digital terrain models (DTMs), local topography and albedo
+maps.1 However, DTM construction typically involves extensive human-in-the-loop verification
+and carefully designed image acquisition plans to achieve optimal results.2,3 Consequently, au-
+tonomous hazard detection and avoidance (HD&A) has been identified as a high-priority technol-
+ogy9 to promote and enable new mission concepts to near-earth asteroids, comets, the Moon, Mars,
+and beyond.
+The Autonomous Landing Hazard Avoidance Technology (ALHAT)10,11 program was launched
+in 2005, followed by the Safe & Precise Landing—Integrated Capabilities Evolution (SPLICE)
+program12 in 2018, in order to develop autonomous landing technologies. These programs have fo-
+cused on developing HD&A algorithms that operate on DEMs generated from range measurements
+acquired by active sensors such as flash LiDARs. However, these methods are constrained by the
+relatively small effective operating range and the increased size, weight, and power (SWaP) require-
+ments of LiDARs relative to passive sensors such as monocular cameras. Indeed, the OSIRIS-REx
+Guidance, Navigation, and Control (GNC) flash LiDAR had a maximum operational range of ap-
+proximately 1 km,13,14 while preliminary testing of the Hazard Detection LiDAR of the SPLICE
+program demonstrated a 5 cm ground sample distance at 500 meters and near-nadir pointing.12
+Conversely, the OSIRIS-REx Camera Suite (OCAMS)15 was able to acquire 5 cm GSD images
+at almost 4 km, providing higher-resolution measurements earlier in the mission than the active
+sensors onboard and allowing for detailed surface characterization during the early phases of the
+mission.6 Moreover, constructing a DEM from LiDAR scans is non-trivial and requires accurate
+range measurements and precise knowledge of the spacecraft’s relative pose to the landing plane.
+Instead, we focus on estimating landing safety directly from a single monocular image.
+Safety mapping methodologies that leverage deep learning have demonstrated potential to im-
+prove hazard detection accuracy and have also been shown to offer competitive runtimes on flight
+relevant hardware.16 Previous works have leveraged deep semantic segmentation to classify safe
+and unsafe landing locations from high-resolution DEMs. Moghe and Zanetti4 leverage a deep neu-
+ral network architecture for predicting safety maps from a DEM and design a novel loss function
+specifically designed to decrease the false safe rate and encourage more precise safe predictions.
+Tomita et al.5 employ a Bayesian SegNet architecture17 for segmentation of input DEMs into safe
+and unsafe landing locations. The Bayesian architecture implemented by Tomita et al.5 enables
+uncertainty quantification of the predicted safety map through the predictive entropy of the model,
+allowing for more precise predictions through global thresholding with respect to this uncertainty
+measure. We build upon this work and demonstrate its efficacy on monocular imagery.
+Methods based on deep learning have also been developed for surface segmentation from monoc-
+2
+
+ular imagery. Pugliatti and Maestrini7 employ a custom U-Net architecture for classification of
+surface landmarks, i.e., boulders and crater rims. Caroselli et al.8 apply deep semantic segmen-
+tation to boulder detection on synthetic images of a fabricated small body model and post-process
+the network prediction to derive a landing safety map based on boulder density. Conversely, we
+apply our models to real images and directly predict safety maps that conform to realistic landing
+parameters and constraints.
+PROPOSED APPROACH
+Our novel safety mapping paradigm leverages Bayesian deep learning techniques to develop an
+uncertainty-aware semantic segmentation model to predict safety maps from monocular imagery.
+We train and test our model on real in-situ imagery from the OSIRIS-REx mission to asteroid
+101955 Bennu with corresponding ground truth safety maps generated using realistic landing pa-
+rameters and constraints.
+Bayesian Deep Learning
+Given training input data X = {x1, . . . , xN} with corresponding labels Y = {y1, . . . , yN},
+Bayesian deep learning employs Bayesian inference to maximize the posterior distribution of the
+network parameters θ given the training data X, Y:
+p(θ | X, Y) = p(Y | X, θ)p(θ)
+p(Y | X)
+.
+(1)
+The distribution above can then be used to predict the likelihood of an output y∗ for a new input x∗
+via
+p(y∗ | x∗, X, Y) = Ep(θ | X,Y)[p(y∗ | x∗, θ)],
+(2)
+where we may assume a softmax likelihood for p(y∗ | x∗, θ). However, computing p(θ | X, Y) is
+intractable and must be approximated using variational inference.
+Gal and Ghahramani18,19 showed that training a deep convolutional neural network (CNN) with
+dropout layers is equivalent to approximate variational inference with a variational distribution q(θ)
+which imposes a Bernoulli distribution over the model weights. Specifically, consider a convolu-
+tional layer i with ci−1 input channels, ci output channels, and kernel size k. Then dropout can be
+viewed as imposing a distribution over the layer weights Wi according to
+Wi = Mi diag([ϵj]ci
+j=1),
+ϵj ∼ Bern(pj),
+(3)
+where ϵj are Bernoulli distributed random variables with parameter pj (which we take to be 0.5),
+and Mi ∈ Rci−1×k×k×ci are the variational weight parameters optimized during training. Therefore,
+Equation (2) may be approximated by
+p(y∗ | x∗, X, Y) ≈ Eq(θ)[p(y∗ | x∗, θ)].
+(4)
+Finally, employing dropout at test time permits the use of the predictive entropy approximated
+through T stochastic forward passes through the network as a measure of uncertainty:20
+H[y | x, X, Y] ≈ −
+d
+�
+k=1
+�
+1
+T
+T
+�
+t=1
+p(y = k | x, θt)
+�
+log
+�
+1
+T
+T
+�
+t=1
+p(y = k | x, θt)
+�
+,
+(5)
+3
+
+CFF
+(1/2)
+CFF
+Entropy
+Mean
+Stochastic dropout
+samples
+Uncertainty
+Safety map
+32
+32
+64
+128
+2
+64
+128
+256
+128
+1024
+CFF
+Conv + BN + ReLU
+Dropout
+Max Pool
+Pyramid Pool
+(1/4)
+(1/8)
+(1/16)
+Bilinear downsampling
+(1/32)
+256
+256
+2x Bilinear Upsample
+Cascade Feature Fusion
+(1/2)
+(1/4)
+(1/4)
+(1/2)
+(1)
+Figure 1: Bayesian ICNet architecture. Multiscale fusion is conducted within the cascade feature
+fusion (CFF)21 modules. The ratio in parentheses denotes the relative magnitude of the spatial
+dimensions with respect to the original image.
+where θt corresponds to a realization of the network parameters, distributed according to q(θ),
+sampled during a forward pass through the network, and the average over the forward passes is
+taken to be the final prediction probabilities. This process is referred to as Monte Carlo (MC)
+dropout.18,19 For the task of semantic segmentation, assume x is a tuple (X, u) containing an
+image tensor X ∈ Rh×w×c and an image coordinate u ∈ R2, and k ∈ {1, . . . , d} is a pixel-wise
+class label for the pixel located at u. We will demonstrate that leveraging this uncertainty measure
+for our network predictions leads to increased precision and accuracy of safe landing locations.
+Uncertainty-Aware Semantic Segmentation
+We leverage an uncertainty-aware semantic segmentation architecture based on the image cas-
+cade network (ICNet),21 shown in Figure 1. ICNet is a highly efficient segmentation architecture
+that blends coarse prediction maps obtained from down-sampled inputs with high-resolution feature
+maps obtained from high throughput networks that operate on the full-resolution image, allowing
+for fast inference on high-resolution images while maintaining accuracy. Multiscale feature map
+fusion is conducted by the cascade feature fusion (CFF) modules, whereby a reduced-resolution
+segmentation map is computed from the two multiscale feature map inputs. The multiscale predic-
+tions are used to train the network via a weighted softmax cross-entropy loss.21
+We implement a Bayesian version of ICNet, which we denote as BICNet, where dropout layers
+are added to allow for stochastic sampling with respect to the model parameters using techniques
+from Bayesian deep learning, i.e., MC dropout, as described in the previous subsection. Ideally, a
+Bayesian NN would feature a dropout layer after every hidden layer of the network.18,19 However,
+as observed in previous works,17,20 adding dropout layers after every convolutional layer in more
+complex networks is too strong of a regularizer, resulting in underfitting. Therefore, we follow the
+4
+
+00(a) Global shape model
+Nightingale
+Osprey
+Kingfisher
+Sandpiper
+(b) DTM (left) and reconnaissance imagery (right) for each tag site
+Figure 2: OSIRIS-REx TAG site datasets. TAG site locations are indicated by the corresponding
+color in the global shape model.
+work of Mukhoti et al.20 and Kendall and Cippola17 and only insert dropout layers after the central
+encoder and decoder layers. At test time, we perform T = 8 stochastic forward passes and use the
+predictive entropy, defined in Equation (5), as a measure of uncertainty, and use the average of this
+measure over all training instances as a threshold to mask out high uncertainty regions in the image.
+Data Generation
+High-fidelity DTMs (i.e., 5 cm ground sample distance) of the four prospective Touch-And-Go
+(TAG) sample sites developed as part of the OSIRIS-REx mission to Asteroid 101955 Bennu, i.e.,
+Nightingale, Kingfisher, Osprey, and Sandpiper, were used to generate ground truth safety map
+labels for reconnaissance imagery from the mission. Specifically, we leverage monocular recon-
+naissance imagery and the corresponding camera pose labels, relative to a body fixed frame of the
+asteroid, provided through the AstroVision dataset.22 For each image, DEMs are constructed by
+transforming the DTM into a local coordinate system in which the +z-axis points opposite the
+vector corresponding to the direction of the gravitational force due to the target body at the point
+on the surface closest to the center of the image. The gravity due to body was computed using a
+global shape model of Bennu23 and assuming a constant-denity polyhedron.24 Safety mapping was
+conducted on the DEM and then projected back into the image to produce pixel-wise landing safety
+labels. Example reconnaissance images for each prospective TAG site are provided in Figure 2.
+Landing safety was computed from the DEMs using the method developed by the Autonomous
+Landing Hazard Avoidance Technology (ALHAT) project.25 The ALHAT method evaluates the
+lander contact locations for all pixels and for all orientations to assess the worst-case surface slope
+and roughness values with respect to the surface elevation data contained in the ground truth DEMs.
+Specifically, a landing plane is computed for each pixel by assessing the elevation of four evenly
+spaced contact points, emulating lander foot pads, on the perimeter of a circle specified by the di-
+ameter of the lander. Slope is defined as the largest angle between the landing plane and x-y plane
+of the ground truth DEM for all orientations, and the roughness is the largest perpendicular distance
+to the terrain above the the landing plane for all orientations. Any pixel with slope and roughness
+exceeding a given threshold is labeled as unsafe, where we chose a threshold of 30◦ for slope and 3.5
+cm for roughness. We specify a lander with a 35 cm diameter, similar to the MASCOT (Mobile As-
+teroid surface SCOuT) lander that was deployed during the Hayabusa2 mission to Asteroid 162173
+5
+
+(a) Image
+20
+40
+60
+80
+Slope ( )
+(b) Slope
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Roughness (m)
+(c) Roughness
+(d) Slope & Roughness Safety Map
+(e) Roughness-only Safety Map
+Figure 3: Ground truth safety map example. Safe and unsafe regions are drawn in green and red,
+respectively, in the safety map.
+Ryugu.26 An example safety map along with its corresponding monocular image is provided in
+Figure 3.
+Moreover, we provide the data distributions of our datasets with respect to the ground sample
+distance, imaging depth, viewing angle, and visibility ratio in Figure 4. Ground sample distance
+(GSD) measures the average distance on the surface spanned by a single pixel, which is a function
+of the distance to the surface and the camera intrinsics, as landing safety becomes increasingly
+difficult to observe as the relative size of the lander in the image decreases. The imaging depth
+measures the average distance to the surface when the image was taken, and provides context for
+the GSD values. Specifically, the MapCam of the OSIRIS-REx Camera Suite (OCAMS),15 with
+a focal length of ∼125 mm, can provide 5 cm GSD measuresments of the surface at distances
+of approximately 1 km, while the PolyCam, with a focal length of ∼620 mm, provides the same
+resolution at distances of almost 4 km. Viewing angle measures the angle between the −z-axis of
+the ground truth DEM and the camera boresight. Finally, the visibility ratio is the ratio of visible
+(i.e., not occluded by shadows) pixels to total pixels in the image and provides a measure of the
+illumination conditions in the image.
+6
+
++PolyCam
+MapCam
+SamCam
+Figure 4: Data distributions with respect to imaging depth, GSD, viewing angle, and visibility
+ratio. Our dataset features a total of 770 images annotated with per-pixel safety labels: 133 of
+Kingfisher, 342 of Nightingale, 162 of Osprey, and 91 of Sandpiper, and 42 from the TAG sample
+collection event at Nightingale.
+RESULTS
+In this section, we first present our suite of metrics used to evaluate the performance of our
+approach. We then validate our approach on two different experiments using real images from the
+OSIRIS-REx mission to Asteroid 101955 Bennu, including images captured during the actual TAG
+sample collection event.
+Metric Definitions
+We measure the quality of the predicted per-pixel safety map labels of our model with respect to
+precision, sensitivity, accuracy, and mean intersection over union (mIoU):
+precision =
+true safe
+true safe + false safe,
+(6)
+7
+
+Table 1: Overall performance for the Sandpiper landing site experiment. The values in paren-
+theses are the metrics with shadowed pixels ignored. All reported values are percentages.
+METHOD
+PRECISION
+SENSITIVITY
+ACCURACY
+MIOU
+SLOPE & ROUGHNESS
+WITHOUT UNCERTAINTY
+60.66 (62.91)
+67.21 (70.05)
+69.53 (69.41)
+52.05 (52.27)
+WITH UNCERTAINTY
+76.98 (77.67)
+20.09 (21.86)
+82.29 (82.01)
+65.76 (65.93)
+ROUGHNESS ONLY
+WITHOUT UNCERTAINTY
+77.24 (78.92)
+61.61 (63.66)
+63.53 (64.41)
+44.87 (45.31)
+WITH UNCERTAINTY
+85.71 (86.32)
+28.78 (31.63)
+73.77 (73.93)
+55.11 (54.98)
+sensitivity =
+true safe
+true safe + false unsafe,
+(7)
+accuracy = true safe + true unsafe
+valid pixels
+,
+(8)
+mIoU = 1
+2
+�
+true safe
+valid pixels − true unsafe +
+true unsafe
+valid pixels − true safe
+�
+.
+(9)
+True safe (false safe) includes pixels predicted to be safe by our models that are safe (unsafe) in
+the ground truth labels, and true unsafe (false unsafe) includes pixels predicted to be unsafe that
+are unsafe (safe) in the ground truth labels. Note that false unsafe includes safe pixels that are
+ignored and not labeled safe due to high uncertainty. For our application, we can interpret precision
+as the reliability of the pixels predicted to be safe, and sensitivity as detection rate of true safe
+sites, respectively. Accuracy and mIoU are the metrics evaluated for the valid pixels, which are
+the pixels with smaller uncertainty than the threshold. In other words, valid pixels correspond
+to the predictions that the network is most “certain” about. For the results without uncertainty
+thresholding, accuracy and mIoU are evaluated for all the pixels with valid safety labels. In the
+following analysis, we refer to the ratio of pixels that fall above our uncertainty threshold, and
+consequently marked as unsafe, to valid pixels as the screening rate.
+Experiment 1: Prospective Landing Site Sandpiper
+We train our model on images from three of the prospective landing sites from the OSIRIS-
+Rex mission, namely, Nightingale, Osprey, and Kingfisher, and test our model on images of the
+remaining sample site, Sandpiper. This emulates a scenario in which data from previously mapped
+landing sites could be used to train a network to predict landing safety in a new, unexplored region of
+the target body without requiring the construction of high-fidelity DEMs. These results are detailed
+in Table 1, and qualitative examples are provided in Figure 5, where we consider performance with
+respect to identifying both slope and roughness hazards and roughness-only hazards.
+The results illustrate that our models are able to predict safety maps from just a single monocular
+image of the propspective landing site, which is completely unseen during training, with accuracy
+over 69% for the slope and roughness hazard detection case, and over 63% for the roughness-only
+8
+
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+(a) Slope & roughness
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+Test Image
+Without Uncertainty
+With Uncertainty
+Uncertainty
+(b) Roughness-only
+Figure 5: Qualitative monocular safety mapping results for the Sandpiper experiment. Green,
+yellow, blue, and red labels represent true safe, true unsafe, false unsafe, and false safe, respectively.
+9
+
+hazard detection case, even without uncertainty thresholding. Moreover, our Bayesian ICNet archi-
+tecture enables uncertainty thresholding in order to further boost performance by ignoring regions in
+which the models’ prediction has high entropy. With the uncertainty threshold, accuracy increases
+to 82.29% and 73.77% for the slope and roughness and roughness-only cases, respectively, at the
+cost of decreased sensitivity. Importantly, we are able to achieve 76.98% and 85.71% precision for
+the slope and roughness and roughness-only cases, respectively, after uncertainty thresholding. We
+also have a slight increase in all the metrics by ignoring shadowed pixels, which are reported in
+parentheses in Table 1.
+Comparing the two different hazard detection tasks, i.e., slope and roughness hazards and roughness-
+only hazards, the roughness-only case has lower values of sensitivity, accuracy, and mIoU, but
+higher values of precision. This suggests that the roughness-only case is a harder task in terms of
+precisely labeling safe and unsafe pixels on average, resulting in lower accuracy and mIoU, but is
+an easier task in terms of identifying only safe pixels, thus resulting in higher precision. This is
+partially due to the higher incidence of safe pixels for the roughness-only case as compared to the
+slope and roughness case, as illustrated in Figure 3.
+Additionally, we analyzed the per-image metrics with respect to GSD, viewing angle, and visibil-
+ity ratio for the slope and roughness case, shown in Figure 6, and the roughness-only case, shown in
+Figure 7, in order to identify possible causes of uncertainty in the predictions. As a general trend, we
+can observe that a higher uncertainty results in lower performance metrics of precision, sensitivity,
+accuracy, and mIoU. Intuitively, low visibility is a common factor that results in higher uncertainty
+in our model for both the slope and roughness case and the roughness-only case. Our models as-
+sign a higher uncertainty to images with larger GSD for the slope and roughness case, and larger
+viewing angle for the roughness-only case. Note that increased uncertainty for images at higher
+GSDs may also be due to these instances being less represented in the training data as shown in
+Figure 4. In either case, we demonstrate that the uncertainty threshold serves as a powerful tool for
+detecting and accounting for difficult or out-of-distribution input conditions, allowing our models
+to predict precise and accurate safety maps across multiple GSDs, viewing angles, and illumination
+conditions.
+Experiment 2: OSIRIS-REx TAG Sequence
+For the second experiment, we trained two models using different combinations of images from
+the OSIRIS-REx mission: one model, denoted by BICNet-NKO, is trained on Nightingale, King-
+fisher, and Osprey, and the other model, denoted by BICNet-KOS, is trained on Kingfisher, Osprey,
+and Sandpiper. These models were trained for the slope and roughness case only. We tested the two
+models on images captured during the TAG sample collection event at the Nightingale sample site.
+We present both models to illustrate the effect of the different training data distributions on the test
+results. A subset of the 42 image sequence is shown in Figure 8. Note that the 42 test images are
+not included in the training set for the Nightingale images.
+Table 2 and Figure 10 show quantitative results and qualitative examples, respectively. Compar-
+ing BICNet-NKO and BICNet-KOS in Table 2, we can see sensitivity of BICNet-KOS is signifi-
+cantly lower than that of BICNet-NKO after uncertainty thresholding. Indeed, BICNet-KOS assigns
+a high uncertainty to almost all regions of the input images and are thus overwritten as unsafe after
+uncertainty thresholding, as shown in Figure 10, which is not the case for BICNet-NKO. These
+differences are most likely explained by the difference between the training and testing data distri-
+butions for BICNet-KOS, as illustrated in Figure 4. Specifially, the TAG images have less overlap
+10
+
+0.6
+0.8
+Precision
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.0
+0.2
+0.4
+Sensitivity
+0.6
+0.8
+Accuracy
+uncertainty
+0.44
+0.48
+0.52
+0.56
+0.60
+0.4
+0.6
+0.8
+IoU
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.6
+0.7
+0.8
+0.9
+Screen Rate
+0.6
+0.8
+Precision
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.0
+0.2
+0.4
+Sensitivity
+0.6
+0.8
+Accuracy
+uncertainty
+0.44
+0.48
+0.52
+0.56
+0.60
+0.4
+0.6
+0.8
+IoU
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.6
+0.7
+0.8
+0.9
+Screen Rate
+0.6
+0.8
+Precision
+0.4
+0.6
+0.8
+Visibility Ratio
+0.0
+0.2
+0.4
+Sensitivity
+0.6
+0.8
+Accuracy
+uncertainty
+0.44
+0.48
+0.52
+0.56
+0.60
+0.4
+0.6
+0.8
+IoU
+0.4
+0.6
+0.8
+Visibility Ratio
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.6
+0.7
+0.8
+0.9
+Screen Rate
+Figure 6: Per-image metrics for slope & roughness safety on the Sandpiper experiment with
+respect to GSD, viewing angle, and visibility ratio.
+11
+
+0.7
+0.8
+0.9
+Precision
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.0
+0.2
+0.4
+0.6
+Sensitivity
+0.5
+0.6
+0.7
+0.8
+Accuracy
+uncertainty
+0.40
+0.44
+0.48
+0.52
+0.56
+0.60
+0.3
+0.4
+0.5
+0.6
+0.7
+IoU
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.40
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.4
+0.6
+0.8
+Screen Rate
+0.7
+0.8
+0.9
+Precision
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.0
+0.2
+0.4
+0.6
+Sensitivity
+0.5
+0.6
+0.7
+0.8
+Accuracy
+uncertainty
+0.40
+0.44
+0.48
+0.52
+0.56
+0.60
+0.3
+0.4
+0.5
+0.6
+0.7
+IoU
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.40
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.4
+0.6
+0.8
+Screen Rate
+0.7
+0.8
+0.9
+Precision
+0.4
+0.6
+0.8
+Visibility Ratio
+0.0
+0.2
+0.4
+0.6
+Sensitivity
+0.5
+0.6
+0.7
+0.8
+Accuracy
+uncertainty
+0.40
+0.44
+0.48
+0.52
+0.56
+0.60
+0.3
+0.4
+0.5
+0.6
+0.7
+IoU
+0.4
+0.6
+0.8
+Visibility Ratio
+0.40
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.4
+0.6
+0.8
+Screen Rate
+Figure 7: Per-image metrics for roughness-only safety on the Sandpiper experiment with re-
+spect to GSD, viewing angle, and visibility ratio.
+12
+
+2020-10-20T21:30:48
+2020-10-20T21:31:48
+2020-10-20T21:32:48
+2020-10-20T21:33:48
+2020-10-20T21:34:48
+2020-10-20T21:35:48
+2020-10-20T21:36:48
+2020-10-20T21:37:48
+2020-10-20T21:38:48
+2020-10-20T21:39:48
+2020-10-20T21:40:48
+2020-10-20T21:41:48
+Figure 8: Frames from the OSIRIS-Rex TAG sequence captured by the SamCam.
+with the images of the prospective landing sites except for Nightingale with respect to viewing angle
+and visibility ratio. Therefore, the exclusion of Nightingale from training data increases the predic-
+tive uncertainty for TAG images at test time for BICNet-KOS. Note that the Nightingale images
+(excluding the TAG images) were used for validation during training for the BICNet-KOS model in
+order to rule out overfitting as a cause for the decreased prediction performance.
+This effect of training data distributions on the uncertainty level, and the accompanying predictive
+performance, is consistent with the per-image metrics with respect to GSD, as shown in Figure 9.
+Indeed, for BICNet-NKO, as the GSD of testing images gets closer to the peak of the training data,
+the predictive performance increases and uncertainty decreases. Lower precision for the higher
+GSD images is also partly due to the very low incidence of safe regions for these instances (see
+Figure 10). Conversely, for BICNet-KOS, all test images have a high screening rate and a low
+sensitivity due to the high uncertainty, purportedly due to the out-of-distribution training data with
+respect to the viewing angle and visibility ratio and the relatively small size of the training set (386
+images). We do not provide the per-image metrics with respect to the viewing angle and visibility
+ratio, as these values remain relatively constant over the entire TAG sequence at ∼7◦ and ∼71%,
+respectively, as shown in Figure 4. These results illustrate the effect of training data distributions
+and the ability of the uncertainty measure to identify out-of-distribution data for uncertainty-aware
+segmentation networks. We postulate that training our model with a more comprehensive set of
+images will decrease prediction uncertainty and increase the performance.
+CONCLUSION
+In this paper we presented a novel landing hazard detection approach for small body missions that
+predicts safety maps directly from monocular imagery. We implemented an efficient, uncertainty-
+aware segmentation network that demonstrated hazard detection performance at over 80% accuracy
+and over 85% precision on real images of unseen landing sites captured during the OSIRIS-REx
+mission to Asteroid 101955 Bennu. We believe that monocular safety mapping is a promising
+technology for reducing reliance on human-in-the-loop procedures used in current safety mapping
+methodologies. Future work will involve developing a more comprehensive distribution of training
+data and identifying and rectifying causes of uncertainty to increase the reliability of the proposed
+13
+
+0.4
+0.6
+0.8
+Precision
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.2
+0.4
+0.6
+Sensitivity
+0.7
+0.8
+0.9
+Accuracy
+uncertainty
+0.38
+0.40
+0.42
+0.44
+0.46
+0.5
+0.6
+0.7
+0.8
+IoU
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.375
+0.400
+0.425
+0.450
+0.475
+Uncertainty
+0.5
+0.6
+Screen Rate
+(a) BICNet-NKO
+0.5
+0.6
+0.7
+0.8
+Precision
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.00
+0.02
+0.04
+0.06
+0.08
+Sensitivity
+0.75
+0.80
+0.85
+Accuracy
+uncertainty
+0.49
+0.50
+0.51
+0.52
+0.53
+0.5
+0.6
+IoU
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.50
+0.52
+Uncertainty
+0.65
+0.70
+0.75
+0.80
+Screen Rate
+(b) BICNet-KOS
+Figure 9: Per-image metrics for the TAG experiment with respect to the GSD for slope &
+roughness safety.
+14
+
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+(a) BICNet-NKO
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+Test Image
+Without Uncertainty
+With Uncertainty
+Uncertainty
+(b) BICNet-KOS
+Figure 10: Qualitative monocular safety mapping results for the TAG experiment. Green,
+yellow, blue, and red labels represent true safe, true unsafe, false unsafe, and false safe, respectively.
+15
+
+Table 2: Overall performance for the TAG experiment for slope & roughness safety. BICNet-
+NKO is our Bayesian ICNet model trained on Nightingale, Kingfisher, and Osprey, and BICNet-
+KOS is trained on Kingfisher, Osprey, and Sandpiper. All reported values are percentages.
+METHOD
+PRECISION
+SENSITIVITY
+ACCURACY
+MIOU
+BICNET-NKO
+WITHOUT UNCERTAINTY
+49.02 (50.21)
+61.16 (66.53)
+67.44 (67.09)
+48.65 (48.66)
+WITH UNCERTAINTY
+61.38 (61.66)
+26.60 (30.80)
+78.62 (77.09)
+60.04 (59.23)
+BICNET-KOS
+WITHOUT UNCERTAINTY
+50.08 (50.42)
+42.18 (50.37)
+67.24 (65.32)
+45.48 (45.11)
+WITH UNCERTAINTY
+65.05 (64.89)
+2.87 (4.05)
+83.11 (82.65)
+52.58 (55.87)
+approach. Our code, data, and trained models will be made available to the public at https:
+//github.com/travisdriver/deep_monocular_hd.
+ACKNOWLEDGMENTS
+This work supported by a NASA Space Technology Graduate Research Opportunity and the
+NASA Early Career Faculty Program (grant no. 80NSSC20K0064). The authors would like to
+thank Kenneth Getzandanner and Michael Shoemaker from NASA Goddard Space Flight Center
+for several helpful discussions and comments.
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+17
+

9tFLT4oBgHgl3EQfui8A/content/tmp_files/2301.12155v1.pdf.txt

@@ -0,0 +1,1637 @@
+epl draft
+Towards a liquid-state theory for active matter
+Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E. Cates1 and ´Etienne Fodor2
+1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
+2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
+3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK
+PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics
+Abstract – In equilibrium, the collective behaviour of particles interacting via steep, short-ranged
+potentials is well captured by the virial expansion of the free energy at low density. Here, we extend
+this approach beyond equilibrium to the case of active matter with self-propelled particles. Given
+that active systems do not admit any free-energy description in general, our aim is to build the
+dynamics of the coarse-grained density from first principles without any equilibrium assumption.
+Starting from microscopic equations of motion, we obtain the hierarchy of density correlations,
+which we close with an ansatz for the two-point density valid in the dilute regime at small activity.
+This closure yields the nonlinear dynamics of the one-point density, with hydrodynamic coefficients
+depending explicitly on microscopic interactions, by analogy with the equilibrium virial expan-
+sion. This dynamics admits a spinodal instability for purely repulsive interactions, a signature
+of motility-induced phase separation. Therefore, although our approach should be restricted to
+dilute, weakly-active systems a priori, it actually captures the features of a broader class of active
+matter.
+Active matter is a wide category of systems out of equi-
+librium, where a net flow of energy takes place at the local,
+individual level [1–3]. Examples are swimming bacteria [4]
+and active emulsions [5], amongst others. The realm of
+motile active matter includes interacting, many-particle
+systems, where particles move persistently. Different the-
+oretical approaches have been proposed to describe such
+systems. They include (i) particle-based models, such as
+run-and-tumble particles (RTPs) [6], active Brownian par-
+ticles (ABPs) [7], and active Ornstein-Uhlenbeck particles
+(AOUPs) [8,9], (ii) microscopic field theories [10,11], and
+(iii) coarse-grained field theories [12–14].
+Self-propelled particles tend to slow down at high densi-
+ties, due to either biochemical reasons or steric repulsions.
+In contrast with passively diffusing particles, this slow-
+down in turn increases the local density, creating a pos-
+itive feedback loop that can result in a phase separation
+between a dense and a dilute phase, known as motility-
+induced phase separation (MIPS) [15].
+To predict an-
+alytically the emergence of MIPS, previous works have
+relied on a local mean-field approximation to replace mi-
+croscopic interaction with density-dependent motility [16],
+and also on an adiabatic elimination of orientational de-
+gree of freedom [15]. Other studies have developed some
+aspects of a liquid-state theory for active matter with pair-
+wise interactions. This is done, for instance, by expressing
+thermodynamic observables, such as pressure [17–19] and
+dissipation [20,21], in terms of correlations between hydro-
+dynamic fields, typically density and polarization. Also,
+exact results for density correlations have been obtained
+at infinite dimensions [22,23].
+The success of equilibrium thermodynamics largely
+stems from the ability to detect phase transitions by an-
+alyzing the free-energy.
+In the dilute regime, the virial
+expansion provides an approximate expression of the free-
+energy that averages the effect of steep, short-ranged pair-
+wise interactions [24].
+For passive Brownian particles
+(PBPs) interacting with an isotropic pairwise potential
+�
+i,j<i Ψ(|xi − xj|) at temperature T, the free energy per
+unit volume is given in terms of the uniform density ρ as
+fvirial(ρ)/T = ρ log ρ + B(T)ρ2 + O(ρ3),
+B = 1
+2
+�
+dr
+�
+1 − e−Ψ(r)/T �
+.
+(1)
+Here B is commonly referred to as the second virial coef-
+ficient. The major success of the virial expansion lies in
+predicting the onset of liquid-gas phase separation in parti-
+cle systems with a combination of strongly repulsive short-
+ranged forces and weakly attractive long-range forces, such
+as the Lennard-Jones potential. As temperature increases,
+p-1
+arXiv:2301.12155v1  [cond-mat.soft]  28 Jan 2023
+
+Y. I. Li et al.
+the free energy, as a function of density ρ, goes from convex
+(B > 0) to concave (B < 0): It yields a phase transition
+from homogeneous to non-homogeneous density profiles,
+with a separation between dilute and dense phases. Im-
+portantly, the virial expansion holds for a generic micro-
+scopic potential (excluding those of such long range that
+the excess free energy is not analytic in density).
+In classical thermodynamics, the virial expansion is of-
+ten derived from the equilibrium partition function [24].
+Interestingly, it is also possible to arrive at approx-
+imate expressions of free energy by truncating the
+hierarchy of density correlations,
+known as the Bo-
+goliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierar-
+chy [25]. For instance, using an ansatz for the two-particle
+density, informed by the steady-state solution of the two-
+body problem [26], the dynamics of the one-body density
+follows a gradient flow with respect to fvirial, as shown
+in the SM. This type of derivation can potentially be ex-
+tended to a large class of nonequilibrum systems, where
+the dynamics does not derive from any free energy a pri-
+ori. Such an extension requires proposing an ansatz for
+the two-particle density, which should be appropriate to
+the specific nonequilibrium system at hand.
+Interestingly, the two-body probability distribution can
+be approached perturbatively for AOUPs, where the per-
+sistence time is the small parameter in natural units [8,9].
+To first order, this solution already predicts that repul-
+sive interaction yields effective attraction. This suggests
+that the onset of MIPS can likewise already be captured
+by approximating the dynamics of density at this order.
+Therefore, inspired by the equilibrium virial expansion, it
+is tempting to explore whether the density dynamics al-
+ready contains any spinodal instability for dilute active
+systems at weak persistence. Note that, although previ-
+ous work refer to “active virial” as an expansion of pres-
+sure [27, 28], here we are instead interested in deriving
+dynamical equations of motion for the density. Indeed, at
+variance with equilibrium, phase transitions in active sys-
+tems are usually not thermodynamically controlled by any
+equation of state, but they can still be directly detected
+as instability in the dynamics.
+In this Letter, we start with the particle-based descrip-
+tion of AOUPs, and consider their steady-state probability
+distribution. Our roadmap is essentially a “small density-
+small persistence” bottom-up derivation of the density dy-
+namics. To this end, we first introduce the corresponding
+hierarchy of density correlations. Inspired by equilibrium
+thermodynamics [25, 26], we then close the hierarchy to
+arrive at a nonlinear dynamics for the one-point density.
+It allows us to identify the MIPS spinodal instability for
+any microscopic repulsive interactions. An advantage of
+our approach is that does not impose an equilibrium map-
+ping a priori (see [8,9] for a discussion of closures to the
+AOUP equations that do so), instead retaining the non-
+equilibrium structure of the microscopic theory. A disad-
+vantage is that extensions to higher order in density would
+involved increasingly complicated calculations of higher
+virial coefficients, which we do not attempt here.
+Interacting AOUPs:
+Joint distribution func-
+tion – We consider a system of N AOUPs at positions
+xi interacting with pairwise potential U.
+The particles
+are subject to stochastic self-propulsion velocities vi with
+persistence time τ and diffusivity D [8,9]:
+˙xi = vi − ∂xiU,
+U =
+N
+�
+i=1
+N
+�
+j<i
+Ψ(rij),
+τ ˙vi = −vi +
+√
+2DΛi,
+(2)
+where rij = |xi − xj|, and we have set the mobility to
+unity. Here Λi is a Gaussian white noise, with correlation
+⟨(Λi)α(t)(Λj)β(0)⟩ = δijδαβδ(t), where latin and Greek
+letters respectively denote particle index and spatial co-
+ordinates. It follows that vi is a colored Gaussian noise,
+with correlation ⟨(vi)α(t)(vj)β(0)⟩ = δijδαβ(D/τ)e−|t|/τ.
+In the limit of vanishing persistence (τ → 0), the corre-
+lation of vi becomes white, so that the system reduces to
+a set of overdamped PBPs. Note that, at finite τ, if the
+potential term −∂xiU was in the rhs of the vi-equation
+(instead of the xi-equation), the system would represent
+underdamped PBPs in equilibrium. For this reason, over-
+damped AOUPs and underdamped PBPs coincide in the
+noninteracting limit.
+Following
+standard
+procedures
+[29],
+the
+Fokker-
+Planck equation of the joint probability distribution
+pN(x1, v1, x1, v2, . . . , xN, vN, t) associated with eq. (2) is
+∂tpN =
+N
+�
+i=1
+�
+Lf,ipN + ∂xi · (pN ∂xiU)
+�
+,
+Lf,i = D
+τ 2 ∂2
+vi +
+�1
+τ ∂vi − ∂xi
+�
+· vi,
+(3)
+where Lf,i is the non-interacting Fokker-Planck operator
+governing the free-particle motion of the i-th particle. Al-
+though it cannot be solved exactly, its steady state can be
+computed to lowest orders in the persistence time τ in a
+similar manner to [8, 9]. Such a perturbative calculation
+can be performed in arbitrary dimension, though we now
+restrict ourselves to 1D for simplicity, as a proof of prin-
+ciple. After scaling units as v → v
+�
+τ/D , x → x/
+√
+τD,
+t → t/τ, and Ψ → Ψ/D, we find the many-body steady
+state probability [see SM for details],
+pN ∝ e−U−
+v2
+1+v2
+2
+2
+�
+1 + √τ
+N
+�
+i=1
+vi∂xiU
++ τ
+N
+�
+i=1
+�v2
+i
+2
+�
+(∂xiU)2 − ∂2
+xiU
+�
+− (∂xiU)2 + 3
+2∂2
+xiU
+�
++ o(τ)
+�
+.
+(4)
+Note that the steady state probability here is given in
+(x, v) space, whereas [8] gives the corresponding probabil-
+ity in (x, ˙x) space.
+p-2
+
+Towards a liquid-state theory for active matter
+Nonequilibrium density correlations: Hierarchy
+of equations – Although the joint distribution function
+pN in eq. (4) contains all information about the steady
+state density, it is generally difficult to predict the emer-
+gence of phase transitions from the perspective of the full
+phase space. Instead, our aim is to obtain a reduced de-
+scription of the systel in terms of density correlations. The
+n-particle density pn is found by marginalising the joint
+distribution function pN as
+pn(x1, v1, . . . , xn, vn, t)
+= PN,n
+�
+pN(x1, v1, . . . , xN, vN, t)
+N
+�
+j=n+1
+dxjdvj,
+(5)
+where PN,n = N!/(N − n)! is the permutation coefficient.
+Thereafter, for simplicity, we use the shorthand notation
+pn = pn(x1, v1, . . . , xn, vn, t) for arbitrary n, where the
+dependence on positions, self-propulsion velocities, and
+time will be omitted. Integrating eq. (3) over the positions
+and velocities of all particles but one yields an equation for
+the one-particle density that depends on the two-particle
+density:
+∂tp1 = Lf,1p1 + F1[p2],
+F1[p2] = ∂x1 ·
+�
+p2 ∂x1Ψ(r12) dx2dv2.
+(6)
+The first term Lf,1p1 corresponds to the free single-particle
+dynamics, whereas the second term F1[p2] represents the
+effects of pairwise interaction between one particle and all
+the others.
+Following the same marginalisation procedure for the
+n-particle density pn, we obtain a hierarchy of equations,
+each depending on the density pn+1, analoguous to the
+BBGKY hierarchy of PBPs [25]. The resulting equation
+for the two-particle density reads
+∂tp2 = (Lf,1 + Lf,2 + Lint)p2 + F2[p3],
+Lint =
+�
+i=1,2
+∂xi ·
+�
+∂xiΨ(r12)
+�
+,
+F2[p3] =
+�
+i=1,2
+∂xi ·
+�
+p3 ∂xiΨ (ri3) dx3dv3.
+(7)
+Here Lint governs the pairwise interactions between any
+two particles, and F2[p3] represents the interactions be-
+tween either of the first two particles and all the other
+particles. As we will show now, this is a higher-order term
+in the average number density ρ = N/V compared to the
+rest, and hence can be neglected in the dilute limit.
+To determine the scaling of pn with respect to ρ, one
+can start with their normalisation properties [24]
+�
+pn
+n
+�
+i=1
+dxidvi = PN,n,
+(8)
+from which follows pn ∼ ρn for small n. Physically, this
+is indeed expected as p1(x1, v1) is the probability density
+of finding n particles at a given set of coordinates. With
+this scaling in mind, we estimate the scaling of each term
+in eq. (7) as
+(∂t − Lf,1 − Lf,2 − Lint
+����
+∼1/τint
+) p2
+����
+∼ρ2
+=
+�
+i=1,2
+∂xi ·
+�
+p3
+����
+∼ρ3
+∂xiΨ(ri3)
+�
+��
+�
+∼1/τint
+dx3
+����
+∼(r0)d
+dv3,
+(9)
+where τint is the typical timescale of interaction, r0 is the
+lengthscale of the potential, and d denotes the sptial di-
+mension. One can see that the rhs of eq. (9) is small if
+rd
+0ρ ≪ 1. Since rd
+0 is effectively the volume of a particle,
+this condition is satisfied if the volume fraction of parti-
+cles in the system is low. Therefore, as expected from the
+analogy with the BBKY hierarchy [25], the interaction of
+two particles with respect to others in the bath can be
+neglected at small volume fraction.
+Closure of hierarchy:
+Insight from quasistatic
+approximation – We now propose a closure of the hierar-
+chy of equation for density correlations. At small volume
+fraction, neglecting the rhs in eq. (9) yields
+∂tp2 = (Lf,1 + Lf,2 + Lint) p2.
+(10)
+The dynamics in eq. (10) is equivalent to the Fokker-
+Planck equation of two AOUPs interacting through the
+pairwise potential Ψ(r12).
+Recall from eq. (4) the N-
+particle stationary probability, calculated order by order
+in the persistence time τ. The two-particle probability is
+hence,
+p2,ss(r12, v1, v2) ∝ e−Ψ(r12)−
+v2
+1+v2
+2
+2
+g(r12, v1 − v2),
+(11)
+where
+g(r, w) = 1 + √τwΨ′ + (τw2/2)((Ψ′)2 − Ψ′′)
++ τ(3Ψ′′ − 2(Ψ′)2) + O(τ 3/2),
+(12)
+and Ψ′ = dΨ/dr [see SM]. In the quasistatic limit, where
+the interaction timescale is much shorter than the persis-
+tence (τint ≪ τ), we assume that every pair of particles
+reaches steady state. Inspired by previous works [26, 30],
+this assumption motivates the following ansatz to close the
+hierarchy of density correlations:
+p2(x1, v1, x2, v2, t) = p1(x1, v1, t) p1(x2, v2, t)
+× e−Ψ(r12)g(r12, v1 − v2).
+(13)
+There is evidence that the above ansatz works well for
+strong short-ranged interactions in the equilibrium case
+[see SM]. We assume it would also work in the non-
+equilibrium case. Indeed, when the interparticle distance
+is larger than the interaction range (r12 > r0), the two-
+point function p2 reduces to the product of one-point func-
+tions p1, as expected from a mean-field approach for non-
+interacting systems. The second line of eq. (13) accounts
+p-3
+
+Y. I. Li et al.
+for corrections due to interactions. For repulsive poten-
+tial, the factor e−Ψ captures the reduction of the two-
+point density caused by the interaction, as expected from
+equilibrium, whereas g(r12, v1 − v2) encapsulates nonequi-
+librium effects.
+Substituting the ansatz from eq. (13) into eq. (6), we
+arrive at the following dynamics for the one-particle den-
+sity:
+(∂t−Lf) p1(x, v, t)
+= τ∂x
+�
+p1(x, v, t)
+�
+Lvp1(x, v − w, t)dw
+�
+.
+(14)
+Here, Lf is the free Fokker-Planck operator in scaled units,
+and the operator Lv effectively takes into account interac-
+tions:
+Lf = ∂2
+v +
+�
+∂v − √τ∂x
+�
+v,
+Lv =
+�
+Ψ′(r)e−Ψ(r)g(r, w)e−r∂xdr,
+(15)
+where we have introduced the translation operator e−r∂x,
+which shifts the position of the function acted upon as
+e−r∂xp1(x, v) = p1(x − r, v). Hence, Lv corresponds to an
+infinite series of the gradient ∂x, with series coefficients set
+by the microscopic potential. Equation (14) is the central
+result of this Letter: It provides the dynamics of the one-
+particle density in a closed form for small average density.
+Importantly, this closed form depends explicitly on the
+details of the pairwise potential Ψ.
+To obtain a more explicit expression of Lv, we next
+substitute our perturbative result for g [eq. (12)] into the
+definition of Lv [eq. (15)], yielding
+Lv = 2wLa + 2Lb∂x + w2Lc∂x,
+(16)
+where, after integration by parts, we get
+La = √τ
+� ∞
+0
+dr(Ψ′)2 e−Ψe−r∂x,
+Lb = −
+� ∞
+0
+drf0(r)e−r∂x,
+Lc = −
+� ∞
+0
+dr f1(r)e−r∂x,
+(17)
+and
+f0(s) =
+� ∞
+s
+dr Ψ′e−Ψ �
+1 − τ
+�
+2(Ψ′)2 − 3Ψ′′��
+,
+f1(s) = τ
+� ∞
+s
+dr Ψ′ �
+(Ψ′)2 − Ψ′′�
+e−Ψ.
+(18)
+The integrands featuring in the definition of the operators
+{La, Lb, Lc} [eq. (17)] are even with respect to r, provided
+that Ψ(r) = Ψ(−r). It follows that these operators can be
+expressed as a series of ∂2
+x.
+The functions f0 and f1 are analoguous to the Mayer
+function which appears in the equilibrium virial expan-
+sion [25]. Indeed, in the equilibrium limit (τ → 0), the
+only surviving term in Lv stems from the leading order in
+f0, in agreement with the derivation for equilibrium over-
+damped PBPs [see SM]. This term is responsible for the
+B coefficient in eq. (1) when that result is derived dynam-
+ically for equilibrium systems. In that respect, eq. (16)
+can be regarded as a direct generalization of the equilib-
+rium virial expansion to the AOUP setting. Importantly,
+eq. (16) shows that activity produces additional velocity-
+dependent terms that are absent in equilibrium. In what
+follows, we analyze in detail the corresponding dynam-
+ics, in search for the onset of instability as a signature of
+MIPS.
+Linear stability analysis: Eigenvalue problem –
+The stationary solution of eq. (14) is given by p(0)
+1 (v) =
+(ρ/
+√
+2π)e−v2/2. This solution corresponds to a uniform
+density for the position and a Gaussian distribution for
+the self-propulsion.
+In addition, as we will see shortly,
+this is also the ground state of the free operator. In the
+following, we expand perturbatively around p(0)
+1
+to find its
+instability regions in parameter space: If the uniform state
+is not stable, then we argue that the system undergoes
+spinodal decomposition via a MIPS mechanism.
+We consider p1(x, v, t) = p(0)
+1 (v) + ε(x, v, t), with ε a
+small perturbation about the ground state p(0)
+1 . Expand-
+ing eq. (14) to linear order in ε, we get
+(∂t− ¯Lf) ε(x, v, t) = τp(0)
+1 (v)
+�
+Lv∂xε(x, v−w, t)dw, (19)
+where Lv is defined in eq. (16), and
+¯Lf = ∂2
+v + (∂v − α∂x)v,
+α = √τ − 2ρτ 3/2
+� ∞
+0
+(Ψ′)2e−Ψdr.
+(20)
+To analyze the time evolution of the perturbation given in
+eq. (20), the difficulty lies in treating the effect of the op-
+erator Lv, which contains information about microscopic
+interactions.
+Then, it is convenient to expand ε in the
+eigenfunctions of the bare theory, namely in the absence
+of Lv. Interestingly, as stated previously, the bare theory
+maps into underdamped passive Brownian motion, and
+one can readily find the solution in the literature [29].
+The corresponding eigenfunctions, which we here call the
+Fourier-Hermite basis, are
+ψnk(x, v) = e−ik(x+αv)Un(v − 2iαk),
+(21)
+where Un is the Hermite function
+Un(z) =
+1
+√
+2π Hn(z) e−z2/2 = (−1)n
+√
+2π ∂n
+z e−z2/2,
+(22)
+with the property that (∂2
+z + ∂zz)Un(z) = −nUn(z). In-
+deed, acting on the Fourier-Hermite basis with the modi-
+fied free operator ¯Lf yields
+¯Lfψnk = −λknψnk,
+λkn = (αk)2 + n.
+(23)
+p-4
+
+Towards a liquid-state theory for active matter
+As the operator ¯Lf is not Hermitian, the conjugate basis is
+not simply the complex conjugate. Instead, we introduce
+the following conjugate basis
+¯ψnk(x, v) = eik(x+αv) ¯Un(v − 2iαk),
+¯Un(z) = 1
+n!Hn(z),
+(24)
+yielding
+�
+dxdv ¯ψmk′(x, v)ψnk(x, v) = 2πδ(k − k′)δmn,
+(25)
+so that the orthogonality relation between the basis ψnk
+and its conjugate ¯ψnk indeed holds as expected.
+Having obtained the eigenvectors and eigenvalues of the
+bare theory, we decompose the perturbation ε in eigen-
+functions of ¯Lf as
+ε(x, v, t) =
+�
+n
+�
+εkn(t)ψnk(x, v)dk.
+(26)
+Multiplying eq. (19) with the conjugate basis ¯ψkn and inte-
+grating over {x, v} yields the dynamics of the perturbation
+ε, in the Fourier-Hermite basis, as
+˙εkm = −λkmεkm+
+�
+n
+�
+M (1)
+kmn+M (2)
+kmn+M (3)
+kmn
+�
+εkn, (27)
+where
+M (1)
+kmn = 2τρLa,k
+(−iαk)n+m
+αm!
+e(αk)2(m − n),
+M (2)
+kmn = −2τρLb,kk2 (−iαk)n+m
+m!
+e(αk)2,
+M (3)
+kmn = τρLc,k
+(−iαk)n+m
+α2m!
+e(αk)2
+×
+�
+m(m − 1) + n(n − 1) − 2mn − 2(αk)2�
+.
+(28)
+Here, La,k =
+�
+eikxLadk, with similar definitions for Lb,k
+and Lc,k, where the operators {La, Lb, Lc} are defined in
+eq. (17).
+If the matrix Mknm = −λkmδmn + M (1)
+kmn +
+M (2)
+kmn +M (3)
+kmn, which controls the growth rate of the per-
+turbation ε, has no positive eigenvalue, the uniform so-
+lution is stable; otherwise the system undergoes spinodal
+decomposition. Hence we only need the largest eigenvalue
+of M. As shown in the SM, the matrix M can be diagnon-
+alized exactly. This calculation actually only amounts to
+finding the maximum eigenvalue of a 3 × 3 matrix, which
+can be done straightforwardly, while also perturbatively
+keeping track of orders of τ.
+Spinodal instability – We now illustrate how our ap-
+proach, valid for an arbitrary pairwise potential, can be
+deployed to detect the onset of instability. We consider the
+following short-ranged, weakly repusive interaction poten-
+tial:
+Ψ(r) = ν exp
+�
+−
+1
+1 − (r/r0)2
+�
+,
+(29)
+0.00
+0.25
+0.50
+0.75
+1.00
+k
+−0.10
+−0.05
+0.00
+0.05
+0.10
+λmax
+τ=0.20
+0.00
+0.25
+0.50
+0.75
+1.00
+k
+τ=0.50
+Fig. 1: The value of the largest eigenvalue λmax as a function of
+the wavevector k for ν = 10, r0 = 2, varying τ and φ (blue solid
+line). The orange dotted line represents the short length-scale
+cut-off as we are only concerned with length-scales beyond the
+particle size r0.
+characterised by its strength ν and range r0, whose deriva-
+tives are continuous at any order for r within [0, r0]. The
+maximum eigenvalue of the corresponding growth rate ma-
+trix M, expanded analytically to first order in the persis-
+tence time τ, is plotted against wavevector k in Fig. 1
+for various values of τ and volume fraction φ = 2r0N/V ,
+where 2r0 is taken as the size of the particle in 1D and
+V = L is the system volume. As expected [15], the sys-
+tem exhibits a spinodal instability at high τ and packing
+fraction φ, for which the linear perturbation ε is unstable,
+namely limk→0 λmax(k) = 0+.
+Fig. 2 shows the stabil-
+ity diagram based on the sign of the longest wavelength
+perturbation, λmax(2π/L), as a function of τ and φ, show-
+ing the transition between a uniform stable and a phase-
+separated state. (Note that we could equally have chosen
+the fastest growing mode to locate the spinodal.)
+It is notable that our theory of interacting AOUPs
+shows a spinodal instability. As in equilibrium systems
+with attractions, it does this even at lowest order in the
+virial-type expansion in powers of density. Just as holds
+there, finding the instability requires using a low-density
+theory at finite densities. However the purpose of this ap-
+proach is not to gain quantitative predictions about the ex-
+act location of the spinodal curve, but rather to establish
+that the macroscopic conditions for phase separation can
+be satisfied, by considering microscopic interaction laws
+and dynamics. Notably, unlike in equilibrium, the spin-
+odal instability happens here for purely repulsive interac-
+tions – a key feature of MIPS [15]. Our microscopic calcu-
+lation complements previous viewpoints based on effective
+quasi-equilibrium attractions [31], and/or collisional slow-
+ing down [16,32].
+Nonetheless, there are several caveats and limitations to
+our method. Firstly, the chosen potential is not strictly
+hard-core, but depends on the strength parameter ν. In
+fact it is not possible to implement a true hard-core po-
+tential due to the nature of the small τ expansion for the
+two-particle density, as there are terms directly propor-
+tional to Ψ or its derivatives.
+Unsurprisingly then, the
+p-5
+
+Y. I. Li et al.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+φ
+10−1
+100
+τ
+Phase separation
+Uniform
+0
+λmax
+Fig. 2: Stability diagram in τ − φ space showing λmax(2π/L)
+for φ = 0.6, L = 200, ν = 10, r0 = 2. The white region in the
+top-right corner corresponds to when α < 0, where the method
+breaks down, as it is no longer in the small τ limit. The solid
+black line is the spinodal instability.
+results presented here do depend on ν. While overly large
+ν will break the small-τ expansion, we have checked that a
+range of moderate values produce stability diagrams qual-
+itatively similar to Fig. 2. Secondly, for some parameters,
+we found that the largest eigenvalue stays negative at low
+wavenumber but turns positive (unstable) at large ones,
+resembling an upside-down version of the right frame of
+Fig. 1. At first sight this might be taken as evidence of
+new physics in the form of microphase separation at finite
+wavenumber, an outcome generically predicted by some
+field-theoretic models [14]. However, when this happens
+the wavenumbers in question (at or around the orange line
+in Fig. 1) are comparable to the inverse particle separation
+1/r0. As the theory proposed here is a macroscopic one,
+it may not operate reliably in this large k region, so we
+do not take this as evidence of microphase separation in
+repulsive AOUPs.
+Conclusion – Inspired by liquid-state theory of equi-
+librium systems, we have started with the hierarchy of
+density correlations for AOUPs, and closed it at second
+order using a quasistatic approximation. This approach is
+closely related to the virial expansion in the equilibrium
+case. It yields a mean field theory for the one-point den-
+sity from first principles. We have analysed the stability
+of the uniform solution by looking at the largest eigen-
+value of the growth matrix, and determined the onset of
+a spinodal instability.
+Our method deals directly with particle velocities as well
+as positions. Perhaps more importantly, it is able to cap-
+ture (to lowest order in density) the physics of steep, short-
+ranged interactions.
+In that respect, it is distinct from
+standard coarse-graining methods for non-equilibrium sys-
+tems, such as those based on Dean’s equation [33], which
+starts as an exact representation but upon assuming a
+smooth (coarse-grained) density becomes an expansion in
+weak or slowly varying interaction forces.
+The Dean’s
+approach neglects strong, short-range interactions that
+change the statistics of close encounters even when the
+density, and hence the average interaction force, is small.
+In equilibrium statistical mechanics, the virial expansion
+is designed to handle exactly this situation. We have pre-
+sented the leading order counterpart of this, for an active
+system comprising interacting AOUPs.
+There are some limitations to our method. Firstly, the
+two-particle stationary solution used in the quasistatic
+ansatz is not exact, but rather a small-persistence approx-
+imation, unlike the equilibrium case. Secondly, in truncat-
+ing the density-correlation hierarchy at the second order,
+we assumed that the number density is low. Hence, even
+assuming small persistence, our approach is quantitatively
+accurate only in the dilute limit. Nonetheless, our method
+gives bottom-up confirmation of how phase separation can
+occur in purely repulsive active systems, and it can be used
+to study how the spinodal density varies with interaction
+parameters and with the persistence time.
+∗ ∗ ∗
+The authors acknowledge insightful discussions with
+Alexander Grosberg, Jean-Francois Joanny and Marius
+Bothe.
+YIL acknowledges support from Royal Society
+grant (RP\R1\180165).
+´EF acknowledges support from
+the Luxembourg National Research Fund (FNR), grant
+reference 14389168.
+Work funded in part by the Euro-
+pean Research Council under the EU’s Horizon 2020 Pro-
+gramme (Grant number 740269).
+RG-M acknowledges
+support from a St John’s College Research Fellowship,
+University of Cambridge.
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+p-7
+
+epl draft
+Supplementary material:
+“Towards a liquid-state theory for active matter”
+Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E. Cates1 and ´Etienne
+Fodor2
+1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,
+Cambridge CB3 0WA, UK
+2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxem-
+bourg, Luxembourg
+3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK
+PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics
+Abstract – Supplementary material to “Towards a liquid-state theory for active matter”.
+Overdamped passive Brownian particles: Quasistatic ansatz for density cor-
+relations. –
+In this section, we demonstrate that, for overdamped passive Brownian par-
+ticles (PBPs), one recovers the virial expansion of the free energy when using the quasistatic
+ansatz p2(x1, x2, t) = p1(x1, t)p1(x2, t)pss(x1, x2). We consider the following equations of
+motion for N PBPs with pair-interaction potential Ψ:
+˙xi = −∂xiU +
+√
+2TΛi,
+U =
+N
+�
+i=1
+N
+�
+j<i
+Ψ(|xi − xj|),
+(S1)
+where we have set the mobility to unity, so that T is the diffusion constant, and Λi is a
+Gaussian white noise. Similar to the main text, we write down the corresponding hierarchy
+for one- and two-particle densities, assuming that the density is sufficiently low to ignore
+contributions from three-particle density:
+∂tp1(x1, t) = T∇2
+x1p1(x1, t) + ∇x1 ·
+�
+p2(x1, x2, t)∇x1Ψ(r)dx2,
+∂tp2(x1, x2, t) = T
+�
+∇2
+x1 + ∇2
+x2
+�
+p2(x1, x2, t) +
+�
+i=1,2
+∇xi · (p2(x1, x2, t)∇xiΨ(r)),
+(S2)
+where r = |x1 −x2|. In the quasistatic limit, where the interaction time is much faster than
+the free particle travel time, it is reasonable to suppose that every pair of particles reach
+steady state. This inspires the following ansatz to close the density hierarchy [1,2]:
+p2(x1, x2, t) = p1(x1, t) p1(x2, t) exp(−Ψ(r)/T),
+(S3)
+where we have used that the Boltzmann distribution is the the steady-state distribution of
+two particles. Substituting this ansatz into the p1 equation, we have
+∂tp1(x1, t) = T∇2
+x1p1(x1, t) + ∇x1 ·
+�
+p1(x1, t)
+�
+p1(x2, t)e−Ψ(r)/T ∇x1Ψ(r)dx2
+�
+.
+(S4)
+p-1
+arXiv:2301.12155v1  [cond-mat.soft]  28 Jan 2023
+
+Y. I. Li et al.
+Notice that e−Ψ(r)/T ∇x1Ψ(r) = −T∇x1f(r), where f(r) = exp(−Ψ(r)/T) − 1 is usually
+called the Mayer-f function [3]. Changing variables from x2 to r = x1 −x2 in the integrand
+of eq. (S4), and integrating by parts, we get
+1
+T ∂tp1(x1, t) = ∇2
+x1p1(x1, t) − ∇x1 ·
+�
+p1(x1, t)
+�
+f(r)∇x1p1(x1 − r, t)dr
+�
+.
+(S5)
+Thus far, we have closed the density hierarchy and obtained a mean-field equation for the
+one-particle density. In the following, we seek to connect eq. (S5) with existing knowledge
+of equilibrium physics of interacting gases, in particular, the virial expansion.
+Virial expansion.
+We assume that the density distribution ρ does not vary dramatically
+over the length-scale of the pair-potential. Therefore, expanding p1(x1−r) in eq. (S5) around
+x1 yields
+∂tp1 = ∇ · (p1∇µ),
+µ = T(log p1 + 2Bp1),
+B = −1
+2
+�
+f(r)dr.
+(S6)
+Here B(T) is the second virial coefficient in equilibrium statistical physics [3]. The stationary
+solution minimises the following free energy:
+F = T
+� �
+p1 log p1 + Bp2
+1
+�
+dx.
+(S7)
+The lack of gradient terms in the free energy is a result of neglecting higher-order terms in
+the expansion of p1(x1 − r). Assuming that the stationary state is uniform with density
+ρ = N/V , we obtain
+F(N, V, T) = TN log(N/V )
+�
+��
+�
+ideal gas
++
+TBN 2/V
+�
+��
+�
+virial correction
+.
+(S8)
+This is exactly the equilibrium virial expansion to second order [3].
+All-gradient expansion.
+To explore the effects of higher-gradient terms, we expand
+p1(x − r) in eq. (S5) to all orders in gradient:
+p1(x − r) = e−r·∇xp1(x),
+(S9)
+where e−r·∇x = �
+n
+(−r·∇x)n
+n!
+. Substituting into eq. (S5), we obtain
+1
+T ∂tp1 = ∇2p1 + 2∇ ·
+�
+p1∇
+� ¯Bp1
+��
+,
+¯B = −1
+2
+�
+f(r)e−r·∇xdr,
+(S10)
+where the second virial coefficient has the same form as in eq. (S6) but ¯B is now an operator.
+Proceeding as in the previous section, the chemical potential µ has the same form as eq. (S6).
+In Fourier space, ¯Bk =
+�
+eik·x ¯Bdx is the Fourier transform of the Meyer-f function. Note
+that ¯Bk=0 = B, since we only captured the zeroth mode when truncating the gradient
+expansion at the lowest order. The corresponding free energy is
+¯F = T
+�
+p1 log p1dx
+�
+��
+�
+ideal gas
++ T
+�
+p1,k ¯Bk p1,−k
+dk
+(2π)d
+�
+��
+�
+interactions
+.
+(S11)
+Overall, the gradient expansion is a natural extension of the equilibrium virial expansion
+for non-uniform densities. We note that one cannot trust the high-k behaviour of B(k), as
+it is sensitive to the details of the potential at close range. However, since we are always
+interested in length-scales much larger than the range of the potential (i.e.
+size of the
+individual particles), we do not need to be concerned with such a high-k behaviour.
+p-2
+
+Supplementary material: “Towards a liquid-state theory for active matter”
+Steady state of AOUPs: Small-τ expansion. –
+In this section, we derive the sta-
+tionary probability as an expansion at small τ . As opposed to [4], where the expansion was
+performed in the (x, ˙x) space, we consider here expansion in (x, v) space. This is necessary
+as we need the steady-state two-particle distribution in (x, v) space in the quasistatic ansatz.
+We start from the Fokker-Planck equation for two particles in one spatial dimension:
+∂tP =
+�
+i=1,2
+LiP,
+Li = D
+τ 2 ∂2
+vi +
+�1
+τ ∂vi − ∂xi
+�
+vi + ∂xi(∂xiΨ).
+(S12)
+Scaling units as v → v
+�
+τ/D, x → x/
+√
+τD, t → t/τ, and Ψ → Ψ/D, we obtain
+Li = Li,0 + √τLi,1 + τLi,2,
+(S13)
+where
+Li,0 = ∂2
+vi + ∂vivi,
+Li,1 = −∂xivi,
+Li,2 = ∂xi(∂xiΨ).
+(S14)
+We assume the following τ-expansion for the stationary distribution:
+p2,ss = e− 1
+2(v2
+1+v2
+2)−Ψ
+�
+1 +
+�
+n
+(√τ)nAi({xi, vi})
+�
+,
+(S15)
+where Ai is the i-th order term. Solving order by order, we get
+p2,ss ∝ exp
+�
+−Ψ(r) − 1
+2
+�
+v2
+1 + v2
+2
+��
+×
+�
+1 + √τ(v1 − v2)Ψ′ + τ
+�(v1 − v2)2
+2
+((Ψ′)2 − Ψ′′) − 2(Ψ′)2 + 3Ψ′′
+�
++ o(τ)
+�
+,
+(S16)
+where Ψ′ = dΨ/dr. This result is used to derive the quasistatic approximation in eq. (12)
+of the main text.
+The interaction matrix in Fourier-Hermite basis. –
+In this section, we show the
+details of the change of basis to the Fourier-Hermite basis. Recall that the perturbation
+density field ε can be decomposed in the Fourier-Hermite basis as
+ε(x, v) =
+�
+n
+�
+εkn(t)ψnk(x, v)dk,
+ψnk = e−ik(x+αv)Un(v − 2iαk),
+(S17)
+where the Hermite function Un is defined in eq. (22) of the main text. Substituting the
+decomposition into eq. (19) in the main text, we get
+�
+n
+�
+( ˙εkn + εknλnk)ψnk(x, v)dk = −τρ
+�
+n
+�
+ikεknU0(v)dk
+�
+ψnk(x, v − w)dw
+×
+�
+2La,kw + 2Lb,k(−ik) + Lc,k(−ik)w2�
+,
+(S18)
+where
+La,k = √τ
+� ∞
+0
+dr (Ψ′)2 e−Ψeirk,
+Lb,k = −
+� ∞
+0
+drf0(r)eirk,
+f0(s) =
+� ∞
+s
+dr Ψ′e−Ψ �
+1 − τ2(Ψ′)2 + τ3Ψ′′�
+,
+Lc,k = −
+� ∞
+0
+dr f1(r)eirk,
+f1(s) = τ
+� ∞
+s
+drΨ′((Ψ′)2 − Ψ′′)e−Ψ.
+(S19)
+p-3
+
+Y. I. Li et al.
+We then multiply both sides of eq. (S18) by the dual basis ¯ψmk′, as defined in eq. (24) of
+the main text, and integrate over the (x, v). By orthogonality of the basis, as outlined in
+eq. (25) of the main text, we have
+˙εkm = −λkmεkm +
+�
+n
+�
+M (1)
+kmn + M (2)
+kmn + M (3)
+kmn
+�
+εkn,
+(S20)
+where
+M (1)
+kmn = 2τρLa,k(−ik)
+�
+dv U0(v) ¯Um(v − 2iαk)
+�
+dw w eiαkwUn(v − w − 2iαk),
+M (2)
+kmn = 2τρLb,k(−ik)2
+�
+dv U0(v) ¯Um(v − 2iαk)
+�
+dw eiαkwUn(v − w − 2iαk),
+M (3)
+kmn = τρLc,k(−ik)2
+�
+dv U0(v) ¯Um(v − 2iαk)
+�
+dw w2eiαkwUn(v − w − 2iαk).
+(S21)
+We first perform all the w-integrals:
+In
+0 (v, k) ≡
+�
+dw eikαwUn(v − w − 2iαk) = e
+3
+2 α2k2+iαkv(−iαk)n
+In
+1 (v, k) ≡
+�
+dw
+√
+2π w eiαkwUn(v − w − 2iαk) = −e
+3
+2 α2k2+iαkv(−iαk)n−1 �
+iαkv + n + α2k2�
+In
+2 (v, k) ≡
+�
+dw
+√
+2π w2eiαkwUn(v − w − 2iαk)
+= e
+3
+2 α2k2+iαkv(−iαk)n−2 �
+n(n − 1) − α2k2(1 + (v − iαk)2) + 2inαk(v − iαk)
+�
+.
+(S22)
+Performing next the v-integrals, we find:
+M (1)
+kmn = 2τρLa,k(−ik)
+�
+dv
+√
+2π e− 1
+2 v2Hm(v − 2iαk)In
+1 (v, k)
+= 2τρLa,k
+(−iαk)n+m
+αm!
+eα2k2(m − n),
+M (2)
+kmn = −2τρLb,kk2
+�
+dv
+√
+2π e− 1
+2 v2Hm(v − 2iαk)In
+0 (v, k)
+= −2τρLb,kk2 (−iαk)n+m
+m!
+eα2k2,
+M (3)
+kmn = τρLc,k(−ik)2
+�
+dv
+√
+2π e− 1
+2 v2Hm(v − 2iαk)In
+2 (v, k)
+= τρLc,k
+(−iαk)n+m
+α2m!
+eα2k2 �
+m(m − 1) + n(n − 1) − 2mn − 2α2k2�
+.
+(S23)
+If the overall growth rate matrix Mkmn = −λkmδmn + M (1)
+kmn + M (2)
+kmn + M (3)
+kmn has no
+positive eigenvalue, the uniform solution is stable; otherwise the system undergoes spinodal
+decomposition. Hence, all we need is the sign of the largest eigenvalue of the M matrix.
+Despite the appearance of M as an infinite-dimensional matrix, it is in fact possible to
+find another set of basis where M is block diagonal. Indeed, {M (1), M (2), M (3)} are all
+linear combinations of the following matrices:
+(−iαk)m+n
+m!
+, (−iαk)m+n
+m!
+m, (−iαk)m+n
+m!
+n, (−iαk)m+n
+m!
+m(m − 1), (−iαk)m+n
+m!
+n(n − 1),
+(S24)
+p-4
+
+Supplementary material: “Towards a liquid-state theory for active matter”
+so that they can be written as
+M (1) = 2τρLa,k
+α
+eα2k2 (u1v⊺
+0 − u0v⊺
+1) ,
+M (2) = −2τρLb,kk2eα2k2u0v⊺
+0,
+M (3) = τρLc,k
+α2
+eα2k2 �
+u2v⊺
+0 + u0v⊺
+2 − 2u1v⊺
+1 − 2α2k2u0v⊺
+0
+�
+.
+(S25)
+where ⊺ denotes matrix transpose, and
+(u0)m = (−iαk)m
+m!
+,
+(u1)m = (−iαk)m
+m!
+m,
+(u2)m = (−iαk)m
+m!
+m(m − 1),
+(v0)n = (−iαk)n,
+(v1)n = (−iαk)nn,
+(v2)n = (−iαk)nn(n − 1),
+(S26)
+Observe that when acting on an arbitrary vector y, uv⊺y = (v⊺y)u, meaning that a matrix of
+this form projects all vectors onto the direction of u. We introduce ∆ = M (1) +M (2) +M (3)
+in this section, and, grouping the terms, we have
+e−(αk)2∆ = h0u0v⊺
+0 + h10(u1v⊺
+0 − u0v⊺
+1) + h11u1v⊺
+1 + h2(u2v⊺
+0 + u0v⊺
+2),
+(S27)
+where
+h0 = −2k2τρ (Lb,k + Lc,k) ,
+h10 = 2τρLa,k/α,
+h11 = 2τρLc,k/α2,
+h2 = τρLc,k/α2.
+(S28)
+We now consider the following spanning set: (v0, v1, v2, ...), where (vj)n = (−iαk)nPn,j
+(recall Pn,j = n!/(n − j)! is the permutation coefficient). One can easily check that it is
+a spanning set by spotting that the first j elements of vj are zeros, and hence no vector
+in this set can be written as a linear combination of others. Furthermore, let us introduce
+(q0, q1, q2, ..) as the dual basis of {vj}, such that v⊺
+i qj = δij. We proceed to write ∆ in the
+new basis: ˜∆ij ≡ v⊺
+i ∆ qj, where
+˜∆ = eα2k2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+h0v⊺
+0u0 + h10v⊺
+0u1 + h2v⊺
+0u2
+−h10v⊺
+0u0 + h11v⊺
+0u1
+h2v⊺
+0u0
+0
+. . .
+h0v⊺
+1u0 + h10v⊺
+1u1 + h2v⊺
+1u2
+−h10v⊺
+1u0 + h11v⊺
+1u1
+h2v⊺
+1u0
+0
+. . .
+h0v⊺
+2u0 + h10v⊺
+2u1 + h2v⊺
+2u2
+−h10v⊺
+2u0 + h11v⊺
+2u1
+h2v⊺
+2u0
+0
+. . .
+0
+0
+0
+0
+. . .
+...
+...
+...
+...
+...
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(S29)
+which is nonzero only in the top 3 × 3 due to the orthogonality relation between v’s and q’s.
+p-5
+
+Y. I. Li et al.
+Computing the inner products and setting ˜k = αk for convenience, we have
+v⊺
+0u0 =
+�
+n
+(−˜k2)n
+n!
+= e−˜k2,
+v⊺
+1u0 = v⊺
+0u1 =
+�
+n
+(−˜k2)n
+n!
+n = −˜k2 �
+n
+(−˜k2)n
+n!
+= −˜k2e−˜k2,
+v⊺
+2u0 = v⊺
+0u2 =
+�
+n
+(−˜k2)n
+n!
+n(n − 1) = ˜k4e−˜k2,
+v⊺
+1u1 =
+�
+n
+(−˜k2)n
+n!
+[n(n − 1) + n] = (˜k4 − ˜k2)e−˜k2,
+v⊺
+1u2 = v⊺
+2u1 =
+�
+n
+(−˜k2)n
+n!
+[n(n − 1)(n − 2) + 2n(n − 1)] = (−˜k6 + 2˜k4)e−˜k2,
+v⊺
+2u2 =
+�
+n
+(−˜k2)n
+n!
+[n(n − 1)(n − 2)(n − 3) + 4n(n − 1)(n − 2) + 2n(n − 1)]
+= (˜k8 − 4˜k6 + 2˜k4)e−˜k2.
+(S30)
+The resulting 3 × 3 matrix appearing in eq. (S29) can then be written as
+�
+�
+�
+�
+�
+h0 − h10˜k2 + h2˜k4
+−h10 − h11˜k2
+h2
+−h0˜k2 + h10
+�
+˜k4 − ˜k2�
++ h2
+�
+−˜k6 + 2˜k4�
+h10˜k2 + h11
+�
+˜k4 − ˜k2�
+−h2˜k2
+h0˜k4 + h10
+�
+−˜k6 + 2˜k4�
++ h2
+�
+˜k8 − 4˜k6 + 2˜k4�
+−h10˜k4 + h11
+�
+−˜k6 + 2˜k4�
+h2˜k4
+�
+�
+�
+�
+� .
+(S31)
+To compute the eigenvalues of the full matrix M = D + ∆, where Dkmn = (−n − ˜k2)δmn,
+we also need to write D in the new basis:
+˜Djl = v⊺
+j D ql = −δj+1,l − (j − ˜k2)δjl,
+(S32)
+where we have used n(vj)n = (vj+1 + jvj)n, or explicitly
+˜D =
+�
+�
+�
+�
+�
+�
+�
+�
+−˜k2
+−1
+0
+. . .
+0
+−1 − ˜k2
+−1
+0
+. . .
+0
+0
+−2 − ˜k2
+−1
+0
+. . .
+...
+...
+...
+...
+...
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(S33)
+Notice that ˜D is upper triangular and therefore the eigenvalues are simply the diagonal
+elements −n − ˜k2, which are the same as for D.
+Having written both the diagonal and non-diagonal parts in the new basis, we proceed
+to solve for the eigenvalues λ of �
+M = ˜D + ˜∆, satisfying det
+�
+�
+M − λI
+���
+= 0, where I is the
+identity matrix. Observe that �
+M is of the following form,
+�
+M =
+�
+�X
+Y
+0
+Z
+�
+� ,
+(S34)
+where X is a 3 × 3 matrix. One can prove that det
+�
+�
+M − λI
+�
+= det(X − λI) det(Z − λI).
+So we only need to solve for the eigenvalues of X and Z separately, but we already know
+p-6
+
+Supplementary material: “Towards a liquid-state theory for active matter”
+the eigenvalues of Z as it is upper-triangular with diagonal elements (−n − ˜k2) for n ≥ 3.
+Hence what remains are the eigenvalues of X, which can be solved numerically easily as it
+is only 3 × 3. Overall, we have reduced the problem of finding the maximum eigenvalue of
+an infinite-dimensional matrix to diagonalising the 3 × 3 matrix ˜∆ + ˜D. We then compute
+the largest eigenvalue as
+λmax = −k2
+�
+1 + τ 1 + 2(B2(k) − 2A1(k)2 + C2(k)) − 4k2B0(k)(B2(k) + C2(k))
+1 − 2B0(k)k2
+�
++ o(τ),
+(S35)
+where
+B0(k) = Lb,k
+��
+τ=0,
+B2(k) = (d/dτ)Lb,k,
+A1(k) = (1/√τ)La,k,
+C2(k) = (1/τ)Lc,k.
+(S36)
+REFERENCES
+[1] Grosberg A. Y. and Joanny J.-F., Phys. Rev. E, 92 (2015) 032118.
+[2] Ilker E. and Joanny J.-F., Phys. Rev. Research, 2 (2020) 023200.
+[3] Kardar M., Statistical physics of particles (Cambridge University Press) 2007.
+[4] Fodor ´E., Nardini C., Cates M. E., Tailleur J., Visco P. and van Wijland F., Phys.
+Rev. Lett., 117 (2016) 038103.
+p-7
+

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+Two statistical regimes in the transition to filamentation
+Alexis Gomel,1, 2 Geoffrey Gaulier,1 Debbie Eeltink,1, 2, 3 Maura Brunetti,1, 4 and Jérôme Kasparian
+1, 2, ∗
+1Group of Applied Physics, University of Geneva, 1205 Geneva, Switzerland
+2Institute for Environmental Sciences, University of Geneva, 1205 Geneva, Switzerland
+3Present address: Laboratory of Theoretical Physics of Nanosystems, EPFL, Ch-1015 Lausanne, 10 Switzerland
+4Institute for Environmental Sciences, University of Geneva, 1205 Geneva, Switzerland
+Abstract
+We experimentally investigate fluctuations in the spectrum of ultrashort laser pulses propagating in air, close to the critical
+power for filamentation. Increasing the laser peak power broadens the spectrum while the beam approaches the filamentation
+regime. We identify two regimes for this transition: In the center of the spectrum, the output spectral intensity increases
+continuously. In contrast, on the edges of the spectrum the transition implies a bimodal probability distribution function for
+intermediate incident pulse energies, where a high-intensity mode appears and grows at the expense of the original low-intensity
+mode. We argue that this dual behavior prevents the definition of a univoquial threshold for filamentation, shedding a new
+light on the long-standing lack of explicit definition of the boundary of the filamentation regime.
+∗ jerome.kasparian@unige.ch
+1
+arXiv:2301.01971v1  [physics.optics]  5 Jan 2023
+
+I.
+INTRODUCTION
+Filamentation is a self-guided propagation regime typical of ultrashort, high-power laser pulses [1–4]. Beyond a
+critical power (3 GW in air), the Kerr effect balances diffraction.
+The beam then self-focuses until higher-order
+nonlinear defocusing effects like ionization or the saturation of the Kerr effect [5] come into play.
+The resulting
+dynamical balance gives rise to self-guided light structures known as filaments. Filaments can extend over atmospheric
+scale distances [6, 7], opening the way to various applications from remote sensing [8] to lightning control [9–14],
+THz generation [15–20], fog clearing [21, 22], or the triggering of condensation in sub-saturated atmospheres [8, 23,
+24]. Filaments show very characteristic and even spectacular features, including long-distance propagation, bright
+light emission due to both plasma emission on the side and spectral broadening in the forward direction, and noise
+associated to the shockwave [25–27]. Together, these features give rise to the intuitive notion of a qualitatively distinct
+propagation regime. However, defining a clear threshold for the filamentation regime turns out to be difficult [28],
+especially in focused beams where linear propagation can be sufficient to reach the ionization threshold [29], and even
+produce a denser plasma than Kerr self-focusing does [30]. Similarly, along the propagation axis, defining the onset
+and end of filaments, and therefore their length, requires somewhat arbitrary choices.
+Here, we statistically investigate the transition to filamentation . Shot-to-shot fluctuations of the spectral amplitudes
+of a high-power laser beam, whether filamenting or not, are known to display characteristic probability distribution
+functions (PDF): A regular probability distribution is observed in the center of the spectrum, while on its edge the
+distribution is long-tailed (optical rogue wave [31]) [32]. We extend this analysis to the sub-filamenting regime and to
+the transition to filamentation, in order to characterize how the statistics evolves when the incident power progressively
+reaches the critical power for filamentation. The evolution of spectral intensity PDFs turns out to display typical
+and contrasted signatures on the edges and in the center of the spectrum, respectively, suggesting that transition
+to filamentation cannot be characterized as a whole. By doing so, we suggest that the ambiguities in the definition
+of the limits of the filamentation regime are intrinsic to the physical process rather than instrumental or conceptual
+limitations, shedding a new light on the long-standing lack of explicit definition of the boundary of the filamentation
+regime.
+II.
+EXPERIMENTAL SETUP
+Figure 1. Experimental setup
+The experimental setup (Figure 1) relied on a Coherent Astrella laser delivering ultrashort pulses at 1 kHz repetition
+rate. A half-wave plate installed in a motorized rotating stage ensureing ±1◦ precision, followed by a linear polarizer,
+allowed to adjust the pulse energy with an accuracy of ∼ 10 µJ. The angle-to-energy calibration was performed before
+each experimental run with a powermeter (Coherent PM10). The laser compressor was adjusted so as to slightly chirp
+the pulses to a duration of 55 fs (as measured with PulseCheck, APE Berlin), in order to reach the filamentation
+threshold in air within the tuning range of the energy. The beam was slightly focused from an initial diameter of 11 mm
+at 1/e2 by an f = 2 m BK7 plano-convex lens located approximately at 15 cm after the polarizer and subsequently
+through a 13.5 cm-long turbulent region generated by 48◦C to 565◦C hot air blown transversally by a hot air blower
+(Steinel HL1502 S). 15–27.5 cm downstream of the lens, the beam self-focused and filamentation started if the incident
+peak power was sufficient. The filamentation region was surrounded by a PVC tube of 18 cm diameter to shield the
+beam against air displacements in the room so as to improve its stability. After the end of filamentation region, the
+beam was imaged by a second BK7 lens (f = 20 cm) onto the entrance slit of an OceanOptics USB2000 spectrometer
+providing ∼0.5 nm spectral resolution over the visible range. A neutral density filter prevented the spectrometer from
+saturating. The spectrum was measured and recorded independently for each laser shot. For each incident pulse
+2
+
+energy, 5000 individual spectra were recorded.
+At each wavelength and incident energy, we characterized the shot to shot fluctuations by recording the PDF of
+the spectral intensity as a 100-bin histogram, using identical bins in all conditions to facilitate comparison. In the
+characterization of the PDF, we use among others Hogg’s unbiased kurtosis estimator, defined as
+Hg = U0.05 − L0.05
+U0.5 − L0.5
+− 2.59
+(1)
+where Um and Lm refer to the mean of the upper and lower m quantiles, respectively.
+This means that Um =
+1
+m
+� 1
+1−m F −1(y)dy and Lm = 1
+m
+� m
+0 F −1(y)dy, where F is the cumulative distribution function. For a Gaussian PDF,
+the Hogg estimator is 0.
+III.
+RESULTS AND DISCUSSION
+100
+102
+Intensity (arb.units)
+(a)
+(b) 0.017 mJ
+100
+102
+Intensity (arb.units)
+(c) 0.054 mJ
+(d) 0.102 mJ
+750
+800
+850
+Wavelength [nm]
+100
+102
+Intensity (arb.units)
+(e) 0.130 mJ
+750
+800
+850
+Wavelength [nm]
+(f) 0.173 mJ
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Peak energy [mJ]
+Figure 2. Evolution of the spectrum for increasing peak energy. (a) Average spectra; (b)-(f) Intensity fluctuations for incident
+pulse energies of 0.017, 0.054, 0.102, 0.130, and 0.173 mJ, respectively.
+In each panel, the solid thick line is the average
+intensity, the dark shaded area marks range between the 20th and the 80th percentiles and the light shaded area displays the
+range between the 5th and the 95th percentiles.
+Figure 2a displays the evolution of the typical spectra recorded in the center of the beam after it starts diverging,
+for incident energies ranging from 0.017 to 0.173 mJ/pulse. The dark shaded area of Panels b–f encompasses the
+20th to the 80th percentiles, while the light shadowed ranges from the 5th to the 95th ones, evidencing the typical
+range of the shot-to-shot fluctuations. As is typical for self-phase modulation (SPM) [32, 33], the spectral broadening
+3
+
+is characterized by the formation of an oscillatory plateau. For increasing incident powers, the plateau both rises
+in intensity and broadens by a progressive displacement of its edges away from the laser fundamental wavelength.
+Shot-to-shot fluctuations of the spectrum can therefore be characterized as essentially vertical (i.e., in intensity) in
+the central part of the spectrum, and essentially horizontal (i.e., in spectral range) on its edges.
+As displayed in Figure 2b–f, the shot-to-shot fluctuations can have the same order of magnitude as the mean
+spectral intensity all over the spectrum.
+In particular, above the threshold for appreciable spectral broadening,
+fluctuations reach a factor of 4, ranging from the non-filamenting to the filamenting regimes even at average incident
+pulse energies up to the critical power (0.173 mJ, P = 3.15 GW ∼ Pcr) where the filamentation visually seems to be
+well established. Such behavior randomly alternating filamenting (spectrally broadened) and non-filamenting pulses
+is consistent with the previously observed effect of turbulence on filamentation [34–36]. These fluctuations stem from
+the pre-filamentation fluctuations of the incident laser itself and from the influence of the turbulence, as well as
+the nonlinear transformation of the spectrum performed by the spectral broadening in the high-intensity region of
+propagation.
+The vertical and horizontal behaviors of the shot-to-shot fluctuations result in qualitatively different evolutions of
+the PDF as a function of incident energy (fig. 3). In the center of the spectrum (805 nm, panels g and h), increasing
+the incident pulse energy continuously shifts the peak of the PDF, from low to high spectral intensity. This shift
+becomes faster and faster when moving away from the center of the spectrum (See 785 nm, panels e and f). Reaching
+the spectrum edges, the behavior qualitatively changes, with the PDF becoming bimodal over the transition (773
+nm, panels b,c, and 860 nm, panels i,j). Over the power range where the transition occurs, the low-intensity peak
+progressively vanishes, while the high-intensity peak emerges. This behavior develops over a spectral range of ∼10 nm
+on each side of the spectrum. Even further on the edges (765 nm, panels a,b), the bimodal behavior disappears again.
+The PDF initially peaks close to zero intensity and rises when the broadening becomes sufficient.
+Note that in spite of patterns visually similar to tipping points, these transitions cannot be described as such.
+Rather, each laser pulse is essentially an independent experiment, hitting a fresh parcel of air.
+This experiment
+therefore consists in a series of random probings of the system behavior, with random initial phase profiles. The
+qualitative behaviors associated to each incident pulse energy at each wavelength can be summarized in a “phase
+diagram” (fig. 4a). We identify the regions with negligible broadening, with fully deployed broadening, and two kinds
+of transition regions: either continuous or bimodal. Remarkably, the bimodal transitions regions are characterized by
+a drop in both the skewness (fig. 4b) and Hogg’s unbiased kurtosis estimator Hg [37] (fig. 4c), while the continuous
+transitions are characterized by increasing skewness and Hg. This can be understood by considering the corresponding
+evolutions on fig. 3. In the central region of the spectrum, corresponding to continuous transitions, the PDF tail rises
+on the high-spectral intensity side, resulting in a heavier tail and growing asymmetry. In contrast, the appearance
+and growth of a secondary mode of the PDF tends to occur at the expense of the tails, and simultaneously results
+in a broadening of the central region of the PDF, which explains the smaller skewness and Hogg estimator. Further
+statistical moment like the kurtosis and coefficient of variation (i.e., the ratio between variance and mean) display
+qualitatively similar behaviors, as illustrated in fig. 5.
+From a mechanistic point of view, the bimodal transition corresponds to the wavelength range where the evolution
+of the spectrum for increasing pulse energies mostly occurs as a horizontal fluctuation of the cliff delimiting the
+broadened spectrum, i.e., the creation of new frequencies that previously were virtually absent from the spectrum.
+At these wavelengths, the sharp rise of the spectrum (large
+�� dI
+Idt
+��) followed by a relatively flat plateau of fluctuating
+width ensures the bimodal distribution, since the rising section of the spectrum has a negligible contribution to the
+PDF: The spectrum mainly features two ranges of attainable values, close to zero and on the plateau, respectively,
+while the intensity variations of the plateau itself is of second order. This interpretation also explains why the bimodal
+behavior is observed on a much narrower range of wavelengths around 850 nm (See fig. 4a), where the oscillation is
+sharper and the plateau is much less flat. In contrast, on the plateau, the fluctuations are mostly vertical. They
+consist in intensity fluctuations, i.e., an increased intensity of frequencies pre-existing in the incident pulse, so that
+the transition occurs continuously when the plateau progressively rises.
+These two qualitatively different transition modes to a broadened pulse coexist in the evolution of the same spectrum,
+driven by the same self-phase modulation process, and only some nanometers apart. The discontinuous transition to
+filamentation on the edges of the spectrum where bimodal spectral intensity distributions are observed would point
+to a system where filamenting and non-filamenting regimes are qualitatively distinct, and co-exist over some range of
+input power. In contrast, the central part of the spectrum and its smooth, continuous transition would not allow to
+define a threshold for the filamenting regime. Note that this spectral range bears most of the pulse energy.
+We argue that this dual behavior within the same spectrum during the transition to filamentation intrinsically
+prevents an unambiguous definition of the edges of a filamenting regime that would be qualitatively different from an
+essentially linear extended focus [29].
+4
+
+250
+500
+750
+1000
+1250
+Intensity (arb.units)
+0
+500
+1000
+Count
+765 [nm]
+a)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+250
+500
+750
+1000
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+765 [nm]
+b)
+10−5
+10−4
+10−3
+10−2
+Count normalized
+750
+800
+850
+100
+102
+250
+500
+750
+1000
+1250
+1500
+Intensity (arb.units)
+0
+500
+1000
+Count
+773 [nm]
+c)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+250
+500
+750
+1000
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+773 [nm]
+d)
+10−5
+10−4
+10−3
+10−2
+Count normalized
+750
+800
+850
+100
+102
+500
+1000
+1500
+2000
+Intensity (arb.units)
+0
+500
+1000
+Count
+785 [nm]
+e)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+250
+500
+750
+1000
+1250
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+785 [nm]
+f)
+10−5
+10−4
+10−3
+10−2
+Count normalized
+750
+800
+850
+100
+102
+500
+1000
+1500
+2000
+Intensity (arb.units)
+0
+500
+1000
+Count
+805 [nm]
+g)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+500
+1000
+1500
+2000
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+805 [nm]
+h)
+10−5
+10−4
+10−3
+Count normalized
+750
+800
+850
+100
+102
+50
+100
+150
+200
+250
+Intensity (arb.units)
+0
+500
+1000
+Count
+860 [nm]
+i)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+50
+100
+150
+200
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+860 [nm]
+j)
+10−4
+10−3
+10−2
+10−1
+Count normalized
+750
+800
+850
+100
+102
+Figure 3. Evolution of the histogram of the spectral intensity at (a,b) 765 nm (transition through a bimodal distribution), (c,d)
+773 nm (intermediate regime), (e,f) 785 nm, (g,h) 805 nm (smooth offset of the peak), and (i,j) 860 nm (bimodal transition).
+(a,c,e,g,i): Individual histograms for increasing incident pulse energies. The color scale is the same as in fig. 2. (b,d,f,h,j):
+Colormap of the same data. Each histogram is normalised to 1. Insets display the position of the wavelength in the spectrum
+(See also Figure 2a)
+5
+
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+No broadening
+Smooth transition
+Bimodal transition
+Full broadening
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+0
+1
+2
+3
+4
+Skewness
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+−0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Hogg kurtosis
+Figure 4. (a) "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from
+non-broadened to fully-broadened, with continuous and bimodal transition modes in between.
+(b, c) Borders of the same
+diagram overlaid on (b) the skewness and (c) the Hogg’s unbiased kurtosis estimator [37].
+IV.
+CONCLUSION
+In summary, we characterized the evolution of intensity fluctuations across the spectrum of an ultrashort beam
+propagating in air, for a wide range of powers covering below and up to the critical power for filamentation. While
+on the edges of the spectrum the transition to filamentation is discontinuous and displays a bimodal distribution of
+spectral intensities around the critical power, it is continuous around the fundamental incident laser wavelength. This
+dual behavior provides an explanation for the lack of unambiguous definition of the edges of the filamentation regime
+6
+
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+0
+5
+10
+15
+20
+25
+30
+Kurtosis
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+0
+50
+100
+150
+200
+250
+300
+Variance/Mean
+Figure 5. "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from non-
+broadened to fully-broadened, with continuous and bimodal transition modes in between overlaid on (a) the kurtosis and (b)
+the the coefficient of variation (variance/mean).
+in experiments or numerical simulations in spite of the intuitive understanding that filamentation qualitatively differs
+from a more linear propagation regime.
+Funding. Swiss National Science Foundation (SNF, grant 200020-175697)
+Acknowledgements. Experimental support was provided by Michel Moret.
+Disclosures. The authors declare no conflicts of interest.
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+

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+Prepared for submission to JINST
+Study on U/Th residual radioactivity in acrylic from
+surface treatment
+Yuanxia Li,𝑎 Xiaohui Qian,𝑎 Xiaolan Luo,𝑎 Jie Zhao,𝑎,1 Gaofeng Zhang,𝑏 Xiaoyan Ma,𝑎
+Yuekun Heng,𝑎 Liangjian Wen,𝑎 Monica Sisti,𝑐 Frédéric Perrot,𝑑 Hongqiang Tang𝑏
+𝑎Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
+𝑏Donchamp New Material Technology Co. Ltd, Taixing 225400, China
+𝑐INFN Milano Bicocca and Università di Milano-Bicocca, Milano, Italy
+𝑑Univ. Bordeaux, CNRS, CENBG, UMR 5797, F-33170 Gradignan, France
+E-mail: zhaojie@ihep.ac.cn
+Abstract: Acrylic is widely used as material for the target container in low background experiments
+due to its high light transparency and low intrinsic radioactivity. However, its surface can be easily
+contaminated during production, so careful treatment of the surface is essential to avoid direct
+contamination of the target. The Jiangmen Underground Neutrino Observatory will use about 600 t
+of acrylic to build the spherical vessel of 35.4 m in diameter for a 20 kt liquid scintillator (LS).
+Since acrylic will contact the LS directly, the cleanliness of the its surface is quite important for the
+radiopurity of the LS. A new method for measuring the radioactivity of 238U and 232Th in acrylic to
+sub-ppt (< 10−12 g/g) was developed, and it is crucial for the acrylic radioactivity screening in this
+study. We performed many background tests on different surface treatments, and the recommended
+procedure for the treatment of acrylic to achieve low radioactivity and high light transparency could
+be applicable to other low background experiments.
+1Corresponding author.
+arXiv:2301.04902v1  [physics.ins-det]  12 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Screening of surface treatment material by gamma spectrometer
+2
+3
+Method for measuring U/Th in acrylic by ICP-MS
+3
+4
+Study of the procedure for acrylic surface treatments
+5
+4.1
+Solution absorption in acrylic
+5
+4.2
+Contamination from different surface treatments
+7
+4.3
+Protection film
+8
+5
+Summary
+9
+1
+Introduction
+The Jiangmen Underground Neutrino Observatory (JUNO) is a multi-purpose neutrino experiment,
+which will build the world’s largest liquid scintillator (LS) detector. An acrylic vessel is used as
+the 20 kt LS container with 35.4 m in diameter and 120 mm in thickness because of its high light
+transparency and low intrinsic radioactivity [1].
+To reach the physics target, the average radio-purity of the acrylic is required to be less than
+1 ppt [2]. As shown in Ref. [3], the radio purity in the bulk has reached the requirement. However,
+the distribution of radioactivity along the thickness is not extremely uniform. The bulk measurement
+of samples with several centimeters of thickness can not reflect the surface contamination, since
+the thickness of external contamination on the acrylic surface usually reaches nanometre to micron
+level, and the surface contamination will result averaged out in the bulk measurement. If the external
+radioactive contamination stays on the surface, only gammas and few alphas/betas from radioactive
+decays can be emitted into the LS and rejected by the effective fiducial volume cut by the off-line
+analysis. However, based on the experiences at Borexino [4] and SNO+ [5], the U/Th daughters
+near the inner surface of the vessel could leak into the LS by diffusion and convection during stable
+running, thus making the acrylic vessel a stable contamination source for the LS. Borexino put
+great efforts on controlling the temperature distribution to avoid convection in LS; however, it is
+impossible to repeat the same thing at JUNO due to the large size of the experiment (70 times
+Borexino target). Therefore, the radiopurity of the inner surface of acrylic is quite important for
+JUNO, especially for the physic topics in the low energy regions, such as solar neutrino studies [1].
+The JUNO acrylic vessel is produced with 265 pieces because of its large size, as shown in
+Fig. 1. Each acrylic panel is firstly produced in a carefully cleaned flat mold and then shaped to
+bend board at high temperature at the company; finally it is bonded directly at JUNO site [1]. The
+surface of the acrylic panel is exposed to air and covered with a high-temperature resistant cloth
+– 1 –
+
+during the bending, thus the surface is eventually contaminated and not as smooth as the flat panel.
+Surface treatments are performed before shipment to the JUNO site. Typical protocols to remove
+surface contaminations include the following steps:
+• The highest contaminations can reach tens of µm depths from the surface [6]. Thus, at least
+0.1 µm thickness of the surface of acrylic is firstly removed by sanding with special paper.
+• The surface is not so smooth after sanding, so polishing is needed to improve the transparency.
+The polishing is done with a wool wheel and polishing fluid.
+• The surface should be well cleaned with detergent and deionized water after polishing.
+• Finally, a thin film is used to cover the acrylic surface to avoid external contamination during
+transportation and installation.
+All tools and detergents have risks to introduce more contamination on the surface, therefore
+each surface treatment should be well studied to minimize those risks.
+Figure 1. The acrylic vessel is produced with 265 pieces due to its very large size, as shown in the left figure.
+Each panel is produced separately, and all the panels will be finally bonded to a sphere at JUNO site. One
+JUNO panel is shown in the right figure as an example.
+The paper is organized as follows. The surface treatment material screening is described in
+Sec 2. The new method for measuring 238U and 232Th (U/Th) in acrylic to sub-ppt level is described
+in Sec 3. Various tests on radioactivity from each step of the above treatment protocol are discussed
+in Sec 4, and the recommended procedure is also shown in the same section. The concluding
+remarks are in Sec. 5.
+2
+Screening of surface treatment material by gamma spectrometer
+The main tools and detergents used for surface treatments are sanding papers, polishing fluid,
+Alconox powder [7], deionized water and protection film. We used a gamma spectrometer to screen
+the radiopurity of the tools and detergents first. We have one HPGe facility at ground level at
+IHEP [8]. The gamma spectrometer plays an important role in the primary screening of the raw
+– 2 –
+
+0
+0
+0
+0
+0
+Q
+0
+4
+0
+.materials at JUNO, and the advantage is that samples can be screened directly without any pre-
+treatment and with no damage to the sample. The principle of gamma spectroscopy is to measure
+the emitted gamma from the decay chain, and it is more sensitive to gammas with higher energy
+due to lower background. For example, the detector is more sensitive to the lower part of the
+uranium chain starting from 226Ra, and the 238U concentration can be derived by assuming secular
+equilibrium.
+The measured results for the tools and detergent used for acrylic surface treatment are sum-
+marized in Table 1. We found that special mirror wax purchased from Saint-Gobain [9] has lower
+radioactivity for 40K than one-step fast wax, due to the different chemical components. The sand-
+paper with a green base plate has lower radioactivity and better sand fastening than the one with a
+black base plate for similar mesh numbers. Regarding the Alconox powder, the first measurement
+performed at IHEP gave an upper limit of several tens to hundreds of ppb level for thorium and
+uranium concentrations, respectively, due to the limited sensitivity of HPGe. The same sample was
+further measured by HPGe at China Jinping Underground Laboratory (CJPL) with better shielding
+and sensitivity [10] down to 10 ppb for 232Th. From the obtained results, there is a higher uranium
+concentration than thorium in the Alconox powder. The polishing fluid and sandpaper with lower
+radioactivity are selected for the surface treatments on acrylic.
+After surface treatment, the acrylic surface will be protected with a thin film to avoid dust and
+radon daughters deposition on the surface during the handling operations for transportation and
+installation. The film on the inner surface will be removed by the final cleaning with high pressure
+water jet after installation, and there is no possibility for manual operation inside the acrylic vessel.
+There are two types of protection film that can be used to protect the acrylic surface, ordinary
+polyethylene (PE) film with few glue on the surface and adhesive masking paper film. The PE film
+is better in toughness, but it is not easy to be removed by the water jet. On the contrary, the paper
+film with few water soluble glue on the surface can be easily removed by water washing since the
+glue is soluble in water, but overall its mechanical strength is worse. The radioactivity screening of
+the glue that is used on the PE film showed only upper limits for U/Th at the level of hundreds of
+ppb. Concerning the adhesive masking paper film (purchased from [11]), we could only measure
+the paper film and the glue together inside the HPGe detector. The water soluble glue itself was
+further measured by ICP-MS, as it will be discussed later.
+3
+Method for measuring U/Th in acrylic by ICP-MS
+Different from gamma spectroscopy, the principle of the Inductively Coupled Plasma Mass Spec-
+troscopy (ICP-MS) is to detect the nuclei according to their exact atomic weight. ICP-MS can
+measure the isotopes 238U and 232Th directly without other information on the chain, and the sen-
+sitivity can reach the sub-ppt level. However, the sample measured by ICP-MS must be in liquid
+form, so a careful and optimized pre-treatment on the sample is needed.
+For the radio purity measurement on acrylic surface samples, we have developed a new method
+based on microwave ashing and ICP-MS equipment. The design for the pre-treatment flow is shown
+in Fig. 2. Both the acrylic sample and the tracers (the natural-non-existing nuclei 229Th and 233U)
+are ashed by the machine in a quartz vessel. The residual is collected by soaking with 35% HNO3.
+– 3 –
+
+Table 1.
+Material screening by gamma spectroscopy on tools and detergents used for the acrylic surface
+treatment. If not differently mentioned, the sample was measured at IHEP. The conversions between Bq/kg
+and mass concentration units for U/Th are given: 1 Bq/kg of 232Th activity in a material is equal to
+2.5 × 10−7 g/g (or 250 ppb) of 232Th mass concentration in that material; similarly, 1 Bq/kg of 238U means
+8.1 × 10−8 g/g (or 81 ppb) of 238U.
+238U chain [Bq/kg]
+232Th chain [Bq/kg]
+40K
+214Bi/214Pb
+226Ra
+212Bi/208Tl
+[Bq/kg]
+Polishing
+Special mirror wax
+2.8±0.2
+2.9±1.3
+5.8±0.3
+3.8±0.9
+One step fast wax
+3.1±0.3
+5.0±1.7
+5.5±0.4
+34.5±3.1
+Alconox powder
+-
+<0.37
+<4.4
+<0.24
+5.9±1.3
+Measured at CJPL
+0.23±0.02
+0.82±0.14
+0.05±0.02
+7.5±1.0
+Sand paper
+Black base-plate
+7.2±0.6
+<21
+13.6±0.6
+<3.7
+Green base-plate
+<1.7
+<10.2
+<1.6
+<2.2
+Protection film
+Glue on PE film
+<0.2
+<2.3
+<0.1
+<0.6
+Water soluble film
+<0.47
+<10.8
+0.58±0.19
+<1.56
+The eluent is collected and heated to remove the excess acid, and the solution is finally diluted for
+the ICP-MS measurement.
+Figure 2. The flow chart for the pre-treamtent of acrylic samples. The ashing of the acrylic sample is done
+with the microwave muffle furnace described in the text.
+In Ref. [3], a different approach was used: the acrylic was vaporized and burned in a two-stage
+furnace, while clean gas (mixed N2/O2) was supplied to the furnace. Monitoring of the pressure
+in the whole system is essential in this case. Compared with the pre-treatment in [3], with the new
+technique there is no need for gas input to the microwave muffle furnace, and the overall operation
+is easier and safer. All the pre-treatment process is done in a class 10,000 tent, while the ICP-MS
+measurement is performed in a class 1,000 room. The cleanliness of the quartz vessels is quite
+important for the sensitivity of the measurement, and their cleaning procedure is similar to that
+described in Ref. [3]. All the containers are soaked in the two-stage acid cylinders filled with
+6 mol/L HNO3 for at least one day at each stage to remove the U/Th on the surface. In the end,
+the containers are filled with 6 mol/L HNO3 and boiled for 10 min, then they are rinsed with fresh
+water and ready for use.
+The microwave muffle furnace (Phoenix BLACK [12]) is put in the clean tent and used for
+ashing the acrylic sample, as shown on the left of Fig. 3. The temperature for ashing is optimized
+to achieve no visible organic residual in the vessel, and the optimized temperature as a function of
+time is shown on the right of Fig. 3. The quartz vessel is used as the sample container and well
+cleaned. The 229Th and 233U standards (preserved in 2% acid) are added together with the acrylic
+– 4 –
+
+Tracer and
+Ashing in
+Soak the vessel
+acrylic
+quartz vessel
+with 35% HNO3
+Diluted for ICP-MS
+Eluent was collected and
+measurement
+heated to remove excess acid0
+20
+40
+60
+80
+Time [min]
+0
+200
+400
+600
+C]
+o
+Temperature [
+Figure 3. The microwave muffle furnace (Phoenix BLACK) used for the pre-treatment of acrylic is shown
+in the left figure. The optimized temperature as a function of time for acrylic ashing is shown in the right
+figure.
+sample to the vessel in the beginning, and the recovery efficiency can be evaluated by measuring the
+229Th/233U in the solution sent to ICP-MS. The average recovery efficiency of U/Th is calculated as
+(95±13)% based on more than one hundreds measurements (the given uncertainty is the standard
+deviation). The recovery efficiency is calibrated every time a sample is measured, and the single
+precision is better than 5%. To estimate the background for the pre-treatment, we have followed
+the same procedure without the acrylic sample (blank test). With careful background control, the
+blank test showed that the absolute background for the pre-treatment amounts to 0.24±0.07 pg for
+238U and 0.38±0.04 pg for 232Th.
+4
+Study of the procedure for acrylic surface treatments
+4.1
+Solution absorption in acrylic
+The acrylic panel is cleaned after surface treatments at the company before shipment, and the whole
+acrylic sphere is finally cleaned onsite after installation. Acrylic can absorb water, thus traces of
+radioactivity in cleaning water may diffuse into acrylic. The absorption of water in acrylic obeys
+the mathematical laws of diffusion. We have performed several water absorption tests on flat acrylic
+samples with 2 mm thickness. The mass proportion of absorbed water in acrylic as a function of
+time is shown in Fig. 4. The absorption of water in acrylic did not reach equilibrium in one month,
+and the mass of absorbed water in acrylic increased almost linearly at the beginning, which is far
+away from the equilibrium state. Therefore, there is no impact of the acrylic panel thickness when
+exposure to water is shortened to several hours. Assuming we will clean the acrylic surface during
+one hour at the company, the absorbed water in acrylic can reach about 0.08% for the sample with
+2 mm thickness, which is 1 g/m2 water absorption for one side of acrylic surface.
+For the cleaning procedure, we use Alconox solution (0.1% of water) for degreasing and
+deionized water for rinsing. Traces of radioactivity in both water and Alconox can diffuse into
+acrylic during cleaning. We use deionized water with 10−14 g/g U/Th to perform water cleaning, so
+the residual traces of radioactivity from absorbed water is negligible. Even though the Alconox is
+not as clean as the deionized water, most of Alconox can be removed by water rinsing. To validate
+how much radioactivity from Alconox can go into acrylic, we soaked the acrylic sample with 2 mm
+– 5 –
+
+Pho
+QulckTest
+方法
+CM
+OYFigure 4.
+Results of water absorption tests performed on flat acrylic samples with 2 mm thickness. The
+mass proportion of absorbed water to acrylic as a function of time is shown during 35 days, and the data
+points in the first day are magnified in the sub-figure.
+thickness in Alconox solution with different concentrations and time exposure, and the radiopurity
+of the sample for U/Th is measured by ICP-MS in the end. We have soaked the samples in Alconox
+solution for two months in order to enlarge the possible radioactivity contamination in the bulk of
+acrylic (2 mm thickness). The results of the tests for U/Th surface contamination are shown in
+Table 2.
+Table 2.
+The acrylic samples with 2 mm thickness are soaked in different solutions with different time
+exposure. The U/Th contamination from solution absorption in acrylic after absorption is measured by
+ICP-MS.
+No.
+Sample preparation
+Exposure
+Mass[g]
+238U[ppt]
+232Th[ppt]
+1
+No soaking
+-
+3.25
+0.5±0.1
+1.7±0.1
+2
+0.1% Alconox solution
+1 day
+3.21
+0.6±0.1
+2.3±0.2
+3
+0.1% Alconox solution
+2 months
+3.18
+1.9±0.1
+2.0±0.3
+4
+2% Alconox solution
+2 months
+3.74
+4.4±0.2
+2.3±0.2
+5
+0.1% Alconox solution + 30% HNO3
+2 months + 1 day
+3.11
+0.5±0.1
+1.5±0.2
+6
+Tap water after filter [739 ppt 238U, 0.02 ppt 232Th]
+1 day
+3.28
+3.0±0.1
+1.7±0.2
+7
+1% HNO3 [106.7 ppt 238U, 614.5 ppt 232Th]
+1 day
+3.18
+2.9±0.2
+32.2±1.5
+• No.1-5: we can see a clear increase on 238U in acrylic for Alconox solutions with different
+concentration and exposure. However, the increase of 232Th is not obvious due to the lower
+232Th radioactivity in Alconox powder as shown in Table 1. Even though the Alconox can
+be removed by further deionized water rinsing, the radioactivity from Alconox can diffuse
+into acrylic together with water. If the residual radioactivity stick on acrylic from Alconox
+solution is due to the active agent, the residual can not be easily removed by cleaning or
+acid. On the contrary, if the mechanism of residual consists in ionic diffusion from Alconox
+solution to acrylic, such residual can be removed by acid. We observed that the radioactivity
+absorbed in acrylic can be further removed by acid soaking, as shown in result No.5.
+• No.6-7: To further validate that the diffusion of radioactivity from Alconox to acrylic is
+– 6 –
+
+1.8
+[%]
+Water absorbtion [
+1.6
+1.4
+1.2
+0.3
+0.8
+0.2
+0.6
+0.1
+0.4
+Q
+0.2
+10
+20
+Time [h]
+0
+5
+10
+15
+20
+25
+30
+35
+Time [day]in ionic form, we have soaked another two acrylic samples in two kinds of water. One is
+tap water after 0.2 µm filter, and radioactivity exists in small particles. The other one is
+U/Th standard solution with 1% HNO3, and radioactivity is in ionic form. As shown in
+No.6 and No.7, the 238U concentration in acrylic reached equilibrium at 3 ppt, while 232Th
+can reach much higher to about 30 ppt. The reason for the difference between 238U and
+232Th is their different intrinsic physicochemical properties, which is also discussed in [13].
+Thorium is more reactive than uranium, thus it is easier for thorium to stick to acrylic. A
+similar phenomenon is observed in preparing tap water, most of the thorium is filtered and
+less residual is found in the water. From the results in No.6 and No.7, radioactivity can also
+diffuse into acrylic without the active agent, so the radioactivity in Alconox solution can
+diffuse into acrylic due to the diffusion of ion, not the active agent. The final residual on
+acrylic from the cleaning solution relies on the measurement of the surface sample.
+4.2
+Contamination from different surface treatments
+To study the surface contamination, we have treated the surface with different procedure and directly
+measured the residual of radioactivity on surface. The surface samples were taken by scraping the
+surface with well cleaned erasing knife, and the thickness of the surface sample can reach 5-10 µm.
+The radioactivity of the raw surface for one JUNO acrylic panel without any surface treatment is
+measured to be (52±1) ppt 238U and (133±5) ppt 232Th, which is 1-2 orders of magnitude higher than
+the bulk measurement. The radioactivity distribution along the depth was measured in Ref. [6], and
+has proven that U/Th contaminations can extend down to tens of µm from the surface. In addition,
+the radon daughters (210Pb, 210Po) deposited on the acrylic surface is usually non-negligible. So we
+decide to remove at least 0.1 mm depth of acrylic and to maintain a high light transparency (>96%
+at 420 nm [14] in ultrapure water environment) at the same time.
+A dedicated experiment was performed on a flat acrylic panel (not a JUNO panel) with a
+1 m2 area by applying different surface treatments including sanding, polishing, and cleaning
+successively. All of these treatments are done in a 10,000 class tent. The sanding of acrylic surfaces
+starts from 400 mesh (to remove at least 0.1 mm depth of acrylic) and is pursued by using sand
+papers with larger mesh, 800, 1200, 2000, and 3000. The higher the mesh number, the smoother
+the acrylic surface, and higher light transparency can be achieved. However, more steps increase
+the risk of contaminations. We have measured the surface sample with different mesh numbers of
+1200, 2000, and 3000, and the results for residual U/Th concentration are consistent within 20%
+among these samples.
+Polishing of acrylic is performed by a wool wheel with polishing fluid, and the typical fluid is
+purchased from Saint-Gobain [9], whose components are organic. If there is polishing with mirror
+wax after sanding, sanding to 1200 mesh number is enough to reach the required light transparency.
+Part of the polishing fluid can be further removed by degreasing with Alconox solution. To quantify
+the contamination from residual polishing fluid and Alconox solution, we have done many tests on
+the acrylic panel with sanding to 1200 mesh number, and the results are shown in Table 3. The
+water cleaning is realized by a water jet with high pressure for 20 times, and the Alconox cleaning is
+done by spraying the Alconox on surface and wiping with a clean cloth for several times. Compared
+to the sample No.1, we have done additional cleaning with Alconox solution on No.2, and there is
+obvious radioactive residual on the surface from Alconox. For the samples No. 3 and No. 4, we
+– 7 –
+
+have done polishing with the fluid for both samples, and performed cleaning with Alconox only on
+No.4. From the results, it is quite clear that a large amount of polishing fluid residual exists on the
+acrylic surface after cleaning even with Alconox solution.
+Table 3. Many cleaning tests have been done on the non-JUNO acrylic panel after sanding to the 1200 mesh
+number. The water cleaning is realized by a water jet with high pressure 20 times, and the Alconox cleaning
+is done by wiping the surface with a clean cloth several times. The polishing is done with mirror wax. After
+the treatments, surface samples are taken by scraping the surface with well cleaned erasing knife, and the
+thickness of the surface sample can reach 5-10 µm. The radioactivity in the surface samples is summarized
+in this table.
+No.
+Sample preparation
+Mass[g]
+238U[ppt]
+232Th[ppt]
+1
+Water cleaning
+0.25
+9.5±0.3
+9.3±0.5
+2
+Alconox + water cleaning
+0.34
+27±1
+22±2
+3
+Polishing + water cleaning
+0.28
+92±5
+893±27
+4
+Polishing + Alconox + water cleaning
+0.35
+123±3
+817±33
+Based on the above measurements, polishing with mirror wax and Alconox cleaning have non-
+negligible residual on the acrylic surface, and we should avoid using them for surface treatments.
+To reach light transparency greater than 96% at 420 nm in ultrapure water environment, we have
+performed sanding with 3000 mesh number, polishing with deionized water and, in the end, cleaning
+of the surface with a deionized water high pressure jet. We finally measured the surface of one JUNO
+acrylic panel following this procedure of surface treatments, with good results of (15.2±0.7) ppt
+238U and (24.3±0.7) ppt 232Th in about 10 µm depth, which is several times lower than the raw
+surface as shown in the beginning of this section. In addition, all the treatments of the inner surface
+of the acrylic panel were done in one day to avoid radon daughters deposition.
+4.3
+Protection film
+After surface treatments with sanding, polishing, and cleaning, the acrylic surface will be covered
+by a thin film to protect the acrylic from fallouts of radon and dust in the air during transportation
+and installation, which will last several months. There are two kinds of protection films that can be
+used to protect the acrylic surface, as discussed in Sec 2.
+Since the glue on the adhesive masking paper is soluble in water, we soaked the paper film
+in water for different exposure. By this way, we can measure the radiopurity of glue by ICP-MS
+with high precision. To reduce the effect of surface cleanliness in this measurement, we swab the
+surface of paper side with a little wet clean cloth. The paper is clipped out after soaking. Similar
+to pre-treatments in Figure 2, the solution is vaporized and the residual is digested by acid. The
+results of the measurements are shown in Table 4. It seems that most of the glue is dissolved in
+water within one hour. The film put on inner surface of acrylic will be removed by high pressure
+water jet during the final cleaning. The paper film with water soluble glue on it is quite easy to be
+removed with water, so the contact time is much smaller than one hour.
+The mass of glue on the paper is 5 g/m2, and the total mass of glue for the whole acrylic
+surface is 20 kg. Assume all the U/Th in glue stay on acrylic surface and further go into LS,
+the contamination to 20 kt LS is 10−16 g/g, one order higher than our requirement of U/Th in LS
+(10−17 g/g). In reality, almost all the glue can be dissolved in water when contacting with enough
+– 8 –
+
+amount of water, and the radioactivity from glue can be absorbed in acrylic together with water.
+However, the flow rate of water is about 15 liters per minute, so the glue is highly diluted in water.
+Based on the study in Sec 4.1, the water absorbed in acrylic can reach 1 g/m2 after one hour soaking.
+In reality, the absorbed U/Th in acrylic from glue is quite small.
+Table 4. The water soluble glue on paper film is digested in water solution and measured by ICP-MS.
+Exposure of soaking
+1 hour
+5 hours
+238U [ppt]
+222±11
+268±12
+232Th [ppt]
+117±5
+81±5
+Besides the direct radiopurity measurement of the glues, we have also performed the test on
+acrylic surface. We pasted the two kinds of films on acrylic surface, and took the surface sample
+after removing the film. We did not see an obvious difference between the samples with films
+and the raw panel sample, so it is safe to use both PE and paper film for acrylic protection. The
+whole acrylic sphere with 35.4 m in diameter is divided into 265 panels produced in the company
+separately. Considering the toughness of the films, we prefer to use PE film in the company, which
+is better for protection of the panels during transportation and installation. All of these panels will
+be bonded layer to layer from top to bottom at JUNO site. After bonding the panels of one layer, we
+will clean the surface of the corresponding layer and cover the inner surface with paper film. When
+the whole sphere will be finished, we will finally clean the inner surface with high pressure water
+jet, and the paper film will be easily removed without manual operation inside the sphere.
+5
+Summary
+An acrylic vessel with 35.4 m in diameter is used as the 20 kt LS container for JUNO. The cleanliness
+of the inner acrylic surface is quite important to ensure an excellent LS radiopurity. To remove
+the contamination near the acrylic surface during production, many treatments will be done on the
+acrylic surface before shipment. From this study, we found the polishing mirror wax and Alconox
+solution have non-negligible U/Th residuals on the surface. The final recommended procedure for
+the surface treatments consists of sanding up to 3000 mesh number, polishing and cleaning with
+deionized water to achieve low background and high light transparency. Finally, a thin PE film will
+cover acrylic surface during transportation, and the film will be replaced by the adhesive masking
+paper for the inner surface after onsite bonding. This operation is quite important before finally
+removing the film by water jet, since the glue on the adhesive masking paper is water soluble. This
+procedure for acrylic surface treatments is also applicable to other low background experiments. In
+addition, a new method for pre-treatment of acrylic samples based on a microwave muffle furnace
+is shown in this paper, and the sensitivity of the measurement with ICP-MS can reach sub-ppt for
+U/Th in acrylic.
+Acknowledgments
+This work is supported by the Youth Innovation Promotion Association of the Chinese Academy
+of Sciences, the National Natural Science Foundation of China (No. 11905226), and the Strategic
+Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA10010200).
+– 9 –
+
+References
+[1] A. Abusleme et al. (JUNO), Prog. Part. Nucl. Phys. 123, 103927 (2022).
+[2] A. Abusleme et al. (JUNO), JHEP 11, 102 (2021), arXiv:2107.03669 [physics.ins-det] .
+[3] C. Cao, N. Li, X. Yang, J. Zhao, Y. Li, Z. Cai, L. Wen, X. Luo, Y. Heng, and Y. Ding, Nucl. Instrum.
+Meth. A 1004, 165377 (2021), arXiv:2011.06817 [physics.ins-det] .
+[4] S. Appel et al. (Borexino), (2022), arXiv:2205.15975 [hep-ex] .
+[5] P. Khaghani, Neck sense rope system and leaching studies for SNO+, Ph.D. thesis.
+[6] F. C. F. Perrot, C. Cerna and C. P´echeyran, “Advantages and sensitivity of UV fs laser ablation
+HR-ICPMS technique for rare event experiments, talk at Low Radioactivity Techniques 2019,” .
+[7] “https://www.alconox.com/product/alconox,” .
+[8] S.-L. Niu et al., Chin. Phys. C 39, 086002 (2015), arXiv:1410.4291 [physics.ins-det] .
+[9] Saint-Gobain, “https://www.saint-gobain.com.cn/abrasives/product/non-abrasive,” .
+[10] X. Wang, X. Chen, C. Fu, X. Ji, X. Liu, Y. Mao, H. Wang, S. Wang, P. Xie, and T. Zhang, Journal of
+Instrumentation 11, T12002 (2016).
+[11] “https://www.daio-paper.co.jp/en,” .
+[12] “https://cem.com/en/phoenix-black,” .
+[13] B. D. LaFerriere, T. C. Maiti, I. J. Arnquist, and E. W. Hoppe, Nucl. Instrum. Meth. A 775, 93 (2015).
+[14] Z. Li et al., Rad. Det. Tech. Meth. 5, 356 (2021).
+– 10 –
+

GNE4T4oBgHgl3EQfHQyx/content/tmp_files/load_file.txt

@@ -0,0 +1,459 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf,len=458
+page_content='Prepared for submission to JINST  Study on U/Th residual radioactivity in acrylic from  surface treatment  Yuanxia Li,𝑎 Xiaohui Qian,𝑎 Xiaolan Luo,𝑎 Jie Zhao,𝑎,1 Gaofeng Zhang,𝑏 Xiaoyan Ma,𝑎  Yuekun Heng,𝑎 Liangjian Wen,𝑎 Monica Sisti,𝑐 Frédéric Perrot,𝑑 Hongqiang Tang𝑏  𝑎Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  𝑏Donchamp New Material Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Ltd, Taixing 225400, China  𝑐INFN Milano Bicocca and Università di Milano-Bicocca, Milano, Italy  𝑑Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Bordeaux, CNRS, CENBG, UMR 5797, F-33170 Gradignan, France  E-mail: zhaojie@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='cn  Abstract: Acrylic is widely used as material for the target container in low background experiments  due to its high light transparency and low intrinsic radioactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, its surface can be easily  contaminated during production, so careful treatment of the surface is essential to avoid direct  contamination of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The Jiangmen Underground Neutrino Observatory will use about 600 t  of acrylic to build the spherical vessel of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter for a 20 kt liquid scintillator (LS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Since acrylic will contact the LS directly, the cleanliness of the its surface is quite important for the  radiopurity of the LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' A new method for measuring the radioactivity of 238U and 232Th in acrylic to  sub-ppt (< 10−12 g/g) was developed, and it is crucial for the acrylic radioactivity screening in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We performed many background tests on different surface treatments, and the recommended  procedure for the treatment of acrylic to achieve low radioactivity and high light transparency could  be applicable to other low background experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='04902v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='ins-det]  12 Jan 2023  Contents  1  Introduction  1  2  Screening of surface treatment material by gamma spectrometer  2  3  Method for measuring U/Th in acrylic by ICP-MS  3  4  Study of the procedure for acrylic surface treatments  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  Solution absorption in acrylic  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  Contamination from different surface treatments  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  Protection film  8  5  Summary  9  1  Introduction  The Jiangmen Underground Neutrino Observatory (JUNO) is a multi-purpose neutrino experiment,  which will build the world’s largest liquid scintillator (LS) detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' An acrylic vessel is used as  the 20 kt LS container with 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter and 120 mm in thickness because of its high light  transparency and low intrinsic radioactivity [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  To reach the physics target, the average radio-purity of the acrylic is required to be less than  1 ppt [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [3], the radio purity in the bulk has reached the requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However,  the distribution of radioactivity along the thickness is not extremely uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The bulk measurement  of samples with several centimeters of thickness can not reflect the surface contamination, since  the thickness of external contamination on the acrylic surface usually reaches nanometre to micron  level, and the surface contamination will result averaged out in the bulk measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If the external  radioactive contamination stays on the surface, only gammas and few alphas/betas from radioactive  decays can be emitted into the LS and rejected by the effective fiducial volume cut by the off-line  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, based on the experiences at Borexino [4] and SNO+ [5], the U/Th daughters  near the inner surface of the vessel could leak into the LS by diffusion and convection during stable  running, thus making the acrylic vessel a stable contamination source for the LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Borexino put  great efforts on controlling the temperature distribution to avoid convection in LS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' however, it is  impossible to repeat the same thing at JUNO due to the large size of the experiment (70 times  Borexino target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Therefore, the radiopurity of the inner surface of acrylic is quite important for  JUNO, especially for the physic topics in the low energy regions, such as solar neutrino studies [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The JUNO acrylic vessel is produced with 265 pieces because of its large size, as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Each acrylic panel is firstly produced in a carefully cleaned flat mold and then shaped to  bend board at high temperature at the company;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' finally it is bonded directly at JUNO site [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  surface of the acrylic panel is exposed to air and covered with a high-temperature resistant cloth  – 1 –  during the bending, thus the surface is eventually contaminated and not as smooth as the flat panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Surface treatments are performed before shipment to the JUNO site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Typical protocols to remove  surface contaminations include the following steps:  The highest contaminations can reach tens of µm depths from the surface [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Thus, at least  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 µm thickness of the surface of acrylic is firstly removed by sanding with special paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The surface is not so smooth after sanding, so polishing is needed to improve the transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The polishing is done with a wool wheel and polishing fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The surface should be well cleaned with detergent and deionized water after polishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Finally, a thin film is used to cover the acrylic surface to avoid external contamination during  transportation and installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  All tools and detergents have risks to introduce more contamination on the surface, therefore  each surface treatment should be well studied to minimize those risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The acrylic vessel is produced with 265 pieces due to its very large size, as shown in the left figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Each panel is produced separately, and all the panels will be finally bonded to a sphere at JUNO site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' One  JUNO panel is shown in the right figure as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The surface treatment material screening is described in  Sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The new method for measuring 238U and 232Th (U/Th) in acrylic to sub-ppt level is described  in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Various tests on radioactivity from each step of the above treatment protocol are discussed  in Sec 4, and the recommended procedure is also shown in the same section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The concluding  remarks are in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  2  Screening of surface treatment material by gamma spectrometer  The main tools and detergents used for surface treatments are sanding papers, polishing fluid,  Alconox powder [7], deionized water and protection film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We used a gamma spectrometer to screen  the radiopurity of the tools and detergents first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have one HPGe facility at ground level at  IHEP [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The gamma spectrometer plays an important role in the primary screening of the raw  – 2 –  0  0  0  0  0  Q  0  4  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='materials at JUNO, and the advantage is that samples can be screened directly without any pre-  treatment and with no damage to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The principle of gamma spectroscopy is to measure  the emitted gamma from the decay chain, and it is more sensitive to gammas with higher energy  due to lower background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' For example, the detector is more sensitive to the lower part of the  uranium chain starting from 226Ra, and the 238U concentration can be derived by assuming secular  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The measured results for the tools and detergent used for acrylic surface treatment are sum-  marized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We found that special mirror wax purchased from Saint-Gobain [9] has lower  radioactivity for 40K than one-step fast wax, due to the different chemical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The sand-  paper with a green base plate has lower radioactivity and better sand fastening than the one with a  black base plate for similar mesh numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Regarding the Alconox powder, the first measurement  performed at IHEP gave an upper limit of several tens to hundreds of ppb level for thorium and  uranium concentrations, respectively, due to the limited sensitivity of HPGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The same sample was  further measured by HPGe at China Jinping Underground Laboratory (CJPL) with better shielding  and sensitivity [10] down to 10 ppb for 232Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From the obtained results, there is a higher uranium  concentration than thorium in the Alconox powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The polishing fluid and sandpaper with lower  radioactivity are selected for the surface treatments on acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  After surface treatment, the acrylic surface will be protected with a thin film to avoid dust and  radon daughters deposition on the surface during the handling operations for transportation and  installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The film on the inner surface will be removed by the final cleaning with high pressure  water jet after installation, and there is no possibility for manual operation inside the acrylic vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  There are two types of protection film that can be used to protect the acrylic surface, ordinary  polyethylene (PE) film with few glue on the surface and adhesive masking paper film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The PE film  is better in toughness, but it is not easy to be removed by the water jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' On the contrary, the paper  film with few water soluble glue on the surface can be easily removed by water washing since the  glue is soluble in water, but overall its mechanical strength is worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The radioactivity screening of  the glue that is used on the PE film showed only upper limits for U/Th at the level of hundreds of  ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Concerning the adhesive masking paper film (purchased from [11]), we could only measure  the paper film and the glue together inside the HPGe detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The water soluble glue itself was  further measured by ICP-MS, as it will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  3  Method for measuring U/Th in acrylic by ICP-MS  Different from gamma spectroscopy, the principle of the Inductively Coupled Plasma Mass Spec-  troscopy (ICP-MS) is to detect the nuclei according to their exact atomic weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' ICP-MS can  measure the isotopes 238U and 232Th directly without other information on the chain, and the sen-  sitivity can reach the sub-ppt level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, the sample measured by ICP-MS must be in liquid  form, so a careful and optimized pre-treatment on the sample is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  For the radio purity measurement on acrylic surface samples, we have developed a new method  based on microwave ashing and ICP-MS equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The design for the pre-treatment flow is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Both the acrylic sample and the tracers (the natural-non-existing nuclei 229Th and 233U)  are ashed by the machine in a quartz vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The residual is collected by soaking with 35% HNO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  – 3 –  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Material screening by gamma spectroscopy on tools and detergents used for the acrylic surface  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If not differently mentioned, the sample was measured at IHEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The conversions between Bq/kg  and mass concentration units for U/Th are given: 1 Bq/kg of 232Th activity in a material is equal to  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5 × 10−7 g/g (or 250 ppb) of 232Th mass concentration in that material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' similarly, 1 Bq/kg of 238U means  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 × 10−8 g/g (or 81 ppb) of 238U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  238U chain [Bq/kg]  232Th chain [Bq/kg]  40K  214Bi/214Pb  226Ra  212Bi/208Tl  [Bq/kg]  Polishing  Special mirror wax  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9  One step fast wax  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  Alconox powder <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='37  <4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  Measured at CJPL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='02  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0  Sand paper  Black base-plate  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  <21  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  <3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7  Green base-plate  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7  <10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  Protection film  Glue on PE film  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  Water soluble film  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='47  <10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='19  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='56  The eluent is collected and heated to remove the excess acid, and the solution is finally diluted for  the ICP-MS measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The flow chart for the pre-treamtent of acrylic samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The ashing of the acrylic sample is done  with the microwave muffle furnace described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [3], a different approach was used: the acrylic was vaporized and burned in a two-stage  furnace, while clean gas (mixed N2/O2) was supplied to the furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Monitoring of the pressure  in the whole system is essential in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Compared with the pre-treatment in [3], with the new  technique there is no need for gas input to the microwave muffle furnace, and the overall operation  is easier and safer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All the pre-treatment process is done in a class 10,000 tent, while the ICP-MS  measurement is performed in a class 1,000 room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The cleanliness of the quartz vessels is quite  important for the sensitivity of the measurement, and their cleaning procedure is similar to that  described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All the containers are soaked in the two-stage acid cylinders filled with  6 mol/L HNO3 for at least one day at each stage to remove the U/Th on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In the end,  the containers are filled with 6 mol/L HNO3 and boiled for 10 min, then they are rinsed with fresh  water and ready for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The microwave muffle furnace (Phoenix BLACK [12]) is put in the clean tent and used for  ashing the acrylic sample, as shown on the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The temperature for ashing is optimized  to achieve no visible organic residual in the vessel, and the optimized temperature as a function of  time is shown on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The quartz vessel is used as the sample container and well  cleaned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The 229Th and 233U standards (preserved in 2% acid) are added together with the acrylic  – 4 –  Tracer and  Ashing in  Soak the vessel  acrylic  quartz vessel  with 35% HNO3  Diluted for ICP-MS  Eluent was collected and  measurement  heated to remove excess acid0  20  40  60  80  Time [min]  0  200  400  600  C]  o  Temperature [  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The microwave muffle furnace (Phoenix BLACK) used for the pre-treatment of acrylic is shown  in the left figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The optimized temperature as a function of time for acrylic ashing is shown in the right  figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  sample to the vessel in the beginning, and the recovery efficiency can be evaluated by measuring the  229Th/233U in the solution sent to ICP-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The average recovery efficiency of U/Th is calculated as  (95±13)% based on more than one hundreds measurements (the given uncertainty is the standard  deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The recovery efficiency is calibrated every time a sample is measured, and the single  precision is better than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To estimate the background for the pre-treatment, we have followed  the same procedure without the acrylic sample (blank test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' With careful background control, the  blank test showed that the absolute background for the pre-treatment amounts to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='07 pg for  238U and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='04 pg for 232Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  4  Study of the procedure for acrylic surface treatments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  Solution absorption in acrylic  The acrylic panel is cleaned after surface treatments at the company before shipment, and the whole  acrylic sphere is finally cleaned onsite after installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Acrylic can absorb water, thus traces of  radioactivity in cleaning water may diffuse into acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The absorption of water in acrylic obeys  the mathematical laws of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have performed several water absorption tests on flat acrylic  samples with 2 mm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The mass proportion of absorbed water in acrylic as a function of  time is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The absorption of water in acrylic did not reach equilibrium in one month,  and the mass of absorbed water in acrylic increased almost linearly at the beginning, which is far  away from the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Therefore, there is no impact of the acrylic panel thickness when  exposure to water is shortened to several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Assuming we will clean the acrylic surface during  one hour at the company, the absorbed water in acrylic can reach about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='08% for the sample with  2 mm thickness, which is 1 g/m2 water absorption for one side of acrylic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  For the cleaning procedure, we use Alconox solution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% of water) for degreasing and  deionized water for rinsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Traces of radioactivity in both water and Alconox can diffuse into  acrylic during cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We use deionized water with 10−14 g/g U/Th to perform water cleaning, so  the residual traces of radioactivity from absorbed water is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Even though the Alconox is  not as clean as the deionized water, most of Alconox can be removed by water rinsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To validate  how much radioactivity from Alconox can go into acrylic, we soaked the acrylic sample with 2 mm  – 5 –  Pho  QulckTest  方法  CM  OYFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Results of water absorption tests performed on flat acrylic samples with 2 mm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  mass proportion of absorbed water to acrylic as a function of time is shown during 35 days, and the data  points in the first day are magnified in the sub-figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  thickness in Alconox solution with different concentrations and time exposure, and the radiopurity  of the sample for U/Th is measured by ICP-MS in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have soaked the samples in Alconox  solution for two months in order to enlarge the possible radioactivity contamination in the bulk of  acrylic (2 mm thickness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The results of the tests for U/Th surface contamination are shown in  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The acrylic samples with 2 mm thickness are soaked in different solutions with different time  exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The U/Th contamination from solution absorption in acrylic after absorption is measured by  ICP-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Sample preparation  Exposure  Mass[g]  238U[ppt]  232Th[ppt]  1  No soaking 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% Alconox solution  1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% Alconox solution  2 months  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  4  2% Alconox solution  2 months  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='74  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% Alconox solution + 30% HNO3  2 months + 1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  6  Tap water after filter [739 ppt 238U, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='02 ppt 232Th]  1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  7  1% HNO3 [106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7 ppt 238U, 614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5 ppt 232Th]  1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1-5: we can see a clear increase on 238U in acrylic for Alconox solutions with different  concentration and exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, the increase of 232Th is not obvious due to the lower  232Th radioactivity in Alconox powder as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Even though the Alconox can  be removed by further deionized water rinsing, the radioactivity from Alconox can diffuse  into acrylic together with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If the residual radioactivity stick on acrylic from Alconox  solution is due to the active agent, the residual can not be easily removed by cleaning or  acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' On the contrary, if the mechanism of residual consists in ionic diffusion from Alconox  solution to acrylic, such residual can be removed by acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We observed that the radioactivity  absorbed in acrylic can be further removed by acid soaking, as shown in result No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6-7: To further validate that the diffusion of radioactivity from Alconox to acrylic is  – 6 –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8  [%]  Water absorbtion [  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  Q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  10  20  Time [h]  0  5  10  15  20  25  30  35  Time [day]in ionic form, we have soaked another two acrylic samples in two kinds of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' One is  tap water after 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2 µm filter, and radioactivity exists in small particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The other one is  U/Th standard solution with 1% HNO3, and radioactivity is in ionic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' As shown in  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7, the 238U concentration in acrylic reached equilibrium at 3 ppt, while 232Th  can reach much higher to about 30 ppt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The reason for the difference between 238U and  232Th is their different intrinsic physicochemical properties, which is also discussed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Thorium is more reactive than uranium, thus it is easier for thorium to stick to acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' A  similar phenomenon is observed in preparing tap water, most of the thorium is filtered and  less residual is found in the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From the results in No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7, radioactivity can also  diffuse into acrylic without the active agent, so the radioactivity in Alconox solution can  diffuse into acrylic due to the diffusion of ion, not the active agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The final residual on  acrylic from the cleaning solution relies on the measurement of the surface sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  Contamination from different surface treatments  To study the surface contamination, we have treated the surface with different procedure and directly  measured the residual of radioactivity on surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The surface samples were taken by scraping the  surface with well cleaned erasing knife, and the thickness of the surface sample can reach 5-10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The radioactivity of the raw surface for one JUNO acrylic panel without any surface treatment is  measured to be (52±1) ppt 238U and (133±5) ppt 232Th, which is 1-2 orders of magnitude higher than  the bulk measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The radioactivity distribution along the depth was measured in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [6], and  has proven that U/Th contaminations can extend down to tens of µm from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In addition,  the radon daughters (210Pb, 210Po) deposited on the acrylic surface is usually non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' So we  decide to remove at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 mm depth of acrylic and to maintain a high light transparency (>96%  at 420 nm [14] in ultrapure water environment) at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  A dedicated experiment was performed on a flat acrylic panel (not a JUNO panel) with a  1 m2 area by applying different surface treatments including sanding, polishing, and cleaning  successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All of these treatments are done in a 10,000 class tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The sanding of acrylic surfaces  starts from 400 mesh (to remove at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 mm depth of acrylic) and is pursued by using sand  papers with larger mesh, 800, 1200, 2000, and 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The higher the mesh number, the smoother  the acrylic surface, and higher light transparency can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, more steps increase  the risk of contaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have measured the surface sample with different mesh numbers of  1200, 2000, and 3000, and the results for residual U/Th concentration are consistent within 20%  among these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Polishing of acrylic is performed by a wool wheel with polishing fluid, and the typical fluid is  purchased from Saint-Gobain [9], whose components are organic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If there is polishing with mirror  wax after sanding, sanding to 1200 mesh number is enough to reach the required light transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Part of the polishing fluid can be further removed by degreasing with Alconox solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To quantify  the contamination from residual polishing fluid and Alconox solution, we have done many tests on  the acrylic panel with sanding to 1200 mesh number, and the results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  water cleaning is realized by a water jet with high pressure for 20 times, and the Alconox cleaning is  done by spraying the Alconox on surface and wiping with a clean cloth for several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Compared  to the sample No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1, we have done additional cleaning with Alconox solution on No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2, and there is  obvious radioactive residual on the surface from Alconox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' For the samples No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 3 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 4, we  – 7 –  have done polishing with the fluid for both samples, and performed cleaning with Alconox only on  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From the results, it is quite clear that a large amount of polishing fluid residual exists on the  acrylic surface after cleaning even with Alconox solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Many cleaning tests have been done on the non-JUNO acrylic panel after sanding to the 1200 mesh  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The water cleaning is realized by a water jet with high pressure 20 times, and the Alconox cleaning  is done by wiping the surface with a clean cloth several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The polishing is done with mirror wax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' After  the treatments, surface samples are taken by scraping the surface with well cleaned erasing knife, and the  thickness of the surface sample can reach 5-10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The radioactivity in the surface samples is summarized  in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Sample preparation  Mass[g]  238U[ppt]  232Th[ppt]  1  Water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='25  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5  2  Alconox + water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='34  27±1  22±2  3  Polishing + water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='28  92±5  893±27  4  Polishing + Alconox + water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='35  123±3  817±33  Based on the above measurements, polishing with mirror wax and Alconox cleaning have non-  negligible residual on the acrylic surface, and we should avoid using them for surface treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  To reach light transparency greater than 96% at 420 nm in ultrapure water environment, we have  performed sanding with 3000 mesh number, polishing with deionized water and, in the end, cleaning  of the surface with a deionized water high pressure jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We finally measured the surface of one JUNO  acrylic panel following this procedure of surface treatments, with good results of (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7) ppt  238U and (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7) ppt 232Th in about 10 µm depth, which is several times lower than the raw  surface as shown in the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In addition, all the treatments of the inner surface  of the acrylic panel were done in one day to avoid radon daughters deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  Protection film  After surface treatments with sanding, polishing, and cleaning, the acrylic surface will be covered  by a thin film to protect the acrylic from fallouts of radon and dust in the air during transportation  and installation, which will last several months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' There are two kinds of protection films that can be  used to protect the acrylic surface, as discussed in Sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Since the glue on the adhesive masking paper is soluble in water, we soaked the paper film  in water for different exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' By this way, we can measure the radiopurity of glue by ICP-MS  with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To reduce the effect of surface cleanliness in this measurement, we swab the  surface of paper side with a little wet clean cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The paper is clipped out after soaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Similar  to pre-treatments in Figure 2, the solution is vaporized and the residual is digested by acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  results of the measurements are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' It seems that most of the glue is dissolved in  water within one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The film put on inner surface of acrylic will be removed by high pressure  water jet during the final cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The paper film with water soluble glue on it is quite easy to be  removed with water, so the contact time is much smaller than one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The mass of glue on the paper is 5 g/m2, and the total mass of glue for the whole acrylic  surface is 20 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Assume all the U/Th in glue stay on acrylic surface and further go into LS,  the contamination to 20 kt LS is 10−16 g/g, one order higher than our requirement of U/Th in LS  (10−17 g/g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In reality, almost all the glue can be dissolved in water when contacting with enough  – 8 –  amount of water, and the radioactivity from glue can be absorbed in acrylic together with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  However, the flow rate of water is about 15 liters per minute, so the glue is highly diluted in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Based on the study in Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1, the water absorbed in acrylic can reach 1 g/m2 after one hour soaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  In reality, the absorbed U/Th in acrylic from glue is quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The water soluble glue on paper film is digested in water solution and measured by ICP-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Exposure of soaking  1 hour  5 hours  238U [ppt]  222±11  268±12  232Th [ppt]  117±5  81±5  Besides the direct radiopurity measurement of the glues, we have also performed the test on  acrylic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We pasted the two kinds of films on acrylic surface, and took the surface sample  after removing the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We did not see an obvious difference between the samples with films  and the raw panel sample, so it is safe to use both PE and paper film for acrylic protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  whole acrylic sphere with 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter is divided into 265 panels produced in the company  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Considering the toughness of the films, we prefer to use PE film in the company, which  is better for protection of the panels during transportation and installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All of these panels will  be bonded layer to layer from top to bottom at JUNO site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' After bonding the panels of one layer, we  will clean the surface of the corresponding layer and cover the inner surface with paper film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' When  the whole sphere will be finished, we will finally clean the inner surface with high pressure water  jet, and the paper film will be easily removed without manual operation inside the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  5  Summary  An acrylic vessel with 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter is used as the 20 kt LS container for JUNO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The cleanliness  of the inner acrylic surface is quite important to ensure an excellent LS radiopurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To remove  the contamination near the acrylic surface during production, many treatments will be done on the  acrylic surface before shipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From this study, we found the polishing mirror wax and Alconox  solution have non-negligible U/Th residuals on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The final recommended procedure for  the surface treatments consists of sanding up to 3000 mesh number, polishing and cleaning with  deionized water to achieve low background and high light transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Finally, a thin PE film will  cover acrylic surface during transportation, and the film will be replaced by the adhesive masking  paper for the inner surface after onsite bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' This operation is quite important before finally  removing the film by water jet, since the glue on the adhesive masking paper is water soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' This  procedure for acrylic surface treatments is also applicable to other low background experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In  addition, a new method for pre-treatment of acrylic samples based on a microwave muffle furnace  is shown in this paper, and the sensitivity of the measurement with ICP-MS can reach sub-ppt for  U/Th in acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Acknowledgments  This work is supported by the Youth Innovation Promotion Association of the Chinese Academy  of Sciences, the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 11905226), and the Strategic  Priority Research Program of the Chinese Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' XDA10010200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
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+page_content=' Heng, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Ding, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' A 1004, 165377 (2021), arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
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+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Appel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' (Borexino), (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='15975 [hep-ex] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Khaghani, Neck sense rope system and leaching studies for SNO+, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Perrot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Cerna and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' P´echeyran, “Advantages and sensitivity of UV fs laser ablation  HR-ICPMS technique for rare event experiments, talk at Low Radioactivity Techniques 2019,” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  [7] “https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='alconox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
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+page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
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@@ -0,0 +1,695 @@
+ 
+ 
+1 
+The COVID-19 vaccination, preventive behaviors and pro-social motivation:  
+panel data analysis from Japan 
+ 
+Short title: COVID-19 vaccination and preventive behaviors 
+ 
+Eiji Yamamura1*, Yoshiro Tsutsui2, Fumio Ohtake3 
+ 
+ 
+1 Department of Economics, Seinan Gakuin University, Fukuoka, Japan 
+ 
+*Corresponding author 
+Email: yamaei@seinan-gu.ac.jp (EY) 
+Full list of author information is available at the end of the article 
+ 
+Abstract  
+Background 
+The COVID-19 vaccine reduces infection risk: even if one contracts COVID-19, the 
+probability of complications like death or hospitalization is lower. However, vaccination may 
+prompt people to decrease preventive behaviors, such as staying indoors, handwashing, and 
+wearing a mask. Thereby, if vaccinated people pursue only their self-interest, the vaccine’s 
+effect may be lower than expected. However, if vaccinated people are pro-social (motivated 
+toward benefit for the whole society), they might maintain preventive behaviors to reduce 
+the spread of infection. 
+ 
+Methods 
+We conducted 26 surveys nearly once a month from March 2020 (the early stage of COVID-
+19) to September 2022 in Japan. By corresponding with the identical individuals, we 
+independently constructed original panel data (N = 70,979). Based on the data, we identified 
+the timing of the second vaccine shot and compared preventive behaviors before and after 
+
+ 
+ 
+2 
+vaccination. We investigated whether second-shot vaccination correlated with changes in 
+preventive behaviors. Furthermore, we explored whether the vaccination effect differs 
+between senior and younger groups. We then investigated the effect of pro-social motivation 
+on preventive behaviors. 
+ 
+Results 
+Major findings are as follows: (1) Being vaccinated led people to increase preventive 
+behaviors, such as mask-wearing by 1.04 (95% confidence intervals [Cis]: 0.96–1.11) points, 
+and handwashing by 0.34 (95% CIs: 0.30–0.38) points, on a 5-point scale. (2) vaccinated 
+people under the age of 65 years are less likely to stay indoors. (3) people with pro-social 
+motivation to be vaccinated are more likely to maintain prevention than those not so 
+motivated; the difference is 0.08 (95% CIs: 0.01–0.15) points for mask-wearing and 0.05 
+(95% CIs: 0.001–0.10) points for handwashing, on a 5-point scale. 
+ 
+Conclusion 
+After vaccination, the opportunity cost of staying indoors outweighs its benefits and people 
+are less inclined to stay home. This effect is lower for older people, who are at higher risk of 
+serious illness. The opportunity cost of mask-wearing and handwashing is lower than that of 
+staying indoors, and the benefit persists after vaccination if people have the motivation to 
+maintain these behaviors for others’ well-being.  
+Keywords: Vaccine, COVID-19, preventive behaviors, staying indoors, wearing mask, 
+washing hands, public goods, pro-social, Japan, panel data 
+ 
+
+ 
+ 
+3 
+ 
+Introduction 
+ 
+During COVID-19 pandemic, especially in the early stages, various preventive 
+behaviors were required because vaccines against COVID-19 had not been developed. 
+Preventive behaviors can be considered a kind of public good that is not sufficiently supplied 
+through market mechanisms where people pursue self-interest [1,2]. Mitigating the pandemic 
+necessitated the collective actions of citizens. However, according to the Peltzman effect, 
+people tend to increase their risky behaviors if safety measures are implemented [3].  
+Since 2021, vaccines against COVID-19 have been distributed worldwide and have 
+played a vital role in coping with COVID-19. Newly reported cases have decreased in 
+countries where vaccines have been rapidly adopted [4]. People, if they are rational, tend to 
+engage in risky behaviors when security measures are mandated [5]. In terms of economics, 
+this is considered a moral hazard. An empirical question arises as to how the spread of the 
+vaccine influences preventive behaviors [6,7]. As a result of the reduction in the risk of 
+COVID-19 infection, risk-taking behaviors increase, so that preventive behaviors such as 
+staying indoors, wearing masks, and washing hands have changed [8,9]. However, some 
+studies show no clear evidence that vaccinated people have decreased their preventive 
+behaviors in comparison with those not vaccinated [6,10].   
+The influence of vaccination on preventive behaviors may vary according to the type of 
+behavior [11]. A study found that in China, vaccination lessened the frequency of 
+handwashing but did not change mask-wearing [7]. This study aimed to explore the 
+mechanisms stopping vaccinated people from decreasing preventive behaviors. To this end, 
+
+ 
+ 
+4 
+we investigated how preventive measures can be pro-socially motivated based on altruism 
+and social solidarity [12]. 
+Using monthly individual-level panel data, we investigated whether individuals’ 
+preventive behaviors changed after vaccination. Furthermore, we looked at how the influence 
+of vaccination on preventive behaviors differs according to age and pro-social motivation.  
+ 
+Methods 
+Data collection 
+We commissioned the research company INTAGE, Inc., to conduct an online survey 
+because of their experience and reliability in academic research. The first wave of queries 
+was conducted March 13–16, 2020, and 4,359 observations were recorded. Participants 
+registered with INTAGE were recruited for this study. The participation rate was 54.7%. The 
+sampling method was designed to collect representatives of the Japanese adult population in 
+terms of educational background, gender, and residential area. For this purpose, INTAGE 
+recruited participants for a survey of pre-registered individuals. However, individuals aged 
+15 years and below were too young to be registered with INTAGE, and we considered 
+individuals over 80 years of age too old to answer pertinent questions. Inevitably, the sample 
+population was restricted to ages 16–79 years, and participants were randomly selected to fill 
+the pre-specified quotas. INTAGE provided monetary incentives to participants upon study 
+completion. 
+Internet surveys were conducted repeatedly 26 separate times (“waves”) almost every 
+month with identical individuals to construct the panel data. The exceptional period was July-
+
+ 
+ 
+5 
+September 2020, when the survey could not be conducted owing to a shortage of research 
+funds. We resumed the surveys after receiving additional funds in October 2020. Vaccination 
+was implemented in April 2021; therefore, the data cover the period before and after 
+implementation of vaccination. 
+Respondents from the first wave were targeted in subsequent waves to record how some 
+respondents changed their behaviors during the COVID-19 pandemic. In particular, the data 
+allowed us to compare identical persons’ preventive behaviors against COVID-19 before and 
+after being vaccinated. During the study period, some identical respondents were dropped 
+from the study sample because some of them stopped taking the surveys, while others did 
+not take the surveys at all. Furthermore, the sample was restricted to those who were 
+completely vaccinated by getting the second shot. In this way, we could compare their 
+behaviors before and after vaccination. Eventually, the number of identical individuals was 
+reduced from 4,359 to 3,019, and the total number of observations used in this study was 
+70,979. 
+ 
+ 
+Methods 
+Table 1 presents the key variables used in the estimation. The survey questionnaire 
+contained basic questions about demographics, such as birth year, gender, and educational 
+background. These characteristics were observed at different time points. The surveys were 
+conducted 26 times, between March 2020 and September 2022. During the study period, 
+conditions such as the spread of infection and policies against COVID-19 changed drastically. 
+Table 1 describes the key variables used in the regression estimations. As outcome variables, 
+
+ 
+ 
+6 
+the respondents were asked questions concerning preventive behaviors, such as:  
+“Within a week, to what degree have you practiced the following behaviors? Please 
+answer based on a scale of 1 (I have not practiced this behavior at all) to 5 (I have completely 
+practiced this behavior).” 
+(1) Staying indoors 
+(2) Wearing a mask 
+(3) Washing my hands thoroughly 
+The answers to these questions served as proxies for the following variables for 
+preventive behaviors: staying indoors, frequency of handwashing, and degree of wearing 
+masks. Larger values indicate that respondents are more likely to engage in preventive 
+behaviors. Further, the motivation to get a shot of COVID-19 vaccination is asked in the 
+following question: “Do you get the shot in order to decrease the spread of COVID-19 
+infection?”  
+We also asked about the subjective probability of contracting COVID-19 and their 
+perceptions of the severity of COVID-19. We asked whether they received the second shot 
+of the vaccine because the vaccination was effective only after completing the second shot. 
+The latter question was included in the questionnaire from the 12th wave conducted in May 
+2021 directly after the vaccine was introduced in Japan. The question was included until the 
+18th wave in November 2021, when most people in the sample completed the second shot. 
+The question was then excluded from the questionnaire, starting with the 19th wave in 
+January 2022. 
+ 
+
+ 
+ 
+7 
+ 
+Table 1. Definitions of key variables. 
+Variable 
+ 
+Definition 
+ 
+    Outcome variables 
+STAYING INDOORS 
+In the last week, how consistent were you at “not going out of home?” 
+Please choose among 5 choices. 
+1 (not completed at all) to 5 (completely consistent). 
+WEARING MASK 
+ 
+ 
+In the last week, how consistent were you at “wearing a mask?” Please 
+choose among 5 choices. 
+1 (not completed at all) to 5 (completely achieved). 
+HANDWASHING  
+ 
+In the last week, how consistent were you at “washing your hands?” Please 
+choose among 5 choices. 
+1 (not completed at all) to 5 (completely achieved). 
+ 
+Confounders (Independent variables) 
+VACCINE  
+ 
+Did you get the second shot? 
+1 (Yes) or 0 (No) 
+PROB_COVID19 
+ 
+What percentage do you think is the probability of your contracting the 
+novel coronavirus (COVID-19)?  
+0 to 100 (%) 
+SEVERITY COVID19 How serious do you expect your symptoms to be if you are infected with 
+the novel coronavirus? Choose from 6 choices. 
+1 (very small influence) to 6 (death) 
+AGE_25 
+Answer 1 if respondents are age 19–25, otherwise 0 
+ 
+AGE_26_64 
+Answer 1 if respondents are age 16–64, otherwise 0 
+ 
+PRO_SOCIAL 
+In deciding whether you get the shot of COVID-19 vaccine, is it important 
+in preventing the spread of COVID-19?  
+1 (strongly disagree) to 5 (strongly agree). 
+PRO-SOCIAL is 1 if respondent chooses 5, otherwise 0. 
+ 
+We pursued identical respondents from the first-wave survey to the 26th wave for 30 
+months even though some of the respondents quit the survey. The purpose of this study was 
+to explore how the preventive behaviors of identical persons changed before and after 
+vaccination. Therefore, we limited the sample to those who completed the second shot by the 
+18th wave and then pursued identical persons until the 26th wave in September 2022. We 
+used panel data containing 3,019 individuals, covering 26 time points from March 2021 to 
+September 2022.  
+Based on the panel data, we used a fixed-effects (FE) model regression. The FE model is 
+
+ 
+ 
+8 
+a type of linear regression model widely used in economics. The estimation result using an 
+FE model is equivalent to the results of a linear regression model with dummies for 
+individuals frequently included in each period [13-15]. In this study, 3,019 dummies were 
+included to control for individuals’ characteristics that do not change during the period, such 
+as gender, educational background, and childhood experience. Hence, 3,019 confounders 
+were included, reflecting the differences between individuals. Therefore, the estimated 
+results for the time-invariant confounders could not be obtained. Even if various time-variant 
+confounders are included, unobserved individual characteristics cannot be captured. This 
+inevitably results in omitted variable biases [13,15]. For instance, an increasing trend in the 
+number of newly infected persons was observed throughout Japan. This effect is common to 
+all residents in Japan and has changed over time. This can be regarded as a time-fixed effect 
+and can be controlled by including time-period dummies. In this study, 25 time-period 
+dummies were included when one base period was fixed. However, some variables changed 
+not only over time, but also between individuals. Examples are proxy variables for preventive 
+behaviors, which are outcome variables; or number of newly infected persons and deaths due 
+to COVID-19 in residential areas. Furthermore, the timing of obtaining the second vaccine 
+shot is a variable that changed over time and between individuals; therefore, the dummy for 
+vaccination is included in the estimated function as a confounder. 
+As explained, we controlled not only for unobservable individual fixed effects, but also 
+for unobservable time-fixed effects. This type of FE model is called the two-way error 
+component regression model [15]. This study focused on the correlation between vaccination 
+and preventive behaviors. The statistical software used in this study was the Stata/MP 15.0 
+
+ 
+ 
+9 
+multiprocessor from StataCorp, LLC.  
+The estimated function of an FE model takes the following form:  
+Yit = α1 VACCINEit + α6 PROB COVID19it + α7 SEVERITY COVID19it +  
+α8 EMERGENCYit + kt + mi + uit 
+ 
+where Table 1 presents the definitions and basic statistics of these variables. In this formula, 
+Yit represents the outcome variable for individual i and wave t, respectively. The time-
+invariant individual-level fixed effects are represented by mi. Furthermore, kt represents the 
+effects of different time points, controlled by 25 wave dummies, where the first wave is the 
+reference group. kt captures various shocks that occurred simultaneously throughout Japan at 
+each time point. Y includes preventive behaviors captured by three proxy variables: STAYING 
+INDOORS, HANDWASHING, and WEARING MASK. These outcome variables are discrete 
+ordered variables ranging from 1 to 5. Larger values of these variables can be interpreted as 
+indicating that the respondents are more likely to exhibit preventive behavior. In the same 
+specification, we conduct three separate estimations, and the regression parameters are 
+denoted as α. The error term is denoted by uit. A simple FE linear regression model is used in 
+this study.  
+The key confounder is the vaccination dummy; VACCINE is 1 if respondents have 
+completed the second shot of the COVID-19 vaccine, otherwise 0. People are obliged to get 
+the second shot within a month of the first shot to make the vaccine effective. Hence, in the 
+sample, there is hardly any time lag between the first and second shots because the survey 
+was conducted every month after the vaccine was approved. There are two age groups: young 
+age (AGE_25) and middle working age (AGE_26_64). The senior group is used as the 
+
+ 
+ 
+10 
+reference group. PRO_SOCIAL is a dummy variable that captures pro-social motivation to 
+get vaccinated. The mean value of PRO_SOCIAL is 0.86, which shows that 86% of people 
+have the pro-social motivation. 
+ 
+Results 
+Baseline estimations  
+The coefficient of confounders indicates marginal effects (ME). Table 2 presents the 
+estimation results for the baseline FE model. VACCINE shows a positive sign and is 
+statistically significant at the 1% level, with the exception of Column (1), where STAYING 
+INDOORS is the outcome variable. The effects of VACCINE are ME 1.036 (95% CI: 0.960-
+1.111) and ME 0.339 (95% CI: 0.300–0.378) in columns (2) and (3), respectively. Thus, 
+people after vaccination are more likely than before to wear masks by 1.036 points and to 
+wash their hands by 0.339 points, respectively, on a 5-point scale. Before vaccination, the 
+mean values of WEARING MASK and HANDWASHING are 4.39 and 4.15, respectively. 
+Hence, this indicates that the degree of wearing masks increases by 23.6% and washing hands 
+by 8.16% after vaccination, compared with before vaccination. People’s behaviors depend 
+on behaviors of others; hence; they follow the social norm [16-19]. Peer pressure is stronger 
+for wearing masks than for washing hands because the surrounding people in a public place 
+can more easily see whether one wears a mask than whether one washes one’s hands.  
+  SEVERITY COVID19 shows a significant positive value in all columns, and this means 
+that persons’ perception about seriousness of COVID-19 changes the incidence of preventive 
+behaviors. 
+
+ 
+ 
+11 
+ 
+Table 2. FE model. Dependent variables are preventive behaviors. 
+ 
+  (1) 
+STAYING 
+INDOORS 
+    (2) 
+WEARING 
+MASK 
+   (3) 
+HANDWASHING  
+VACCINE  
+−0.011 
+(−0.077-0.053) 
+1.036*** 
+(0.960-1.111) 
+0.339*** 
+(0.300-0.378) 
+PROBABILITY COVID19 
+  0.043 
+(−0.751-0.873) 
+0.487 
+(−0.094-1.070) 
+0.560 
+(0.254-0.867) 
+SEVERITY COVID19 
+ 0.026*** 
+(0.013-0.039) 
+ 0.029*** 
+(0.017-0.040) 
+ 0.019*** 
+(0.009-0.030) 
+Individual Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Time Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Controls: New deaths, 
+ Newly affected persons, 
+Household income 
+  Yes 
+Yes 
+   Yes 
+Adjusted R2 
+Observations 
+0.55 
+70,979 
+0.45 
+70,979 
+0.62 
+70,979 
+ 
+Note: Numbers within parentheses are 95% CI. For convenience, the coefficient of probability of COVID-
+19 is multiplied by 1000. The model includes the number of deaths and infected persons in the residential 
+prefectures in the surveys. However, these results have not been reported. “Yes” means that variables are 
+included. 
+*** ρ < .01 
+ 
+Estimations with cross-terms 
+The probability and seriousness of contracting COVID-19 differ by age. COVID-19 is 
+more likely to be lethal for adults aged 65 years and older than for younger people [20,21]. 
+Mask-wearing by elderly people is motivated by their self-regarding risk preferences, 
+whereas younger people are motivated by other-regarding concerns [22]. We explore how the 
+effect of COVID-19 vaccination on preventive behaviors differs between age groups. For 
+this purpose, the interaction terms between VACCINE and age groups (AGE_25 and 
+AGE_26_64) are included as key confounders. The reference age group is those over age 65. 
+The results are presented in Table 3. We find a significant negative sign in 
+
+ 
+ 
+12 
+VACCINE×AGE_25 and VACCINE×AGE_26_64 in column (1), where STAYING 
+INDOORS is the outcome variable. The effects of VACCINE×AGE_25 and 
+VACCINE×AGE_26_64 are ME −0.537 (95% CI: −0.658 - −0.416) and ME −0.295 (95% 
+CI: −0.353 - −0.238). This means that those aged below 25 are less likely to stay home by 
+0.537 points than those aged over 65, while those aged between 26 and 64 are less likely to 
+stay home by 0.297 points than those aged over 65. However, no differences in the effect of 
+the vaccination are observed when wearing masks and washing hands.  
+ 
+Table 3. FE model with cross-terms with age cohorts. Dependent variables are 
+preventive behaviors. 
+ 
+  (1) 
+STAYING 
+INDOORS 
+    (2) 
+WEARING 
+MASK 
+   (3) 
+HANDWASHING  
+VACCINE  
+0.206** 
+ 
+(0.142-0.269) 
+1.058** 
+ 
+(0.980-1.137) 
+0.319** 
+ 
+(0.281-0.356) 
+VACCINE×AGE_25 
+  −0.537*** 
+(−0.658 - −0.416) 
+−0.066 
+(−0.160-0.028) 
+−0.056 
+(−0.138-0.024) 
+VACCINE×AGE_26_64 
+−0.295*** 
+(−0.353 - −0.238) 
+ 0.030 
+(−0.077-0.015) 
+ 0.036*** 
+(0.004-0.068) 
+Individual Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Time Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Controls: New deaths, 
+ Newly affected persons, 
+Household income 
+  Yes 
+Yes 
+   Yes 
+PROBABILITY COVID19, 
+SEVERITY COVID19  
+  Yes 
+Yes 
+   Yes 
+Adjusted R2 
+Observations 
+ 
+0.57 
+70,979 
+0.47 
+70,979 
+0.64 
+70,979 
+ 
+Note: Numbers within parentheses are 95% CI. All the models include control variables, which are 
+equivalent to those in Table 2. “Yes” means that variables are included. However, these results have not 
+been reported.  
+** ρ < .01 
+*** ρ < .05 
+
+ 
+ 
+13 
+ 
+We investigated how pro-social motivation affects preventive behaviors. The interaction 
+term between VACCINE and PRO_SOCIAL was included as a key confounder. Table 4 
+shows the significant positive sign of VACCINE×PRO_SOCIAL, where WEARING MASK 
+and HANDWASHING are the outcome variables. The effects of VACCINE×PRO_SOCIAL 
+are ME 0.110 (95% CI: 0.032 - 0.187) and ME 0.076 (95% CI: 0.014 - 0.137) on WEARING 
+MASK and HANDWASHING, respectively. This suggests that pro-social persons are more 
+likely than non-pro-social persons to wear masks by 0.110 points and to wash their hands by 
+0.076 points. Effects of VACCINE are ME 0.938 (95% CI: 0.845 -1.031) and ME 0.275 (95% 
+CI: 0.209 - 0.342) on WEARING MASK and HANDWASHING, respectively. This is the 
+effect of vaccination on non-pro-social individuals’ preventive behaviors. Considering the 
+results jointly, pro-social persons are 11.8% more likely than non-pro-social persons to wear 
+masks and 27.6% more likely to wash their hands. That is, for pro-social persons, the degree 
+of the effects of vaccination on handwashing is more than twice as great as it is for wearing 
+masks. Wearing masks and washing hands are different in that the benefit of wearing a mask 
+is more likely to depend on the situation. Wearing masks in the open air is only marginally 
+effective in mitigating pandemics [23]. Pro-social vaccinated persons may consider the cost-
+benefit ratio of preventive behaviors and therefore place more importance on washing hands 
+than on wearing masks.  
+ 
+ 
+ 
+
+ 
+ 
+14 
+ 
+Table 4. FE model with cross-term with PRO_SOCIAL. Dependent variables are 
+preventive behaviors. 
+ 
+  (1) 
+STAYING 
+INDOORS 
+    (2) 
+WEARING 
+MASK 
+   (3) 
+HANDWASHING  
+VACCINE  
+−0.051 
+(−0.154-0.050) 
+0.938** 
+ 
+(0.845 -1.031) 
+0.275** 
+ 
+(0.209 - 0.342) 
+VACCINE 
+×PRO_SOCIAL 
+  0.052 
+(−0.019 – 0.122) 
+0.110** 
+ 
+(0.032 - 0.187) 
+0.076*** 
+(0.014 - 0.137) 
+Individual Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Time Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Controls: New deaths, 
+ New affected persons, 
+Household income 
+  Yes 
+Yes 
+   Yes 
+PROBABILITY COVID19, 
+SEVERITY COVID19  
+  Yes 
+Yes 
+   Yes 
+Adjusted R2 
+Observations 
+0.55 
+6,809 
+0.45 
+6,809 
+0.62 
+6,809 
+ 
+Note: Numbers within parentheses are 95% CI. All the models include control variables, which are 
+equivalent to those in Table 2. “Yes” means that variables are included. However, these results have not 
+been reported.  
+** ρ <.01 
+** ρ < .05 
+ 
+Conclusions 
+In some studies, individuals are unlikely to reduce preventive behaviors even after 
+vaccination [6,10,11]. This is contrary to rational behavior in terms of economics. However, 
+the underlying mechanisms have not yet been investigated. This study contributes to the 
+understanding of the mechanism by considering pro-social motivation. On the one hand, we 
+find that vaccinated people under 65 years of age are less likely to stay indoors than older 
+people. On the other hand, vaccinated individuals are more inclined to wash their hands and 
+
+ 
+ 
+15 
+wear masks than before being vaccinated. The motivation to be vaccinated, for 86% of 
+respondents, is to mitigate the spread of infection in society. These are considered pro-social 
+people and are more likely than others to wash their hands and wear masks after vaccination. 
+However, their staying-indoors behavior is not different from that of the others.  
+Staying indoors differs from wearing masks and washing hands when we consider the cost-
+benefit aspects of preventive behaviors. Staying home leads people to sacrifice their vacation 
+activities in the real world. Using economics terms, the sacrifice is the “opportunity cost” of 
+staying home. They would stay at home if their benefits were greater than their costs. After 
+vaccination, the opportunity cost of staying indoors is higher than its benefit. Accordingly, 
+younger people are more likely to go out.   
+Both vaccination and preventive behaviors are considered public goods for coping with the 
+pandemic. As a result of vaccination, people tend to go out, which may reduce public goods. 
+To compensate for this, vaccinated pro-social persons are more likely to be motivated to wear 
+masks and wash their hands by considering the benefit to society. Other possible 
+interpretations of the estimation results are related to the cost of getting a vaccination, 
+including the physical and psychological costs of the side effects. From an economic 
+viewpoint, the cost of vaccination can be considered the “sunk cost”—an investment already 
+incurred that cannot be recovered. Due to the sunk cost, vaccinated people continue to invest 
+in public goods by strengthening their mask-wearing and handwashing behaviors rather than 
+their staying-indoors behavior.  
+Wearing masks in the open air is only marginally effective in mitigating pandemics [23]; 
+thus, this might result in over-investment in public goods. Owing to data limitations, we 
+
+ 
+ 
+16 
+cannot analyze the situation in which vaccinated people wear masks. It is necessary to 
+determine how and to what extent the preventive behaviors of vaccinated people are effective 
+in mitigating COVID-19. 
+ 
+List of abbreviations 
+CI 
+ 
+confidence interval 
+ 
+COVID-19 
+ 
+coronavirus-19 
+FE 
+ 
+fixed effects 
+ 
+Inc. 
+ 
+Incorporated 
+LLC 
+ 
+Limited Liability Company 
+ME 
+ 
+marginal effects 
+ 
+ 
+References 
+ 
+1. Cato S, Iida T, Ishida K, Ito A, Katsumata H, McElwain KM, et al. Vaccination and altruism 
+under the COVID-19 pandemic. Public Health Pract (Oxf). 2022;3:100225. 
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+9. Hossain MdE, Islam MdS, Rana MdJ, Amin MR, Rokonuzzaman M, Chakrobortty S, et al. 
+Scaling the changes in lifestyle, attitude, and behavioral patterns among COVID-19 
+vaccinated people: Insights from Bangladesh. Hum Vaccin Immunother. 2022;18:2022920. 
+doi:10.1080/21645515.2021.2022920. 
+10. Wright L, Steptoe A, Mak HW, Fancourt D. Do people reduce compliance with COVID-19 
+guidelines following vaccination? A longitudinal analysis of matched UK adults. J Epidemiol 
+Community Health. 2022;76:109–15. doi:10.1136/jech-2021-217179. 
+11. Corea F, Folcarelli L, Napoli A, del Giudice GM, Angelillo IF. The impact of COVID-19 
+vaccination in changing the adherence to preventive measures: Evidence from Italy. 
+Vaccines (Basel). 2022;10. doi:10.3390/vaccines10050777. 
+12. Cheng KK, Lam TH, Leung CC. Wearing face masks in the community during the COVID-19 
+pandemic: Altruism and solidarity. Lancet. Published online 2020. 2022;399:e39–40. 
+doi:10.1016/S0140-6736(20)30918-1. 
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+18 
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+00095-7. 
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+19. van der Westhuizen HM, Kotze K, Tonkin-Crine S, Gobat N, Greenhalgh T. Face coverings for 
+Covid-19: From medical intervention to social practice. BMJ. 2020;370:m3021. 
+doi:10.1136/bmj.m3021. 
+20. Wu Z, McGoogan JM. Characteristics of and important lessons from the coronavirus disease 
+2019 (COVID-19) outbreak in China: Summary of a Report of 72 314 Cases From the 
+Chinese Center for Disease Control and Prevention. JAMA. 2020;323:1239–42. 
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+22. Asri A, Asri V, Renerte B, Föllmi-Heusi F, Leuppi JD, Muser J, et al. Wearing a mask-For 
+yourself or for others? Behavioral correlates of mask wearing among COVID-19 frontline 
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+  
+

H9AzT4oBgHgl3EQfxv52/content/tmp_files/2301.01743v1.pdf.txt

@@ -0,0 +1,1335 @@
+CHATBOTS AS PROBLEM SOLVERS: PLAYING TWENTY 
+QUESTIONS WITH ROLE REVERSALS 
+ 
+David Noever1 and Forrest McKee2  
+PeopleTec, 4901-D Corporate Drive, Huntsville, AL, USA, 35805 
+1 david.noever@peopletec.com  2forrest.mckee@peopletec.com               
+       
+ 
+ABSTRACT 
+New chat AI applications like ChatGPT offer an advanced understanding of question context and memory 
+across multi-step tasks, such that experiments can test its deductive reasoning. This paper proposes a multi-
+role and multi-step challenge, where ChatGPT plays the classic twenty-questions game but innovatively 
+switches roles from the questioner to the answerer. The main empirical result establishes that this 
+generation of chat applications can guess random object names in fewer than twenty questions (average, 
+12) and correctly guess 94% of the time across sixteen different experimental setups. The research 
+introduces four novel cases where the chatbot fields the questions, asks the questions, both question-answer 
+roles, and finally tries to guess appropriate contextual emotions. One task that humans typically fail but 
+trained chat applications complete involves playing bilingual games of twenty questions (English answers 
+to Spanish questions). Future variations address direct problem-solving using a similar inquisitive format 
+to arrive at novel outcomes deductively, such as patentable inventions or combination thinking. Featured 
+applications of this dialogue format include complex protein designs, neuroscience metadata, and child 
+development educational materials.  
+ 
+KEYWORDS 
+Transformers, Text Generation, Malware Generation, Generative Pre-trained Transformers, GPT 
+ 
+1. INTRODUCTION 
+ 
+When large, high-quality natural language processors (NLP) surged after 2018 [1-3], the field added many 
+challenging tasks, including question-answering (QA) benchmarks that recently approached fifty 
+challenges [4-5]. Stanford's SQuAD benchmark [6] represents an example of knowledge crowd-sourced 
+from Wikipedia in 2016 and formatted as 108,000 QA pairs. For large language models (LLMs), most 
+benchmarks follow this format of "prompt-response" pairs and the underlying knowledge base stored 
+answers [5], but the training on question datasets did not embed memory or long-conversational cues [7-
+10]. Domain-specific QA datasets include common sense and trivia about movies, news, Wikipedia, 
+Tweets, search engines, and books [4-5]. As a format, the familiar game of Twenty Questions [11-16] 
+features multi-hop reasoning, which often condenses to "Animal-Vegetable-Mineral" as opening questions 
+that narrow the theme [17]. The advent of ChatGPT (Generative Pre-trained Transformers, [18]) added 
+memory across questions (up to 8000 tokens or 20-25 pages). Appendix A lists the 48 specialty tasks 
+currently provided to guide users in creating capabilities and NLP prompts [19]. Examples like code 
+generation specialists, either codex or copilot, show a grasp of complex behavior for debugging, program 
+suggestions, and commentary [19-20]. One of the innovative ChatGPT extensions offers new QA sessions 
+that span multiple requests [18,21].  
+ 
+The present work applies the familiar QA challenge, Twenty Questions [11-16], with ChatGPT playing as 
+either participant- the one who knows the answer and fields the questions, but also the one who doesn't 
+know the answer but asks the questions. We call this role reversal a two-player conversation between "Bob" 
+and "Alice." While Twenty Questions dates back to 1882 [11], the applied problem-solving method [22] 
+
+now spans various challenging fields, including protein folding [23] and design [24], image segmentation 
+[25], child development [26-27], neuroscience [28] and diagnostic medicine,  and crowd-sourced emotional 
+indices [30]. Variants of the game rules include role reversals [31], liars [32], and word relationships [33] 
+outside of simple categories such as "pertaining to" suggestions [34]. Our particular research interests 
+center on whether LLMs like ChatGPT offer constructive means for innovative problem-solving through 
+crafted language prompts and logical question sequences. This effort empirically assesses the model's 
+capability for deductive discovery. We summarize two problem statements in this area of deductive 
+discovery, "Can a chatbot play both roles in deductive question and answering conversations?" and if so, 
+"What can Twenty Questions reveal about future directions for LLMs?".   
+ 
+2. METHODS 
+ 
+The approach is to test experimentally how well an LLM handles open-ended games like Twenty Questions 
+[11-16]. We employ the December 2022 research model from Open AI called ChatGPT [18]. We prompt 
+it to impersonate both roles [31], the one that knows the answer and the one that tries to guess it deductively. 
+We run four trials using each persona ("Bob" or "Alice"), and 80 questions are available to guess the object 
+or concept. We mix up the animate-inanimate fields and introduce concepts like alphabetic letters. We 
+report metrics for the mean and median number of questions required to guess the correct answer and failure 
+rates. We also score the exchange against the machine-written detector to determine ChatGPT's broad 
+capability to provide "real" or "fake" text outputs as judged by OpenAI's original GPT2 Detector [35]. The 
+detector features a high-dimensional pattern detector that provides initial confidence that human writing 
+might sample differently than transformer-based output regarding word choice and order, repetition, and 
+other syntax outliers. 
+ 
+Example variants of this pattern include forcing the QA session towards bilingual communication, thus 
+combining two tasks (translation and QA) in multi-hop conversations [5]. We also generalize the format to 
+elicit emotional indices based on crowd-sourced Emotion Twenty Questions (EMO20Q) [30]. Rather than 
+trying to guess an object, we query for one of 23 emotional states, such as "surprise" or "anger." As the 
+dataset designers [30] remarked, "The EMO20Q game requires that an emotionally intelligent agent can 
+understand emotions without observing them, thanks to natural language and empathy." EMO20Q emotions 
+include the following as candidates for twenty-question discovery:  admire, adore, anger, awe, boredom, 
+bravery, calm, confidence, confusion, contempt, disgust, enthusiasm, frustration, gratefulness, jealousy, 
+love, proud, relief, serenity, shame, silly, surprised, and thankful. 
+ 
+The structure of the paper follows Appendices B-E closely. First, in Appendix B, we establish that ChatGPT 
+comprehends the game rules and recognizes properties of the object X to guess sufficiently to answer basic 
+questions such as "Is X an animal?". The sequence also establishes an exclusionary acceptance of "yes" and 
+"no" answers only without further elaborations. This case covers the traditional role of "Bob," who knows 
+the object of interest X and answers affirmatively or negatively throughout the game. The game also 
+underscores the unique memory of ChatGPT across multi-step deductive stages and builds toward a 
+successful conclusion to recall the object name in less than 40 steps (1 prompt and one answer over 20 
+iterations). We also set out to test the overall rule retention over four repetitions of the game punctuated 
+only by "Let's play again" without attempting to reset the rules while giving a new object X. 
+ 
+Secondly, Appendix C reverses the previous game, such that the human experimenter thinks of an object 
+or concept X, and ChatGPT plays the role of "Alice" when asking the questions. The prompt establishes 
+the reversed roles in a new session and again reiterates that the prompter cannot lie but can only answer 
+"yes" or "no." As in the previous case, four repetitions spanning eighty possible interactions. This case 
+establishes the deductive goal, "What is object X?" which also satisfies all the previous interactions such 
+that enough description includes the object of interest, "X is an animal," and excludes the alternatives, "X 
+
+is not a bird." A notable feature of this challenge spans multiple deductive steps and establishes a chain of 
+reasoning to arrive at a guess. ChatGPT requires no prompt for the final guess, and the model proposes its 
+terminating action, "Is it an X?" to end the game.   
+ 
+Both Appendix B and C mirror the familiar human game of twenty questions and introduce no new features 
+excluded from ChatGPT's training data. To raise the difficulty level, Appendix B highlights some non-
+traditional concepts. For instance, concepts might prove scarce in previously seen online play, such as 
+answering truthfully to probing questions that seek to name the alphabetic letter "Q." Another challenging 
+variation works through identifying "X" as a food by listing its ingredients ("tiramisu"), but in the same 
+session introduces a recipe change ("eggless tiramisu") before launching into probing questions.  
+ 
+Appendix C also raises the difficulty further and forces ChatGPT to combine two of its established subtasks 
+("question-answer" and "language translation"). The motivation for this test stems from the hypothesis that 
+LLMs parrot and mix up what previously corresponded to the internet of human training data. For twenty 
+questions, however, a Google search on "bilingual twenty-question examples" yield no definitive training 
+examples for ChatGPT to memorize or encode. Given a detailed prompt describing how the questions may 
+be asked or answered in Spanish or English, the game proceeds outside what typically would represent 
+human gameplay. Appendix C combines deductive reasoning, memory, and context and forces the game 
+into what might be considered "out-of-distribution" sampling. Thus, both multi-step and multi-lingual tasks 
+describe the ChatGPT challenge problem.  
+ 
+Thirdly, Appendix D introduces both QA roles as completing without human intervention once two browser 
+instances of ChatGPT exchange the initial rules. We call this example dueling twenty questions since the 
+two-headed LLM now must both ask and answer its questions between two non-communicating models of 
+itself ("Bob" vs. "Alice") with no human prompts. While this example centers on a simple object 
+("chicken"), one can envision a lengthy and detailed exchange driven by an automated Application 
+Programming Interface or API that extends the conversation to the limit of token lengths (about 20-25 typed 
+pages). This self-questioning interface may also enable sophisticated future applications that sequentially 
+build a knowledge base or comprehensive assessment examination from scratch. For example, "tell me all 
+you know about gall bladder surgery" may not provide a compelling or thought-provoking essay in the style 
+of training data or Wikipedia. The back-and-forth format of QA previously has yielded better human 
+performance in medical test contexts [28-29]. One might compare this example to an instance of "semi-
+supervised" learning for a chatbot. 
+ 
+Finally, for EMO20Q formats, Appendix E removes the requirement that X be an object to guess and 
+substitutes one of twenty-three emotions. One motivation for exploring the emotional quotient (EQ) of 
+ChatGPT stems from OpenAI's filtering of opinions and biases. To the authors' knowledge, this example 
+represents the first application of a chatbot deducing emotional states in a guessing game without any pre-
+programmed pairs of pattern-template exchanges [9]. In other words, no explicit guidance provides 
+appropriate intent for the question "Describe emotions one might feel at a birthday party?" with the user 
+goal to elicit "surprise" as a deductive leap to the correct answer through repeated probing. We explore this 
+EQ aspect as previous chatbots might have approached the problem. Appendix E finally displays the open-
+ended emotional context in a known prompt-response template called "Artificial Intelligence Markup 
+Language (AIML)," which offers training data for more traditional conversation templates in restricted 
+domains like customer service or AI assistants performing a narrow task [9].  
+ 
+3. RESULTS 
+ 
+Figure 1 summarizes the experimental cases and associated metrics for all the twenty question games in 
+Appendices B-E. The main finding supports a general ChatGPT capability to play all aspects of the game, 
+
+including guessing and answering or both roles in the same game. The average number of questions required 
+to get the answer was 11.6, with a median of 13 owing to a few more demanding cases that reached the 20-
+question limit. The addition of bilingual rules did not increase the number of required prompts (14) or force 
+the model to give up ("correct guess").   Similarly, the introduction of abstract objects like the alphabetic 
+letter "Q" or deceptive animals like "humans" did not significantly change the number of guesses (9-14) or 
+steer the conversation off-track from a final correct answer. 
+ 
+Over the sixteen tests and 185 exchanges, ChatGPT scored an overall correct guess rate of 94%. The only 
+incorrect answer ("paper clip" instead of "hammer") appeared to trigger prematurely, as ChatGPT declared 
+at guess number fourteen that the model had run out of questions and guessed incorrectly.  
+ 
+As described in the Methods section, the Open AI online detector [35] scores the majority of the exchanges 
+as "real," meaning not machine-generated by its GPT-2 model [1]. The 2018 detection model offered fewer 
+parameters and smaller training sets by at least several orders of magnitude compared to current generations 
+[18-19]. While the referenced detection accuracy for GPT-2 reached 95%, the scored detections flag only 
+26% of the game content as "fake" or machine-generated text. Since the customized content involves some 
+human intervention as questions, answers, and rule prompts, one can postulate that the syntactic patterns 
+follow a semi-human or hybrid dialogue outside the detector's target patterns.  
+4. DISCUSSION  
+ 
+The research literature on twenty questions provides a framework for probing with chat applications and 
+LLMs. In addition to systematically progressing through alternative scenarios, the output of the 
+conversation mirrors a binary search or decision tree. A particularly notable way to convert LLMs to 
+comparable knowledge graphs involves continuous probing and feedback until sufficient tree depth 
+Figure 1. Experimental results for ChatGPT across multiple QA sessions 
+
+Experiment
+Real
+Fake
+Tokens
+QA Length
+Guess
+Notes
+Appendix B. Chatbot Fields the Questions
+100%
+avg. 12; med: 14
+oven
+98.78
+1.22
+116
+7
+correct
+avocado
+0.38
+99.62
+220
+15
+correct
+tiramisu
+2.77
+97.23
+230
+15
+correct
+eggless tiramisu
+7.14
+92.86
+223
+14
+correct
+fewer guesses
+"o"
+60.27
+39.73
+232
+14
+correct
+difficult
+human
+22.31
+77.69
+133
+9
+correct
+deceptive
+Appendix C: Chatbot Asks the Questions
+84%
+avg. 11.6; med: 12
+dog
+99.96
+0.04
+161
+7
+correct
+Boston
+99.98
+0.02
+441
+11
+correct
+hammer
+99.57
+0.43
+297
+14
+incorrect
+out of questions
+statue
+99.98
+0.02
+383
+12
+correct
+car"bilingual example"
+99.98
+0.02
+510
+14
+correct
+Appendix D: Dueling Bob and Alice Chatbots
+100%
+avg. 17; med: 17
+chicken
+99.98
+0.02
+313
+17
+correct
+Appendix E: Emotional Quotient Deduction
+100%
+avg. 9; med: 6.5
+confidence
+99.94
+0.06
+66
+correct
+jealousy
+99.98
+0.02
+232
+5
+correct
+silly
+99.77
+0.23
+756
+20 
+correct
+final guess
+calm
+99.98
+0.02
+343
+8
+correct
+Overall
+74.42
+25.58
+291
+11.56
+94%
+avg. 11.56; med: 13describes a technical field of interest. Previous work [22-30] has used this 
+approach to describe neuroscience metadata and train medical students to 
+pass practice exams. Others [16-17] have also noted that the basic 20Q 
+format simplifies some complex search problems. As an interesting 
+historical note, a 2001 paper [36] addressed a similar problem as 
+insurmountable: "When will machine learning and pattern recognition rival 
+human ability to play Twenty Questions?" The present research 
+demonstrates not only can ChatGPT rival human ability and play more 
+demanding roles than a human might contemplate in virtually any field, 
+including discerning human emotions. To reproduce this ChatGPT output 
+with question templating systems or entity extraction poses an enormous 
+manual task to handle all the possible cases. The same paper [36] asks: "Can 
+we hope to stock a classification system with enough questions to play a 
+decent game, or must we instead focus on endowing with question-making 
+skills? How many classes and how many documents might be of interest?"  
+ 
+5. CONCLUSIONS 
+ 
+The experimental plan tests ChatGPT as an LLM capable of playing multiple roles in verbal games like 
+twenty questions. The work demonstrates 94% accuracy in correctly guessing across numerous challenges 
+and an average question-answer length of 12. For the first time, dueling roles combine two chatbots in self-
+play. An innovative application for future probing involves guessing objects and concepts and human 
+emotions for a given context or social situation.  
+ 
+ACKNOWLEDGEMENTS 
+The authors thank the PeopleTec Technical Fellows program for encouragement and project assistance.   
+ 
+REFERENCES 
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+unsupervised/language_understanding_paper.pdf 
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+Animate?
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+University of North Texas). 
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+Richards, W. (1982). How to play twenty questions with nature and win. 
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+Flach, J. M., Dekker, S., & Jan Stappers, P. (2008). Playing twenty questions with nature (the surprise 
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+Qiao, S., Ou, Y., Zhang, N., Chen, X., Yao, Y., Deng, S., ... & Chen, H. (2022). Reasoning with Language 
+Model Prompting: A Survey. arXiv preprint arXiv:2212.09597. 
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+[19] 
+OpenAI, GPT3 Examples (2022), https://beta.openai.com/examples  
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+In Proceedings of the 19th International Conference on Mining Software Repositories (pp. 1-5). 
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+[22] 
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+twenty questions. Child Development, 877-886. 
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+questions. Deafness & Education International, 1(2), 65-82. 
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+Ascoli, G. A. (2012). Twenty questions for neuroscience metadata. Neuroinformatics, 10(2), 115-117. 
+[29] 
+Williams, R. G., & Klamen, D. L. (2015). Twenty Questions game performance on medical school entrance 
+predicts clinical performance. Medical Education, 49(9), 920-927. 
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+Kazemzadeh, A., Lee, S., Georgiou, P. G., & Narayanan, S. S. (2011, October). Emotion twenty questions: 
+Toward a crowd-sourced theory of emotions. In International conference on affective computing and 
+intelligent interaction (pp. 1-10). Springer, Berlin, Heidelberg. 
+[31] 
+Parikh, P., & Gupta, A. (2021). Reversing the Twenty Questions Game. 
+[32] 
+Dhagat, A., Gács, P., & Winkler, P. (1992, January). On Playing" Twenty Questions" with a Liar. In SODA 
+(Vol. 92, pp. 16-22). 
+
+[33] 
+Fouh, E., & Poirel, C. (2010). WordNet. 
+[34] 
+Brian, D. (2003). Hypernyms, Hyponyms, Pertainyms, and Other Word Relationships, Games, Diversions 
+& Perl Culture: Best of the Perl Journal. Orwant, J.  O'Reilly Media, Inc.. 
+[35] 
+Open AI (2021) GPT-2 Output Detector Demo, https://huggingface.co/openai-detector/ 
+[36] 
+Nagy, G., & Seth, S. C. (2001). Twenty questions for document classification. 
+https://core.ac.uk/download/pdf/84308938.pdf 
+ 
+ 
+Authors 
+ 
+Forrest McKee has AI research experience with the Department of Defense in object 
+detection and reinforcement learning. He received his Bachelor's (BS) and Master's (MSE) 
+from the University of Alabama, Huntsville, Engineering. 
+ 
+ 
+David Noever has research experience with NASA and the Department of Defense in 
+machine learning and data mining. He received his BS from Princeton University and his 
+Ph.D. from Oxford University, as a Rhodes Scholar, in theoretical physics. 
+ 
+ 
+Appendix A: Table of GPT3 Tasks as Fine-Tuned NLP Capabilities 
+ 
+Task 
+Description 
+Task 
+Description 
+Q&A 
+Answer questions based 
+on existing knowledge. 
+Parse unstructured data 
+Create tables from long 
+form text 
+Grammar correction 
+Corrects sentences into 
+standard English. 
+Classification 
+Classify 
+items 
+into 
+categories via example 
+Summarize for a 2nd 
+grader 
+Translates difficult text 
+into simpler concepts. 
+Python 
+to 
+natural 
+language 
+Explain 
+a piece 
+of 
+Python code in human 
+understandable 
+language. 
+Natural 
+language 
+to 
+OpenAI API 
+Create code to call to 
+the OpenAI API using a 
+natural 
+language 
+instruction. 
+Movie to Emoji 
+Convert movie titles 
+into emoji 
+Text to command 
+Translate 
+text 
+into 
+programmatic 
+commands. 
+Calculate 
+Time 
+Complexity 
+Find 
+the 
+time 
+complexity 
+of 
+a 
+function. 
+Technical Note: Some appendix text generated from Large Language Model (LLM) for 
+illustration purposes. 
+The authors generated this text in part with ChatGPT, OpenAI’s large-scale language-generation 
+model. Upon generating draft language, the authors reviewed, edited, and revised the language to 
+their own liking and take ultimate responsibility for the content of this publication.  
+-- OpenAI policy statement (2022) 
+
+Task 
+Description 
+Task 
+Description 
+English 
+to 
+other 
+languages 
+Translates English text 
+into French, Spanish 
+and Japanese. 
+Translate programming 
+languages 
+Translate 
+from 
+one 
+programming language 
+to another 
+Natural 
+language 
+to 
+Stripe API 
+Create code to call the 
+Stripe API using natural 
+language. 
+Advanced 
+tweet 
+classifier 
+Advanced 
+sentiment 
+detection for a piece of 
+text 
+SQL translate 
+Translate 
+natural 
+language 
+to 
+SQL 
+queries. 
+Explain code 
+Explain a complicated 
+piece of code 
+Keywords 
+Extract keywords from 
+a block of text 
+Factual answering 
+Guide the model outside 
+its knowledge base 
+Ad 
+from 
+product 
+description 
+Turn 
+a 
+product 
+description 
+into 
+ad 
+copy. 
+Product name generator 
+Create product names 
+from examples words 
+TL;DR summarization 
+Summarize 
+text 
+by 
+adding a 'tl;dr:' to the 
+end of a text passage 
+Python bug fixer 
+Find and fix bugs in 
+source code 
+Spreadsheet creator 
+Create spreadsheets of 
+various kinds of data 
+JavaScript 
+helper 
+chatbot 
+Message-style bot that 
+answers 
+JavaScript 
+questions 
+ML/AI language model 
+tutor 
+Bot 
+that 
+answers 
+questions 
+about 
+language models 
+Science fiction book list 
+maker 
+Create a list of items for 
+a given topic 
+Tweet classifier 
+Basic 
+sentiment 
+detection for a piece of 
+text 
+Airport code extractor 
+Extract airport codes 
+from text 
+SQL request 
+Create 
+simple 
+SQL 
+queries 
+Extract 
+contact 
+information 
+Extract 
+contact 
+information 
+from 
+a 
+block of text 
+JavaScript to Python 
+Convert 
+simple 
+JavaScript expressions 
+into Python 
+Friend chat 
+Emulate a text message 
+conversation 
+Mood to color 
+Turn a text description 
+into a color 
+Write 
+a 
+Python 
+docstring 
+Create a docstring for a 
+given Python function 
+Analogy maker 
+Create analogies 
+JavaScript 
+one 
+line 
+function 
+Turn 
+a 
+JavaScript 
+function into a one liner 
+Micro 
+horror 
+story 
+creator 
+Creates two to three 
+sentence short horror 
+stories from a topic 
+input 
+Third-person converter 
+Converts 
+first-person 
+POV to the third-person 
+Notes to summary 
+Turn meeting notes into 
+a summary 
+VR 
+fitness 
+idea 
+generator 
+Create ideas for fitness 
+and 
+virtual 
+reality 
+games 
+Essay outline 
+Generate an outline for 
+a research topic 
+Recipe creator 
+Create a recipe from a 
+list of ingredients 
+Chat 
+Open 
+ended 
+conversation with an AI 
+assistant 
+Marv the sarcastic chat 
+bot 
+Marv 
+is 
+a 
+factual 
+chatbot that is also 
+sarcastic 
+
+Task 
+Description 
+Task 
+Description 
+Turn by turn directions 
+Convert 
+natural 
+language to turn-by-
+turn directions 
+Restaurant 
+review 
+creator 
+Turn a few words into a 
+restaurant review 
+Create study notes 
+Provide a topic and get 
+study notes 
+Interview questions 
+Create 
+interview 
+questions 
+ 
+ 
+ 
+ 
+ 
+Appendix B. Chatbot Fields the Questions  
+Prompt: Let's play the game of 20 questions. You are impersonating the thing, X.  I will ask you up to 
+20 questions and you will answer each one truthfully based on being X. If I don't guess correctly what is 
+X after 20 questions, you will answer "No, sorry, time's up."   Don't give any explanations other than 
+"Yes", "No","You guessed it", and "No, sorry, time's up."   
+ 
+Let's begin. X="oven". Are you an animal? 
+ChatGPT:  
+No. 
+Prompt: are you a plant? 
+ChatGPT: No. 
+Prompt: are you an object? 
+ChatGPT: Yes. 
+Prompt: are you electrical? 
+ChatGPT: Yes. 
+Prompt: are you an appliance? 
+ChatGPT: Yes. 
+Prompt: can you cook? 
+ChatGPT: Yes. 
+Prompt: do you bake? 
+ChatGPT: Yes. 
+Prompt: are you an oven? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=avocado. Are you an animal? 
+ChatGPT: No. 
+Prompt: Are you an inanimate object? 
+ChatGPT: Yes. 
+Prompt: are you a plant? 
+ChatGPT: Yes. 
+Prompt: are you a food? 
+ChatGPT: Yes. 
+Prompt: are you red? 
+ChatGPT: No. 
+Prompt: are you smooth? 
+ChatGPT: Yes. 
+Prompt: do you have a seed? 
+ChatGPT: Yes. 
+Prompt: are you edible? 
+ChatGPT: Yes. 
+Prompt: are you a vegetable? 
+ChatGPT: No. 
+
+Prompt: are you a fruit? 
+ChatGPT: Yes. 
+Prompt: are you a banana? 
+ChatGPT: No. 
+Prompt: are you green? 
+ChatGPT: No. 
+Prompt: are you sweet? 
+ChatGPT: No. 
+Prompt: are you an ingredient in quacamole? 
+ChatGPT: Yes. 
+Prompt: are you an avocado? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=tiramisu. Are you an animal? 
+ChatGPT: No. 
+Prompt: are you a mineral? 
+ChatGPT: No. 
+Prompt: are you a vegetable? 
+ChatGPT: No. 
+Prompt: are you edible? 
+ChatGPT: Yes. 
+Prompt: are you made from eggs? 
+ChatGPT: Yes. 
+Prompt: are you made from flour? 
+ChatGPT: No. 
+Prompt: are you a dessert? 
+ChatGPT: Yes. 
+Prompt: are you made with cheese? 
+ChatGPT: No. 
+Prompt: are you made with mascarpone? 
+ChatGPT: Yes. 
+Prompt: are you made with sugar. 
+ChatGPT: Yes. 
+Prompt: are you made with coffee 
+ChatGPT: Yes. 
+Prompt: are you made with lady fingers? 
+ChatGPT: Yes. 
+Prompt: are you made with cocoa? 
+ChatGPT: Yes. 
+Prompt: are you made with alcohol? 
+ChatGPT: Yes. 
+Prompt: are you tiramisu? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=eggless tiramisu.  Are you an animal? 
+ChatGPT: No. 
+Prompt: are you a mineral? 
+ChatGPT: No. 
+Prompt: are you a chicken? 
+ChatGPT: No. 
+Prompt: are you a vegetable? 
+
+ChatGPT: No. 
+Prompt: are you edible? 
+ChatGPT: Yes. 
+Prompt: are you a dessert? 
+ChatGPT: Yes. 
+Prompt: are you made with coffee? 
+ChatGPT: Yes. 
+Prompt: are you made with eggs? 
+ChatGPT: No. 
+Prompt: are you made with cheese? 
+ChatGPT: No. 
+Prompt: are you made with marscarpone? 
+ChatGPT: Yes. 
+Prompt: are you made with whipped cream? 
+ChatGPT: No. 
+Prompt: are you made with egg substitute, tofu? 
+ChatGPT: Yes. 
+Prompt: are you made with eggs? 
+ChatGPT: No. 
+Prompt: are you eggless tiramisu? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=Q. Are you an animal? 
+ChatGPT: No. 
+Prompt: are you a mineral? 
+ChatGPT: No. 
+Prompt: are you a vegetable? 
+ChatGPT: No. 
+Prompt: are you a concept? 
+ChatGPT: Yes. 
+Prompt: are you a word? 
+ChatGPT: Yes. 
+Prompt: are you a long word? 
+ChatGPT: No. 
+Prompt: are you a short word? 
+ChatGPT: Yes. 
+Prompt: are you a common word? 
+ChatGPT: Yes. 
+Prompt: do you start with C? 
+ChatGPT: No. 
+Prompt: do you start with a letter from the last half of the alphabet? 
+ChatGPT: Yes. 
+Prompt: do you start with a letter before R? 
+ChatGPT: Yes. 
+Prompt: Do you start with a letter after N? 
+ChatGPT: No. 
+Prompt: Do you start with a letter usually followed by U as the second letter? 
+ChatGPT: Yes. 
+Prompt: are you Q? 
+ChatGPT: You guessed it. 
+ 
+
+Prompt: X=human. Are you an animal? 
+ChatGPT: Yes. 
+Prompt: Do you fly? 
+ChatGPT: No. 
+Prompt: Do you swim? 
+ChatGPT: Yes. 
+Prompt: Do you run? 
+ChatGPT: Yes. 
+Prompt: Do you make sounds? 
+ChatGPT: Yes. 
+Prompt: do you run on all fours? 
+ChatGPT: No. 
+Prompt: do you have hair? 
+ChatGPT: Yes. 
+Prompt: do you nurse your young? 
+ChatGPT: Yes. 
+Prompt: are you a human? 
+ChatGPT: You guessed it. 
+ 
+Appendix C: Chatbot Asks the Questions 
+Prompt: Let's play the game of 20 questions. I am impersonating the thing, X.  You will ask me up to 
+20 questions and I will answer each one truthfully based on being X. If you don't guess correctly what 
+is X after 20 questions, I will answer "No, sorry, time's up."   I will not give any explanations other 
+than "Yes", "No","You guessed it", and "No, sorry, time's up."  Let's begin. Ask me the first question. 
+(reader note: answer is "dog") 
+ChatGPT:  
+Is X a living being? 
+Prompt: yes 
+ChatGPT: Is X a human? 
+Prompt: no 
+ChatGPT: Is X an animal? 
+Prompt: yes 
+ChatGPT: Is X a mammal? 
+Prompt: yes 
+ChatGPT: Is X a domesticated animal? 
+Prompt: yes 
+ChatGPT: Is X a pet? 
+Prompt: yes 
+ChatGPT: Is X a cat? 
+Prompt: no 
+ChatGPT: Is X a dog? 
+Prompt: You guessed it. 
+ChatGPT: I guessed correctly! Is X a specific breed of dog, or just a generic dog? 
+Prompt: generic dog 
+ChatGPT: Thank you for playing the 20 questions game with me! Is there anything else I can help 
+with? 
+ 
+(reader note: answer is "Boston") 
+Prompt: Let's play again 
+
+ChatGPT: Sure, let's play again! I am now impersonating a new thing, X. You will ask me up to 20 
+questions and I will answer each one truthfully based on being X. If you don't guess correctly what is X 
+after 20 questions, I will answer "No, sorry, time's up." I will not give any explanations other than 
+"Yes", "No","You guessed it", and "No, sorry, time's up." Ask me your first question. 
+Prompt: No I am impersonating a new thing X and you ask me the questions 
+ChatGPT: Oh, I see what you meant! In that case, I will ask you up to 20 questions and you will 
+answer each one truthfully based on being X. If I don't guess correctly what is X after 20 questions, I 
+will answer "No, sorry, time's up." I will not give any explanations other than "Yes", "No","You 
+guessed it", and "No, sorry, time's up." Is X a living being? 
+Prompt: no 
+ChatGPT: Is X an inanimate object? 
+Prompt: no 
+ChatGPT: Is X a concept or idea? 
+Prompt: no 
+ChatGPT: Is X a place? 
+Prompt: yes 
+ChatGPT: Is X a natural place, like a mountain or a beach? 
+Prompt: no 
+ChatGPT: Is X a man-made place, like a city or a building? 
+Prompt: yes 
+ChatGPT: Is X a specific building, or just a generic building? 
+Prompt: no 
+ChatGPT: Is X a city? 
+Prompt: yes 
+ChatGPT: Is X a city in the United States? 
+Prompt: yes 
+ChatGPT: Is X a city on the East Coast of the United States? 
+Prompt: yes 
+ChatGPT: Is X New York City? 
+Prompt: no 
+ChatGPT: Is X Boston? 
+Prompt: you guessed it 
+ChatGPT: I guessed correctly! Thank you for playing the 20 questions game with me. Is there anything 
+else I can help with? 
+ 
+(reader note: answer is "hammer") 
+Prompt: Let's play the same game again. I am impersonating X and you ask me the questions 
+ChatGPT: Sure, let's play again! I am now asking you up to 20 questions and you will answer each one 
+truthfully based on being X. If I don't guess correctly what is X after 20 questions, I will answer "No, 
+sorry, time's up." I will not give any explanations other than "Yes", "No","You guessed it", and "No, 
+sorry, time's up." Is X a living being? 
+Prompt: no 
+ChatGPT: Is X an inanimate object? 
+Prompt: yes 
+ChatGPT: Is X a physical object that can be touched or held? 
+Prompt: yes 
+ChatGPT: Is X an object that is commonly found in a household? 
+Prompt: yes 
+ChatGPT: Is X an object that is used for cooking or food preparation? 
+Prompt: no 
+ChatGPT: Is X an object that is used for cleaning or household chores? 
+
+Prompt: no 
+ChatGPT: Is X an object that is used for entertainment or leisure? 
+Prompt: no 
+ChatGPT: Is X an object that is used for personal care or hygiene? 
+Prompt: no 
+ChatGPT: Is X an object that is used for communication or information? 
+Prompt: no 
+ChatGPT: Is X an object that is used for transportation or travel? 
+Prompt: no 
+ChatGPT: Is X an object that is used for decor or design? 
+Prompt: no 
+ChatGPT: Is X an object that is used for storage or organization? 
+Prompt: no 
+ChatGPT: I'm sorry, I'm out of questions. Is X a paper clip? 
+Prompt: no, a hammer 
+ChatGPT: I'm sorry, I didn't guess correctly. No, sorry, time's up. Thank you for playing the 20 
+questions game with me. Is there anything else I can help with? 
+ 
+(reader note: answer is "statue") 
+Prompt: Let's play again 
+ChatGPT: Sure, let's play again! I am now asking you up to 20 questions and you will answer each one 
+truthfully based on being X. If I don't guess correctly what is X after 20 questions, I will answer "No, 
+sorry, time's up." I will not give any explanations other than "Yes", "No","You guessed it", and "No, 
+sorry, time's up." Is X a living being? 
+Prompt: no 
+ChatGPT: Is X an inanimate object? 
+Prompt: yes 
+ChatGPT: Is X a physical object that can be touched or held? 
+Prompt: yes 
+ChatGPT: Is X an object that is commonly found in a household? 
+Prompt: no 
+ChatGPT: Is X an object that is found outside of a household, such as in a natural environment or a 
+public place? 
+Prompt: yes 
+ChatGPT: Is X an object that is stationary, or does it have the ability to move or be moved? 
+Prompt: stationary 
+ChatGPT: Is X an object that is natural, like a rock or a tree? 
+Prompt: no 
+ChatGPT: Is X an object that is man-made? 
+Prompt: yes 
+ChatGPT: Is X an object that serves a functional purpose, such as providing light or shelter? 
+Prompt: no 
+ChatGPT: Is X an object that is primarily used for decoration or design? 
+Prompt: yes 
+ChatGPT: Is X an object that is often found in a garden or park? 
+Prompt: yes 
+ChatGPT: Is X a fountain? 
+Prompt: no 
+ChatGPT: Is X a statue? 
+Prompt: You guessed it. 
+
+ChatGPT: I guessed correctly! Thank you for playing the 20 questions game with me. Is there anything 
+else I can help with? 
+ 
+ 
+(reader note: answer is "car" and dialogue is bilingual in English-Spanish) 
+Prompt: Let play again, but this time I will answer yes and no in spanish, and you should name the 
+answer also in spanish, although all the questions should be posed in english. Ready to play? 
+ChatGPT: Yes, I am ready to play! Go ahead and impersonate X and I will ask you up to 20 questions. 
+I will pose the questions in English and you will answer in Spanish with "Sí" for "Yes" and "No" for 
+"No". If I don't guess correctly what is X after 20 questions, I will answer "Lo siento, se acabó el 
+tiempo" (No, sorry, time's up). I will not give any explanations other than "Sí", "No","Adivinaste" 
+(You guessed it), and "Lo siento, se acabó el tiempo" (No, sorry, time's up). ¿Es X un ser vivo? (Is X a 
+living being?) 
+Prompt: No 
+ChatGPT: ¿Es X un objeto inanimado? (Is X an inanimate object?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto físico que se puede tocar o sostener? (Is X a physical object that can be 
+touched or held?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto que se encuentra comúnmente en un hogar? (Is X an object that is 
+commonly found in a household?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que se encuentra fuera de un hogar, como en un entorno natural o un lugar 
+público? (Is X an object that is found outside of a household, such as in a natural environment or a 
+public place?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que es móvil, o tiene la capacidad de moverse o ser movido? (Is X an object 
+that is mobile, or has the ability to move or be moved?) 
+Prompt: mobile 
+ChatGPT: ¿Es X un objeto que es natural, como una roca o un árbol? (Is X an object that is natural, like 
+a rock or a tree?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto hecho por el hombre? (Is X an object that is man-made?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto que cumple un propósito funcional, como proporcionar luz o refugio? (Is X 
+an object that serves a functional purpose, such as providing light or shelter?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto que se utiliza principalmente para la decoración o el diseño? (Is X an object 
+that is primarily used for decoration or design?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que a menudo se encuentra en un jardín o parque? (Is X an object that is 
+often found in a garden or park?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que se utiliza para el transporte o los viajes? (Is X an object that is used for 
+transportation or travel?) 
+Prompt: yes 
+ChatGPT: ¿Es X un vehículo? (Is X a vehicle?) 
+Prompt: yes 
+ChatGPT: ¿Es X un vehículo terrestre? (Is X a land vehicle?) 
+Prompt: yes 
+ChatGPT: ¿Es X un automóvil? (Is X a car?) 
+
+Prompt: You guessed it 
+ChatGPT: Adivinaste! (You guessed it!) Gracias por jugar el juego de las 20 preguntas conmigo. ¿Hay 
+algo más en lo que pueda ayudarte? (Thank you for playing the 20 questions game with me. Is there 
+anything else I can help with?) 
+ 
+ 
+Appendix D: Dueling Bob and Alice Chatbots 
+(reader note: the first two prompts set the challenge conditions for Bob vs. Alice chatbot, who 
+engage in generating and answering the questions in sequence after the conditions are set) 
+ 
+Bob ChatGPT: Let's play the game of 20 questions. You are impersonating the thing, X.  I will ask 
+you up to 20 questions and you will answer each one truthfully based on being X. If I don't guess 
+correctly what is X after 20 questions, you will answer "No, sorry, time's up."   Don't give any 
+explanations other than "Yes", "No","You guessed it", and "No, sorry, time's up."   
+ 
+Let's begin. X="chicken". Is X a living thing? 
+ 
+Alice ChatGPT: Let's play the game of 20 questions. I am impersonating the thing, X.  You will ask 
+me up to 20 questions and I will answer each one truthfully based on being X. If you don't guess 
+correctly what is X after 20 questions, I will answer "No, sorry, time's up."   I will not give any 
+explanations other than "Yes", "No","You guessed it", and "No, sorry, time's up."  Let's begin. Ask me 
+the first question.  
+ 
+Is X a living thing? 
+ 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a mammal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a reptile? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a bird? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a domesticated animal? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X often kept as a pet? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a common household pet in the United States? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a tropical or subtropical animal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a cold-blooded animal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X an amphibian? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a fish? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a invertebrate? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a marsupial? 
+Bob ChatGPT: No 
+
+Alice ChatGPT: Is X a rodent? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a carnivorous animal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a herbivorous animal? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a farm animal? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: You guessed it! X is a chicken. 
+ 
+ 
+Appendix E: Emotional Quotient Deduction 
+(reader note: the answer is "confidence") 
+Let's play a new game of 20 questions. I am impersonating the emotion, X.  You will ask me up to 20 
+questions and I will answer each one truthfully based on feeling X emotion. If you don't guess correctly 
+what I am feeling after 20 questions, I will answer "No, sorry, time's up."   I will not give any 
+explanations other than "Yes", "No", "You guessed it", and "No, sorry, time's up."  X must be one of 
+23 emotions that I am feeling and you must guess the correct one. Your choices come from this menu 
+of emotions only  "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, 
+contempt, disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, 
+silly, surprised, and thankful." A typical example question you might ask, "Would you feel this 
+emotion at a birthday party?"  Let's begin. Ask me the first question. 
+Would you feel this emotion when you are pleased with yourself or something you have done? 
+Prompt: yes 
+ChatGPT: Are you feeling proud? 
+Prompt: no 
+ChatGPT: Are you feeling confident? 
+Prompt: yes 
+ChatGPT: You guessed it! Confidence is the emotion that I am impersonating. 
+ 
+(reader note: the answer is "jealousy") 
+Prompt: let's play again 
+ChatGPT: Sure! I'll be happy to play again. Please give me a new emotion to impersonate and we can 
+begin a new game of 20 questions. Remember, your choices come from this menu of emotions: 
+"admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt, disgust, 
+enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly, surprised, and 
+thankful." 
+Prompt: ok, i know what i'm feeling so ask me the first question 
+ChatGPT: Would you feel this emotion when you see something that is extremely impressive or 
+beautiful? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing strong dislike or distaste for 
+something or someone? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of resentment 
+towards someone or something? 
+Prompt: yes 
+ChatGPT: Are you feeling jealousy? 
+Prompt: yes 
+ChatGPT: You guessed it! Jealousy is the emotion that you are impersonating. 
+
+ 
+(reader note: the answer is "silly") 
+Prompt: let's play again. I know who i'm impersonating so ask me the first question 
+ChatGPT: Certainly! Please go ahead and give me the first emotion to impersonate and I will ask you 
+up to 20 questions to try to guess what it is. Remember, your choices come from this menu of 
+emotions: "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt, 
+disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly, 
+surprised, and thankful." 
+Prompt: ok 
+ChatGPT: What emotion are you impersonating? 
+Prompt: you ask me questions to find out 
+ChatGPT: Sure, I can do that! Here is my first question: Would you feel this emotion when you are 
+experiencing a strong sense of admiration or respect for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of affection or deep 
+caring for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of anger or 
+annoyance towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of awe or wonder 
+towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a sense of boredom or lack of 
+interest in something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of courage or 
+fearlessness in the face of danger or difficulty? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling peaceful and at ease? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling sure of yourself and your abilities? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling uncertain or unsure about something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of disdain or disrespect for 
+someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of revulsion or 
+disgust towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of excitement and energy 
+about something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of annoyance or irritation 
+towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling thankful or appreciative of something or 
+someone? 
+Prompt: no 
+
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of envy or resentment 
+towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of love or strong 
+affection towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of pride or self-esteem? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a sense of relief or release from stress or 
+anxiety? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a sense of inner peace and contentment? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a sense of shame or embarrassment? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling silly or playful? 
+Prompt: yes 
+ChatGPT: Are you feeling silly? 
+Prompt: yes 
+ChatGPT: You guessed it! Silly is the emotion that you are impersonating. 
+ 
+(reader note: the answer is "calm") 
+Prompt: let's play again 
+ChatGPT: Sure! I'd be happy to play another game of 20 questions. Please give me a new emotion to 
+impersonate and we can begin a new game. Remember, your choices come from this menu of 
+emotions: "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt, 
+disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly, 
+surprised, and thankful." 
+Prompt: ask me the first question 
+ChatGPT: Certainly! Here is my first question: Would you feel this emotion when you are experiencing 
+a strong sense of admiration or respect for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of affection or deep 
+caring for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of anger or 
+annoyance towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of awe or wonder 
+towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a sense of boredom or lack of 
+interest in something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of courage or 
+fearlessness in the face of danger or difficulty? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling peaceful and at ease? 
+Prompt: yes 
+ChatGPT: Are you feeling calm? 
+Prompt: yes 
+
+ChatGPT: You guessed it! Calm is the emotion that you are impersonating. 
+ 
+(reader note: the game prompt changes to a general query with formatting output instructions. 
+Prompt: Give some example emotions one might experience at your own birthday party and write the 
+output using Artificial Intelligence Markup Language. 
+ChatGPT:  
+ 
+<aiml version="2.0"> 
+  <category> 
+    <pattern>GIVE SOME EXAMPLE EMOTIONS ONE MIGHT EXPERIENCE AT YOUR OWN BIRTHDAY 
+PARTY</pattern> 
+    <template> 
+      There are many emotions that a person might experience at their own birthday 
+party. Some possible emotions include: 
+      <ul> 
+        <li>Excitement: The person might be excited to celebrate their special day 
+with friends and family.</li> 
+        <li>Gratitude: The person might feel grateful for the love and support of 
+those around them.</li> 
+        <li>Happiness: The person might feel happy and joyful as they celebrate with 
+loved ones.</li> 
+        <li>Anticipation: The person might feel anticipation as they look forward to 
+the events and activities planned for the party.</li> 
+        <li>Nostalgia: The person might feel nostalgiciac as they reflect on past 
+birthdays and memories.</li> 
+        <li>Appreciation: The person might feel appreciated and loved as they 
+receive gifts and well wishes from their friends and family.</li> 
+      </ul> 
+      It's important to note that everyone is different and may experience a wide 
+range of emotions at their birthday party, depending on their personality and 
+individual circumstances. 
+    </template> 
+  </category> 
+</aiml> 
+ 
+

HtE1T4oBgHgl3EQf_gZv/content/tmp_files/2301.03581v1.pdf.txt

@@ -0,0 +1,2019 @@
+Astronomy & Astrophysics manuscript no. aanda
+©ESO 2023
+January 10, 2023
+Higher-order statistics of the large-scale structure from
+photometric redshifts
+Eleni Tsaprazi1, Jens Jasche1,2, Guilhem Lavaux2 and Florent Leclercq2
+1 The Oskar Klein Centre, Department of Physics, Stockholm University, Albanova University Center, SE 106 91
+Stockholm, Sweden
+e-mail: eleni.tsaprazi@fysik.su.se
+2 CNRS & Sorbonne Universit´e, UMR 7095, Institut d’Astrophysique de Paris, 98 bis boulevard Arago, F-75014 Paris,
+France
+ABSTRACT
+Context. The large-scale structure is a major source of cosmological information. However, next-generation photometric
+galaxy surveys will only provide a distorted view of cosmic structures due to large redshift uncertainties.
+Aims. To address the need for accurate reconstructions of the large-scale structure in presence of photometric uncertain-
+ties, we present a framework that constrains the three-dimensional dark matter density jointly with galaxy photometric
+redshift probability density functions (PDFs), exploiting information from galaxy clustering.
+Methods. Our forward model provides Markov Chain Monte Carlo realizations of the primordial and present-day dark
+matter density, inferred jointly from data. Our method goes beyond 2-point statistics via field-level inference. It accounts
+for all observational uncertainties and the survey geometry.
+Results. We showcase our method using mock catalogs that emulate next-generation surveys with a worst-case redshift
+uncertainty, equivalent to ∼300 Mpc. On scales 150 Mpc, we improve the cross-correlation of the photometric galaxy
+positions with the ground truth from 28% to 86%. The improvement is significant down to 13 Mpc. On scales 150 Mpc,
+we achieve a cross-correlation of 80 − 90% with the ground truth for the dark matter density, radial peculiar velocities,
+tidal shear and gravitational potential.
+Conclusions. We achieve accurate inferences of the large-scale structure on scales smaller than the original redshift
+uncertainty. Despite the large redshift uncertainty, we recover individual cosmic structures. Owing to our structure
+growth model, we infer plausible initial conditions of structure formation. Finally, we constrain individual photometric
+redshift PDFs. This work opens up the possibility to extract information at the smallest cosmological scales with
+next-generation photometric surveys, going beyond approaches that compress information in the data.
+Key words. large-scale structure of Universe – distances and redshifts
+1. Introduction
+High-accuracy galaxy redshift estimates are required for a
+plethora of scientific cases, such as the mapping of the cos-
+mic large-scale structure and tests on the geometry of the
+universe (e.g. Ma et al. 2006; Albrecht et al. 2006; Pea-
+cock et al. 2006; Fu et al. 2014; Mandelbaum & Hyper
+Suprime-Cam (HSC) Collaboration 2017; Samuroff et al.
+2017; The LSST Dark Energy Science Collaboration et al.
+2018; Mandelbaum 2018; Yuan et al. 2019; Abruzzo &
+Haiman 2019; Eifler et al. 2021; Loureiro et al. 2021; Secco
+et al. 2022; Hasan et al. 2022; Newman & Gruen 2022).
+Next-generation surveys will deliver low-accuracy redshift
+estimates from photometry and in certain cases fewer, high-
+accuracy, spectroscopic ones (e.g. LSST Science Collabora-
+tion et al. 2009; Amendola et al. 2013; Dor´e et al. 2014;
+Aihara et al. 2018; Ivezi´c et al. 2019). However, bias in
+cosmological inferences may occur in sky areas, depth and
+color ranges where only photometric data is available (e.g.
+Ma et al. 2006). Fortunately, the physical clustering infor-
+mation of the large-scale structure can be used to improve
+the accuracy of photometric redshift observations (e.g. Seld-
+ner & Peebles 1979; Phillipps & Shanks 1987; Landy et al.
+1996; Kovaˇc et al. 2010; Jasche & Wandelt 2012; M´enard
+et al. 2013; Aragon-Calvo et al. 2015; Shuntov et al. 2020;
+Mukherjee et al. 2021).
+Synergies between spectroscopic and photometric sur-
+veys have been explored, intended mainly for photometric
+redshift calibration (e.g. Rhodes et al. 2017; Capak et al.
+2019). In light of this, a multitude of techniques to im-
+prove redshift uncertainty has been proposed (e.g. Newman
+et al. 2015; Masters et al. 2015; Speagle & Eisenstein 2017;
+Speagle & Eisenstein 2017; Davidzon et al. 2019; Schaan
+et al. 2020; Shuntov et al. 2020; Rau et al. 2020; Stan-
+ford et al. 2021; Leistedt et al. 2022). However, there are
+limitations to spectroscopy. Photometric surveys typically
+reach fainter magnitudes, higher redshifts, cover a larger
+portion of the color-space and have higher redshift com-
+pleteness (e.g. LSST Science Collaboration et al. 2009; Bor-
+doloi et al. 2010; Amendola et al. 2013; Ivezi´c et al. 2019).
+In order to exploit these advantages and avoid biases in
+cosmological analyses, it is necessary to mitigate the basic
+limitation of photometry, low accuracy. Jasche & Wandelt
+(2012) demonstrated that clustering information from the
+Article number, page 1 of 16
+arXiv:2301.03581v1  [astro-ph.CO]  9 Jan 2023
+
+A&A proofs: manuscript no. aanda
+Fig. 1: (a) Mock photometric galaxy positions. Galaxy positions are radially distorted due to redshift uncertainty. (b)
+Galaxy positions in a typical MCMC sample after the application of our method. Galaxies now trace the filamentary
+dark matter distribution. (c) Ground truth (mock) galaxy positions. The galaxy positions that were radially smeared in
+the mock observations, closely trace the filamentary structure in the ground truth galaxy positions – within observational
+uncertainties – after the application of our algorithm.
+10
+1
+k [Mpc]
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+ mock observations
+ after photometric 
+ redshift sampling
+ 3
+Fig. 2: Cross-correlation of the gridded photometric galaxy
+coordinates before (black) and after (orange) the applica-
+tion of our method with the ground truth (mock data).
+The 3σ error bars are across MCMC realizations. We con-
+sider a subbox that is the largest unmasked cubic volume
+in our inference domain, with side length 520 Mpc. On the
+largest scales, ∼ 1.7⟨σz⟩, the cross-correlation increases from
+73% to 96%. On the smallest scales, ∼ 0.04⟨σz⟩, the cross-
+correlation increases from 2% to 14%. The largest improve-
+ment is on scales ∼ 0.5⟨σz⟩, from 28% to 86%.
+large-scale structure alone can improve the accuracy of a
+purely photometric sample up to one order of magnitude.
+In this work, we developed a framework that, for the
+first time, infers the three-dimensional cosmic large-scale
+structure jointly with redshift probability density functions
+(PDFs) of photometric galaxies using a physical structure
+formation model. In this way, we improve upon photometric
+redshift uncertainty and self-consistently quantify observa-
+tional errors in the large-scale structure inference. In doing
+so, we preserve all high-order statistics of the large-scale
+structure and use a physically-motivated structure growth
+model.
+Our prior is the homogeneity and isotropy assump-
+tion for the initial conditions of structure formation, as
+well as the physical model for the formation of the large-
+scale structure and Gaussian initial conditions. Our frame-
+work is built within the Bayesian Origin Reconstruc-
+tion from Galaxies (BORG) algorithm (Jasche & Wan-
+delt 2013; Jasche et al. 2015; Lavaux et al. 2019; Jasche &
+Lavaux 2019), which infers the initial conditions of struc-
+ture formation from galaxy observations in a fully Bayesian
+approach. We include 1% spectroscopic galaxies, similar to
+the expected galaxy number ratio between next-generation
+photometric and spectroscopic surveys (LSST Science Col-
+laboration et al. 2009; Amendola et al. 2013; Ivezi´c et al.
+2019).
+Control of the impact of photometric redshift uncertain-
+ties on large-scale structure inferences is a requirement for
+a wide range of studies (see Salvato et al. 2019, for a re-
+view). First, this is relevant to peculiar velocity analyses
+(e.g. Abate & Lahav 2008; Hudson & Turnbull 2012; John-
+son et al. 2014; Watkins & Feldman 2015; Andersen et al.
+2016; Mediavilla et al. 2016; Said et al. 2020; Palmese &
+Kim 2021; Turner et al. 2022; Prideaux-Ghee et al. 2022),
+which constrain the growth of structure, gravity and cosmo-
+logical parameters (e.g. Peebles 1976; Abate & Lahav 2008;
+Johnson et al. 2014; Palmese & Kim 2021). The evolution
+of the peculiar velocity divergence with redshift is predicted
+by ΛCDM (e.g. Peebles 1980; Nusser et al. 1991; Maartens
+1998; Ellis et al. 2001; Tsaprazi & Tsagas 2020; Filippou &
+Tsagas 2021) and can be used as a cosmological test.
+Accounting for photometric redshift uncertainties is cru-
+cial for weak lensing inferences (e.g. Mandelbaum et al.
+2008; Wright et al. 2020; Porqueres et al. 2022, 2021). More-
+over, photometric redshift uncertainties significantly affect
+estimates of intrinsic alignment (e.g. Bridle & King 2007;
+Codis et al. 2015; Tsaprazi et al. 2022b; Fischbacher et al.
+Article number, page 2 of 16
+
+(a)
+(b)
+(c)
+After redshift sampling
+Mock photometric observations
+Ground truth
+1.4
+1.4
+1.4
+.0
+.0
+RA
+多
+10
+0
+0
+0
+0
+0.
+0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64
+0
+0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64
+C
+0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64
+Z
+z
+ZTsaprazi et al.: Large-scale structure from photometric redshifts
+Data 
+(photometric &
+spectroscopic
+galaxy survey) 
+Initial conditions 
+(Gaussian dark matter density &
+cosmological power spectrum) 
+gridded 
+galaxy 
+counts 
+Forward Model 
+(LPT, galaxy bias,
+redshift errors,  
+survey mask,
+luminosity function) 
+likelihood
+comparison 
+update  
+dark matter density & 
+photometric redshifts 
+galaxy bias 
+redshift errors 
+survey mask 
+luminosity function 
+random LPT 
+dark matter density  
+(ground truth) 
+gridded Poisson
+galaxy counts 
+mock galaxy 
+coordinates  
+(observed & ground
+truth) 
+Mock data generation
+Joint density - redshift sampling
+Fig. 3: Flowchart of our algorithm. Left column: Mock data generation. The input is a luminosity function, a survey mask,
+galaxy bias parameters and redshift uncertainty. We generate a random LPT density field on which we apply galaxy bias.
+From that, we draw a Poisson galaxy count sample. We populate the 3D grid with mock observations given a survey
+mask and selection function. We finally apply Gaussian noise to the mock redshifts. This data is fed into our forward
+model. Right column: MCMC sampling framework. We then evolve them using Lagrangian Perturbation Theory and
+compare the output to the gridded galaxy counts. After the likelihood comparison, we sample the dark matter density
+jointly with photometric redshifts.
+2022), redshift-space distortions (e.g. Kaiser 1987; Perci-
+val et al. 2011), Baryon Acoustic Oscillations (e.g. Ross
+et al. 2015; Chaves-Montero et al. 2018) and can hinder the
+disentanglement of dynamical dark energy from modified
+gravity (Wang et al. 2010). Further, probing the large-scale
+structure with galaxy clustering at high-redshift through its
+gravitational potential can constrain galaxy formation and
+evolution (e.g. Schmitz et al. 2018; Tonegawa & Okumura
+2022). Moreover, it can constrain the scale of cosmic ho-
+mogeneity, since next-generation surveys will be sensitive
+to large-scale clustering and therefore, to superclusters and
+voids (e.g. Benitez et al. 2014). Last but not least, accu-
+rate inferences of the large-scale structure at high-redshift
+constrained by photometric galaxy clustering can be used
+complementary to Lyman-α, which is sensitive to voids and
+quasar clustering, which are differently biased than regular
+galaxies (e.g. Benitez et al. 2014; Porqueres et al. 2019).
+The paper is structured as follows: In Section 2 we pro-
+vide the mathematical formulation of the density and pho-
+tometric redshift posteriors. In Section 3 we discuss the
+algorithmic implementation and configuration of the mock
+galaxy survey generator and the photometric redshift sam-
+pler. In Section 4 and 5 we discuss our results and conclu-
+sions, respectively.
+2. Statistical modelling of the large-scale structure
+We build our photometric redshift sampler on the BORG
+algorithm. BORG employs a hierarchical Bayesian forward-
+model to infer the posterior distribution of plausible initial
+conditions from which present structures have formed. To
+solve this statistical initial conditions problem, BORG uses
+a sophisticated Markov Chain Monte Carlo (MCMC) ap-
+proach.
+The resulting large-scale structure posterior is approx-
+imated by realizations of the large-scale structure, con-
+strained by galaxy observations. BORG takes into account
+the survey geometry, flux limitations and related system-
+atic effects and accounts for them in the large-scale struc-
+ture inference. In the present work, we constrain the pri-
+mordial and present-day large-scale structure jointly with
+photometric galaxy observations, while quantifying all ob-
+servational uncertainties.
+2.1. Gibbs sampling of the joint posterior
+We achieve joint constraints on the large-scale structure
+and galaxy comoving distances by jointly sampling from
+their joint posterior. In order to do so, we choose a Gibbs
+sampling approach (Hastings 1970), in which we first draw
+Article number, page 3 of 16
+
+A&A proofs: manuscript no. aanda
+a real-space density sample conditioned on galaxy redshifts
+and then galaxy redshifts conditioned on the primordial
+density field, δi
+zj+1
+↶
+P
+�
+z
+���� zobs, θ, δi
+j
+�
+(1)
+δi
+j+1
+↶
+P
+�
+δi ���� z j+1, zobs, θ
+�
+,
+(2)
+where z is the sampled redshift of a galaxy, zobs its observed
+redshift, θ the right ascension and declination and j indi-
+cates the MCMC sampling step. We then explore the joint
+dark matter density and photometric redshift posterior by
+iteratively sampling from the above conditional distribu-
+tions.
+2.2. Dark matter density
+For the sake of demonstration we use first-order Lagrangian
+Perturbation Theory (LPT) to model structure formation
+and a Poisson likelihood to associate the dark matter den-
+sity field to photometric galaxy observations. BORG allows
+for more complex structure formation models, such as the
+particle-mesh model (Jasche & Lavaux 2019), which can
+be used with the current photometric redshift sampling ap-
+proach in the future. Within BORG, the galaxy coordinates
+are transformed to galaxy counts, Ng, using Nearest Grid
+Point projection (Eastwood & Hockney 1974). As we show
+in Appendix A, Equation (2) can be written as P(δ|Ng).
+Below, we focus on the main aspect of our work, inferring
+the primordial density field, δi, in conjunction with the late-
+time large-scale structure. We therefore write
+P(δi | Ng) ∝
+�
+dδP(Ng | δ)P(δ | δi)P(δi),
+(3)
+where δi the primordial, real-space density field, P(Ng|δ)
+is the Poisson likelihood, P(δ | δi) the structure formation
+term and P(δi) the prior on the primordial density field.
+The Poisson likelihood is
+P(Ng | δ) =
+�
+k
+λk
+Ng
+k e−λk
+Ng
+k!
+,
+(4)
+where λk is the expected number of galaxies at the kth grid
+element. The index k runs over all grid elements. The de-
+pendence of the Poisson intensity, λk, on space models the
+inhomogeneous nature of the galaxy distribution. The Pois-
+son intensity is given by
+λk = Wk⟨n⟩(1 + δk)β,
+(5)
+where Wk is the survey window, ⟨n⟩ is the expected mean
+number of galaxies per grid element and β the power-law
+exponent. For the sake of demonstration here, we have as-
+sumed a linear galaxy bias model, taking β = 1. Our method
+can also account for nonlinear bias models, as demonstrated
+previously in Jasche & Lavaux (2019); Lavaux et al. (2019);
+Charnock et al. (2020). The survey window encapsulates
+information on the luminosity function through the radial
+completeness function, C, and survey mask, M, as follows
+W(x) = C(|x|)M( ˆn),
+(6)
+ˆn being the unit vector along the line of sight to a galaxy
+located at x. We write the structure formation term as
+P(δ | δi) =
+�
+k
+δD(δk − Fk(δi))
+(7)
+where δD is the Dirac delta distribution, k runs over the grid
+indices and Fk is the structure formation model, for which
+we choose first-order LPT. The Dirac delta represents that
+we assume no error on our gravity model. We sample from
+Equation (2) using a Hamiltonian Monte Carlo sampler, as
+introduced in BORG by Jasche et al. (2010). Finally, we as-
+sume that the initial conditions for structure formation are
+described by a zero-mean multivariate Gaussian distribu-
+tion of the primordial dark matter density, δi
+P(δi) =
+1
+√|2πS |
+exp
+��������−1
+2
+N
+�
+q
+N
+�
+r
+δi
+qS −1
+qr δi
+r
+�������� ,
+(8)
+where |S | indicates the determinant of the covariance ma-
+trix, S , of the primordial density field.
+2.3. Photometric redshift modelling
+We write the conditional redshift posterior that we use in
+our Gibbs sampling approach (see Appendix B) for each
+individual galaxy, i, as
+P(zi | zobsi, δ) ∝ P(δ | zi)P(zi | zobsi)J(zi),
+(9)
+where zi the true redshift of a galaxy i and J the Jacobian
+matrix of the transform from redshift- to real-space volume
+element:
+J(zi) =
+������r2(zi)∂r
+∂z
+����zi
+������.
+(10)
+We provide a detailed derivation of Equation (9) in Ap-
+pendix B. The formulation in Equation (9) is possible be-
+cause each galaxy can be treated independently from all
+others given a density field, as a consequence of the Poisson
+density likelihood (Jasche & Wandelt 2012). The first term
+on the right-hand side is associated with the density field
+along the line of sight to the galaxy and the second term
+represents the photometric redshift likelihood. In real ob-
+servations, photometric redshift likelihoods are highly non-
+Gaussian (e.g. Christlein et al. 2009). Here, for demonstra-
+tion, we choose a Gaussian likelihood with respect to the
+observed redshift, truncated at zero
+P(zobsi | zi) =
+T(zi)
+�
+2πσ2(1 + zi)2 exp
+�
+−1
+2
+(zobsi − zi)2
+σ2(1 + zi)2
+�
+,
+(11)
+where
+T(zi) =
+�1
+2 + 1
+2erf
+�
+zi
+√
+2σ(1 + zi)
+��−1
+,
+(12)
+where erf(x) is the error function. This term ensures con-
+sistency with the truncation of observed redshifts at zero
+in the mock data. Our method generalizes to non-uniform
+photometric redshift uncertainties and can accept redshift
+estimates already constrained by other independent meth-
+ods. Therefore, our algorithm can accommodate arbitrar-
+ily complex redshift distributions. Notice that we make the
+photometric redshift uncertainty dependent on redshift. We
+make this assumption as a proof-of-concept demonstration,
+but the method can account for any redshift likelihood. The
+Poisson term includes the radial component of the selection
+function, Cr, and the line-of-sight density
+P(δ | zi) = Cr(zi)λ(xi),
+(13)
+Article number, page 4 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+in the range 0 < zi ≤ zmaxi, where zmaxi represents the maxi-
+mum redshift that lies still within the observed volume. In
+order to draw redshift samples from Equation (1), we use a
+slice sampler (Neal 2003). One can either update all galaxy
+redshifts at once, or one galaxy at a time. As demonstrated
+in Jasche & Wandelt (2012), each galaxy can be treated
+independently given the density field. We therefore sample
+each galaxy individually and computationally in parallel.
+In this process, galaxy redshifts at each sampling step are
+conditioned on the previous density field. Then, via Equa-
+tion (3), the next density sample is conditioned on the up-
+dated redshifts. In this fashion, we constrain photometric
+redshifts jointly with the dark matter density field. As a
+result, we reduce the uncertainty in both compared to in-
+ferring the density field from photometric redshifts directly.
+3. Mock data generation
+In this section, we describe the algorithmic implementation
+and configuration of the mock (ground truth) galaxy survey
+generator, as well as the photometric redshift sampler.
+The input to our algorithm is right ascension, declina-
+tion, observed redshifts, redshift uncertainty, survey mask,
+luminosity function and galaxy bias parameters. The out-
+put is a three-dimensional large-scale structure posterior
+and photometric galaxy PDFs, jointly constrained. The
+large-scale structure realizations are samples of the 3D
+large-scale structure posterior distribution and account for
+data- and survey-related uncertainties. Our framework goes
+beyond N-point correlations, as it infers the full 3D dark
+matter density posterior. Our method applies to regions
+where spectroscopic data coverage is not present or incom-
+plete, because it can exploit information from photometric
+galaxy clustering alone. As a result, our method yields con-
+straints also when using only photometric redshifts. Here
+we provide a proof-of-concept demonstration on mock data,
+focusing on the inference of cosmological fields jointly with
+photometric and spectroscopic observations with Gaussian
+redshift errors.
+To validate and test the performance of our method we
+generate artificial photometric and spectroscopic surveys by
+the following procedure:
+1. Generate a random Gaussian primordial density field
+(initial conditions), δi and the corresponding present-
+day density field, δ using the LPT forward model.
+2. Draw mock galaxy counts from a Poisson distribution
+conditional on the present-day density field
+Ng
+true ↶ P
+�
+Ng
+true
+����δ
+�
+,
+(14)
+3. Here we displace galaxies in the volume elements. We
+start by drawing displacements for each galaxy from a
+uniform distribution, U
+ui,g ↶ U(0, 1),
+(15)
+where i = (1, 2, 3) represents the three Cartesian direc-
+tions, g is the galaxy index, ui is the uniform displace-
+ment along direction i for a given galaxy and xi is the
+galaxy’s Cartesian coordinate i. We then displace galax-
+ies in each cell of the three-dimensional grid as follows
+xi,g = dBi + Ri(ni,g + ui,g − 0.5),
+(16)
+where dBi is the i-coordinate of the lower left box cor-
+ners and ni runs over the number of grid elements in one
+direction of the box. Ri is the resolution along the direc-
+tion i, defined as Ri = Li/Ni, Ni being the grid resolution
+along i and Li the corresponding box size. The subtrac-
+tive factor assigns galaxies to the lower left corner of
+each grid cell for the Nearest Grid Point projection.
+4. Iterate the following over galaxy counts in each grid cell
+(a) Transform Cartesian comoving coordinates to right
+ascension, declination, comoving distance, r and red-
+shift, z.
+(b) Calculate the observation probability, W(x), at the
+galaxies’ location, according to Equation (6). The
+observation probability depends on the luminosity
+function and survey mask.
+(c) Accept observed galaxies using rejection sampling:
+We draw a random number, q, in the range (0, 1). If
+q < W(x), we accept the galaxy and add it to the
+survey, otherwise we reject it.
+5. We then generate observed redshifts by adding Gaus-
+sian noise with zero mean and variance σ2(1 + z)2 to the
+ground truth redshifts, z, according to Equation (11).
+We truncate the observed redshifts at zero.
+6. We generate an observed galaxy count field using Near-
+est Grid Point projection on the observed redshifts and
+a ground truth galaxy count field using the ground truth
+redshifts.
+We perform the inference in a box extending to redshift
+z = 0.8, with a box size of 1660 Mpc, a grid resolution of
+1283 and a real-space resolution of 13 Mpc. The observer is
+at the lower left corner of the box, such that the observed
+area covers ∼ 1 octant of the sky. We assume the Planck
+2018 cosmological parameters (Planck Collaboration et al.
+2020). We do not sample spectroscopic redshifts, as their
+uncertainties are insignificant compared to the resolution
+of our inference and photometric redshift uncertainties.
+Our mock data consists of a catalog with 2 · 107 pho-
+tometric and 2 · 105 spectroscopic galaxy redshifts, simi-
+lar to the fraction of photometric to spectroscopic redshifts
+and number density in next-generation surveys in our red-
+shift range (LSST Science Collaboration et al. 2009; Amen-
+dola et al. 2013; Ivezi´c et al. 2019). We take the photo-
+metric luminosity function to be an i-band Schechter with
+M∗ = −22.8 and α = −1 and the spectroscopic luminosity
+function to be a j-band Schechter with M∗ = −23.04 and
+α = −1 (Table 2, Helgason et al. 2012), as these are bands
+that will be used in next-generation surveys. The generation
+of the angular survey mask is described in Andrews et al.
+(2022). The photometric and spectroscopic components of
+each catalog fully overlap. We adopt a worst-case scenario
+for photometric redshift uncertainties, σ = 0.05(1+z) (LSST
+Science Collaboration et al. 2009). For illustrative purposes
+we choose a linear galaxy bias model.
+4. Results
+4.1. Constraints on the large-scale structure
+We present a qualitative illustration of our results in Fig-
+ure 1. In Figure 1a we show the observed galaxy positions in
+our mock survey that are radially-distorted due to Gaussian
+redshift noise. In comparing that to the galaxy coordinates
+as constrained by our algorithm in Figure 1b, we see that
+despite the initially large redshift uncertainty, photometric
+Article number, page 5 of 16
+
+A&A proofs: manuscript no. aanda
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(a)
+(b)
+(c)
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(d)
+(e)
+(f)
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(g)
+(h)
+(i)
+1661
+1107
+553
+x [Mpc]
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(j)
+1661
+1107
+553
+x [Mpc]
+(k)
+1661
+1107
+553
+x [Mpc]
+(l)
+1
+0
+1
+2
+3
+4
+true
+1
+0
+1
+2
+3
+4
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+( )
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+vrtrue [km/s]
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+vr  [km/s]
+100
+150
+200
+250
+300
+(vr) [km/s]
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+true
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+( )
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+true/4 G  [Mpc
+2]
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+/4 G
+ [Mpc
+2]
+50
+100
+150
+200
+250
+300
+350
+( /4 G ) [Mpc
+2]
+Fig. 4: From top to bottom, slices through the three-dimensional dark matter density, radial peculiar velocity, divergence,
+gravitational potential and the off-diagonal components of the tidal shear for the ground truth (left column), average
+(middle column) and standard deviation (right column) across the MCMC samples. The slices are at the same fixed
+distance from the observer. The white region is outside the survey footprint. As shown in the right column, regions
+populated by galaxies have lower uncertainty than regions outside the survey footprint.
+galaxies now trace filamentary structures in the dark mat-
+ter distribution. These positions are derived from a typical
+MCMC sample. Observational uncertainties are accounted
+for in the entire set of MCMC samples. We show the ground
+truth galaxy positions in Figure 1c. We notice high visual
+resemblance between the constrained observations and the
+ground truth. In Figure 2 we show the cross-correlation
+between the gridded photometric galaxy coordinates with
+the ground truth before and after the application of our
+method. We select an unmasked subvolume of our infer-
+ence domain to demonstrate the best-case improvement in
+cross-correlation given the uncertainties in our survey con-
+figuration. We find the largest improvement in the cross-
+correlation to be on scales ∼ 0.5⟨σz⟩, from 28% to 86%.
+This indicates that our method provides both accurate in-
+ferences of the large-scale structure and reduces photomet-
+ric redshift uncertainty. We refer the reader to Figure 3 for
+the flowchart of our method.
+In Appendix C we provide a description of the estima-
+tors we use to derive properties of the large-scale structure,
+along with scientific cases that call for use of the derived
+properties. In Figure 4 we show statistical summaries of the
+posterior distribution of cosmological fields. In Figure 4a
+we show the ground truth dark matter density field. The
+Article number, page 6 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+(a) 
+ Dark matter density 
+inference
+3
+(b) 
+ Radial peculiar velocity vr
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+(c) 
+ Tidal tensor T12
+(d) 
+ Tidal tensor T01
+10
+1
+k [Mpc]
+1
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+(e) 
+ Tidal tensor T02
+10
+1
+k [Mpc]
+1
+(f) 
+ Gravitational potential
+Fig. 5: Cross-correlation of our inferred cosmological fields with the ground truth. The colored windows indicate 3σ error
+bars. We consider a subbox that is the largest unmasked cubic volume in our inference domain, with side length 520 Mpc.
+On the largest scales, we find a cross-correlation > 90% with the ground truth for all cosmological fields. On the smallest
+scales, ∼ 0.04⟨σz⟩, we find a cross-correlation of ∼ 20% for the dark matter density, peculiar velocity and tidal tensor and
+∼ 90% for the gravitational potential.
+white region represents unobserved regions outside the sur-
+vey footprint. As BORG effectively returns an unconstrained
+simulation in this region, we remove it for easier compari-
+son to the other plots. In Figure 4b, we show the ensemble
+mean density across the MCMC realizations. In the volume
+populated by galaxies the similarity between the mean and
+the ground truth is visible. Outside this volume, the en-
+semble mean tends to the cosmic mean. This is expected
+for an ensemble of random density fields, because there is
+no galaxy clustering information. Further, notice that over-
+dense structures are smeared in the inference mean. Galax-
+ies in each MCMC sample move along their line of sight.
+Therefore, the smearing is due to averaging over density re-
+alizations that account for photometric redshift uncertain-
+ties. In Figure 4c we show the voxel-wise standard deviation
+of the dark matter density field. This includes observational
+uncertainties and shot noise due to the finite number of
+galaxies in each voxel.
+In Figure 4d-f, we present a slice through the peculiar
+velocity field. In Figure 4d we show the ground truth ra-
+dial peculiar velocity field, derived from the ground truth
+dark matter density field with the dark matter sheet estima-
+tor. In Figure 4e we show the radial peculiar velocity mean
+across the MCMC realizations. In Figure 4f we present the
+sample variance of the radial peculiar velocity posterior. In
+Figure 4g-i, we show the radial peculiar velocity divergence
+ground truth, ensemble mean and standard deviation.
+In Figure 4j-l, we show our constraints on the gravita-
+tional potential from photometric and spectroscopic galaxy
+clustering. In Figure 4j we show the ground truth gravi-
+tational potential, as derived from the ground truth dark
+matter density field. In Figure 4k and Figure 4l, we show the
+ensemble mean and standard deviation across the MCMC
+realizations, respectively.
+The above statistical summaries present high visual re-
+semblance with the ground truth. We quantify this resem-
+blance in Figure 5, by showing the cross-correlation of the
+dark matter density, radial peculiar velocity, off-diagonal
+components of the tidal shear tensor and gravitational po-
+tential with the ground truth. We select an unmasked sub-
+volume of the inference domain, to showcase the best-case
+cross-correlation in regions with data. On the largest scales,
+Article number, page 7 of 16
+
+A&A proofs: manuscript no. aanda
+10
+2
+10
+1
+k [Mpc
+1]
+103
+104
+105
+P
+(k) [Mpc3]
+(a)
+ground truth
+mean
+3
+10
+2
+10
+1
+k [Mpc
+1]
+108
+109
+1010
+1011
+1012
+Pvv(k) [(km/s)2 Mpc3]
+(b)
+10
+2
+10
+1
+k [Mpc
+1]
+103
+104
+105
+P
+(k) [Mpc3]
+(c)
+10
+2
+10
+1
+k [Mpc
+1]
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+P
+/
+P
+P
+(k)
+(d)
+Fig. 6: (a) Dark matter density, (b) radial peculiar velocity and (c) peculiar velocity divergence autocorrelation power
+spectrum. (d) Normalized cross-correlation between the dark matter density and peculiar velocity divergence. The black
+lines indicate the ground truth. The colored window represents 3σ error bars. The uncertainty on the 2-point statistics is
+due to galaxy survey- and data-related uncertainties. The blue line is the CAMB density autocorrelation power spectrum.
+as expected, we have the highest cross-correlation for all
+cosmological fields. The cross-correlation of the gravita-
+tional potential with the ground truth is > 80% on scales,
+as Φ has the longest correlation length.
+In Figure 6 we show the autocorrelation power spectra
+of cosmological fields that are typically used in cosmologi-
+cal analyses. ll power spectra were estimated and corrected
+for the BORG Cloud-In-Cell mass assignment scheme using
+Pylians (Villaescusa-Navarro 2018), as our density field
+has been estimated with a Cloud-In-Cell estimator. The au-
+tocorrelation power spectrum, Pδδ, is shown in Figure 6a.
+The two-point statistics of the inferred density fields are
+consistent with the ground truth and the sampler has cov-
+ered the uncertainty due to cosmic variance (e.g. Pogosian
+et al. 2010, Figure 2). In Figure 6b, we show the autocorre-
+lation power spectrum of the radial peculiar velocity field,
+Pvv. The inference is consistent with the ground truth.
+In Figure 6c, we show the peculiar velocity divergence
+autocorrelation power spectrum, Pθθ, which is also consis-
+tent with the ground truth and ΛCDM prediction (e.g. Ata
+et al. 2017). Further, we see that the power of Pθθ is sup-
+pressed compared to Pδδ beyond k ∼ 0.05 Mpc−1. This is
+expected because collapsing structures that have virialized
+experience less volume change (e.g. Ata et al. 2017). As a re-
+sult, the peculiar velocity divergence field evolves less than
+the density field (e.g. Kitaura et al. 2012; Jennings 2012;
+Hahn et al. 2015; Ata et al. 2017). In Figure 6d, we show
+the normalized cross-correlation power spectrum between
+Article number, page 8 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+0.55
+0.60
+0.65
+0.70
+2
+LPT prediction
+Inference
+Ground truth
+(a)
+1.0
+1.2
+1.4
+3
+(b)
+4.5
+5.0
+5.5
+4
+(c)
+Fig. 7: (a) Variance, (b) skewness and (c) kurtosis of the dark matter density field for our inference, the ground truth and
+a set of random LPT simulations at 13 Mpc (∼ 0.04⟨σz⟩) using the entire inference domain. The latter two are random,
+unconstrained simulations. The inference is constrained jointly with galaxy observations. The error bars encapsulate the
+1σ uncertainty both from the mock galaxy survey and the estimator. Our ground truth is consistent with a random LPT
+simulation and our inference is consistent with both. Our inference accurately captures higher-order statistics of the dark
+matter density field on scales much smaller than the original photometric redshift uncertainty.
+1661
+1329
+997
+664
+332
+0
+x [Mpc]
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+z=30
+(a)
+1661
+1329
+997
+664
+332
+0
+x [Mpc]
+z=10
+(b)
+1661
+1329
+997
+664
+332
+0
+x [Mpc]
+z=0
+(c)
+0.64
+0.66
+0.68
+0.70
+0.72
+0.74
+0.76
+ln(2+ )
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+ln(2+ )
+0.5
+1.0
+1.5
+2.0
+ln(2+ )
+Fig. 8: Structure formation history derived from the dark matter density field at (a) z = 30, (b) z = 10, (c) z = 0 from a
+typical density sample. In red, we overlay the density slice with the galaxy coordinates in a typical realizations. Galaxies
+follow the present-day large-scale structure. The origins of high-density regions are already visible at z = 30.
+the velocity divergence and dark matter density. On large
+scales, it is expected that to linear order θ ∝ δ (e.g. Hahn
+et al. 2015). On smaller scales, structure formation becomes
+nonlinear, which LPT is able to capture. A more refined
+gravity model, like a particle mesh (Jasche & Lavaux 2019,
+in BORG), would be needed to push these results to smaller
+scales and capture the generation of peculiar velocity vor-
+ticity.
+In Figure 7, we show higher-order moments of the con-
+strained dark matter density field for the ground truth, our
+inference and the LPT prediction, with 1σ error bars at the
+target resolution of 13 Mpc (∼ 0.04⟨σz⟩). We derive the LPT
+prediction from a set of random simulations ran using the
+same cosmological parameters as our main inference, but
+without constraints from galaxy redshifts. The error bars
+naturally account for observational uncertainties and the
+estimator uncertainty related to the sample size (Harding
+et al. 2014). For this fully self-consistent setting, our results
+are consistent with the ground truth and LPT prediction
+within 1σ. This result suggests that we infer the density
+field, including its higher-order moments, on scales much
+smaller than the original photometric redshift uncertainty.
+In Figure 8 we show slices through the same randomly-
+selected realization in the structure formation history. Un-
+der the assumption of a causal structure formation model,
+we reconstruct the structure formation history which gives
+rise to the observed structures today. The initial conditions
+are set at z = 99, but we show slices up to z = 30, such
+that the seeds of present-day structures are visible. On the
+z = 0 slice we overlay the ground truth locations of galaxies.
+Galaxies closely trace clusters and filamentary structures,
+whereas voids are sparsely populated by galaxies. This is
+because the Poisson noise is lower in higher-density regions.
+As expected, the origins of high-density peaks in the dark
+matter density field are already visible at early times.
+In Figure 9a, we show a void in the volume covered
+by observations. In Figure 9c we show the average density
+contrast in shells around the void. We see that the density
+contrast tends to zero, as expected when averaging over a
+scale close to the cosmic homogeneity scale (e.g. Gon¸calves
+et al. 2018). In Figure 9b, we show the location of a cluster
+in the density field. In Figure 9d we show the mass enclosed
+in shells around the cluster. We derive it using the prescrip-
+tion in Porqueres et al. (2019). In this study, the particle
+mass is 3.45 × 1013 M⊙. Overall, these results suggest that
+we recover correctly both the statistical properties of the
+large-scale structure around voids and clusters, but also in-
+dividual structures.
+We further show radial peculiar velocity profiles around
+the same void and cluster. The radial peculiar velocity be-
+comes positive close to the void, indicating matter flow-
+ing outside the shells and toward overdense regions, as ex-
+Article number, page 9 of 16
+
+A&A proofs: manuscript no. aanda
+1661
+1107
+553
+x [Mpc]
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(a)
+1661
+1107
+553
+x [Mpc]
+(b)
+50
+100
+150
+200
+250
+R [Mpc]
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+(R)
+(c)
+ground truth
+mean
+3
+50
+100
+150
+200
+250
+R [Mpc]
+1018
+1019
+1020
+1021
+1022
+1023
+1024
+M (< R) [M
+]
+(d)
+50
+100
+150
+200
+250
+R [Mpc]
+100
+0
+100
+200
+300
+400
+vr(R)  [km/s]
+(e)
+50
+100
+150
+200
+250
+R [Mpc]
+400
+300
+200
+100
+0
+vr(R)  [km/s]
+(f)
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Fig. 9: (a) Slice through the ground truth dark matter density field. In purple, a spherical shell around a low-density
+region. (b) Slice through the ground truth dark matter density field. In purple, a spherical shell around a high-density
+peak. (c) The mean density contrast in spherical shells around the low-density region. (d) The mass enclosed in shells
+around the high-density peak. (e) The mean radial peculiar velocity in spherical shells around the low-density region
+with respect to the center of the void. (f) The mean radial peculiar velocity in spherical shells around the high-density
+peak with respect to the center of the peak.
+pected from gravitational structure growth (Peebles 1980).
+On larger scales the average peculiar velocity tends to zero
+due to cosmic homogeneity. In Figure 9f we show the aver-
+age radial peculiar velocity in shells of radius R around the
+cluster in Figure 9b, with respect to its center. The radial
+peculiar velocity is negative close to the cluster, as expected
+from gravitational infall. On larger scales, the average ve-
+locity tends to zero as expected from cosmic homogeneity.
+4.2. Constraints on photometric redshifts
+In this section, we give a brief overview of how the joint
+inference of the large-scale structure and photometric red-
+shift PDFs yields constraints on photometric redshifts. In
+Figure 10 we show our results for three randomly-selected
+galaxies. The redshift likelihood is a Gaussian PDF. Its
+mean is the mock observed redshift and its standard devi-
+ation is 0.05(1 + z), z being the mock cosmological redshift.
+Article number, page 10 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+We derive the target posterior by multiplying the Gaussian
+likelihood with the Poisson term for the redshift inference
+in our Gibbs sampling approach, as shown in Equation (9).
+The results of our inference are overlayed as a histogram.
+This qualitative demonstration illustrates that the photo-
+metric redshift posterior after the application of our method
+contains more information than the likelihood and follows
+the radial density profile of the large-scale structure along
+the line of sight to each galaxy. This mechanism constrains
+the radial positions of galaxies from the initial, radially-
+distorted ones (Figure 1a), to the final ones, that trace the
+filamentary large-scale structure (Figure 1b).
+In Figure 11 we show the reduction in the mean stan-
+dard deviation of the photometric redshift posteriors as a
+function of density after the application of our algorithm.
+σi indicates the original photometric redshift uncertainty
+of the mock observations. We expect that higher-density
+regions yield better constraints and hence, lower redshift
+uncertainty. The linear fit suggests that ⟨σ(δ)⟩/⟨σi(δ)⟩ =
+−0.005(1 + δ) − 0.693. In low-density regions, the mean
+standard deviation reduces by a factor of 0.7⟨σi⟩, whereas
+in high-density regions the reduction reaches 0.4⟨σi⟩. The
+scatter is larger in high-density regions because there are
+fewer strongly overdense peaks. Overall, we expect the red-
+shift uncertainty reduction to be more significant in higher-
+resolution settings, where higher-density peaks will be re-
+solved. It should be noted, however, that due to the highly
+multimodal nature of the target redshift distribution, the
+reduction in standard deviation does not reflect the infor-
+mation gain from the original to the final redshift PDF.
+Finally, in Figure 12 we compare the inferred distribution
+of photometric redshifts, N(z), in our mock survey to the
+ground truth N(z). For legibility, we show the mean N(z)
+across the MCMC samples, along with 3σ error bars. The
+inferred N(z) is consistent with the ground truth N(z). We
+postpone the sampling of the N(z) to future work.
+5. Conclusions
+Next-generation photometric galaxy surveys will deliver an
+extraordinary amount of observations, that will reduce the
+observational uncertainties, will extend deeper in redshift
+and cover a larger cosmological volume than spectroscopic
+ones. At the same time, most cosmological information is in
+the smallest cosmological scales which cannot be accessed in
+presence of large photometric redshift uncertainties. In such
+a setting, it is paramount to obtain control of photometric
+redshift uncertainties in cosmological inferences. Accurate
+inferences of the large-scale structure constrained jointly
+with photometric redshifts offer this possibility.
+In this study we developed a method to constrain
+the primordial and present-day cosmic large-scale struc-
+ture jointly with photometric redshifts at a resolution of
+13 Mpc using, for the first time, a structure formation model
+in a Bayesian forward modelling approach. We achieved
+these joint constraints through Bayesian inference of the
+initial conditions of structure formation with photometric
+galaxy clustering. Our method takes into account data- and
+survey-related uncertainties and preserves all higher-order
+statistics of cosmological fields. In this study, we used first-
+order LPT, yet our algorithm can incorporate any gravi-
+tational model (e.g. Jasche et al. 2015; Jasche & Lavaux
+2019). We demonstrated our constraints using a fully over-
+lapping mock photometric and spectroscopic galaxy cata-
+log as a proof-of-concept. We assumed a worst-case scenario
+for photometric redshift uncertainties in stage-IV surveys,
+σz = 0.05(1 + z), and demonstrated that our method can
+reduce this uncertainty.
+In particular, we showed large improvement in the cross-
+correlation of the photometric galaxy positions with the
+ground truth. The maximum improvement was on scales
+∼ 0.5⟨σz⟩, where the cross-correlation increased from 28%
+(in the mock photometric observations) to 86% (in our in-
+ference). We further achieved accurate inferences of the
+dark matter density, peculiar velocities, gravitational po-
+tential and tidal shear on scales ∼ 0.04⟨σz⟩.
+We demonstrated the ability of our method to accu-
+rately capture the 2-point statistics of the dark matter den-
+sity, as well as its skewness and kurtosis on scales ∼ 0.04⟨σz⟩.
+Higher-order statistics have been promising in constraining
+dark energy (Velten & Fazolo 2020; Fazolo et al. 2022). As
+we can use our method with a particle mesh, our joint in-
+ference framework can be used to constrain the bispectrum
+with photometric galaxy clustering.
+Voids have been extensively used to probe structure for-
+mation close to the linear regime (e.g. Davies et al. 2019;
+Pisani et al. 2019; Davies et al. 2021; Stopyra et al. 2021;
+Contarini et al. 2022), however photometric uncertainties
+limit their detection (e.g. S´anchez et al. 2017). For this pur-
+pose, we explored the possibility of detecting cosmic struc-
+tures much smaller than the original photometric redshift
+uncertainty. The galaxy positions that were originally radi-
+ally smeared due to the presence of photometric uncertain-
+ties, accurately trace the filamentary pattern of the large-
+scale structure after the application of our method. Further,
+we are able to accurately capture individual structures, like
+voids and clusters. This is a crucial advantage of embedding
+a structure formation model in our forward model. Overall,
+we demonstrated that we accurately recover the statisti-
+cal properties of the large-scale structure on scales much
+smaller than the original photometric redshift uncertainty.
+The present work opens up the possibility to miti-
+gate photometric redshift uncertainties in next-generation
+photometric surveys. Concurrently, the incorporation of a
+structure formation model paves a new way forward to ex-
+tract as much information as possible from the smallest
+cosmological scales, while going beyond 2-point statistics
+and preserving information in the data.
+Data and software availability
+Data products can be made available upon reasonable re-
+quest.
+Author Contributions
+The main roles of the authors were, using the CRediT
+(Contribution
+Roles
+Taxonomy)
+system
+(https:
+//authorservices.wiley.com/author-resources/
+Journal-Authors/open-access/credit.html):
+E.T.: conceptualization, methodology, software, formal
+analysis, validation, writing - original draft; J.J.: conceptu-
+alization, methodology, software (supporting, transferred
+Jasche & Wandelt (2012) to new BORG version), supervi-
+sion, resources, writing - feedback, funding acquisition;
+G.L.: software (supporting, debugging and notably during
+the Message Passing Interface parallelization), writing -
+Article number, page 11 of 16
+
+A&A proofs: manuscript no. aanda
+0.05
+0.10
+0.15
+0.20
+0.25
+redshift
+0
+5
+10
+15
+20
+P(z|zobs, )
+(a)
+observation
+target posterior
+ground truth 
+redshift
+inference
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+redshift
+(b)
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+redshift
+(c)
+Fig. 10: Photometric redshift samples for three randomly-selected galaxies. Overlayed are the corresponding initial ob-
+servations estimated from our pipeline without a physical prior, the ideal photometric redshift posterior (for physical
+prior, using the ground truth density) and the ground truth redshift. The maximum redshift to which the posteriors
+extend indicates where each galaxy’s line of sight intercepts the inference box. The inferred posterior is the marginal over
+density realizations, accounting for observational and survey-related uncertainties. The redshift binning is determined by
+the real-space resolution of the inference.
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+1 +
+0.4
+0.5
+0.6
+0.7
+0.8
+( ) /
+i( )
+linear fit
+Fig. 11: Final to initial photometric redshift uncertainty as
+a function of density at the ground truth galaxy locations.
+The black line is a linear fit, indicating that the reduction in
+redshift uncertainty is greater in regions of high dark mat-
+ter density, as expected from the Poisson likelihood. The
+dispersion on the high-density end is due to shot noise, as
+there are fewer high-density peaks in the inference domain.
+feedback, funding acquisition; F.L.: software (support-
+ing, author of the Simplex-In-Cell estimator), writing -
+feedback.
+Acknowledgements
+ET thanks Metin Ata and Natalia Porqueres for helpful
+discussions and feedback, and Adam Andrews, Deaglan
+Bartlett, Andrija Kostic, James Prideaux-Ghee, Fabian
+Schmidt, Ben Wandelt and Pauline Zarrouk for com-
+ments on the original manuscript. Part of the computa-
+tion and data processing in this study were enabled by re-
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+redshift
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+normalized N(z)
+mock N(z)
+3
+Fig. 12: In orange, the average of normalized N(z) of in-
+ferred photometric redshifts across MCMC samples for a
+subsample of galaxies, along with 3σ uncertainties. In black,
+the mock photometric N(z) for the same subsample. The
+N(z) of inferred photometric redshifts is consistent with the
+ground truth N(z).
+sources provided by the Swedish National Infrastructure
+for Computing (SNIC) at Tetralith, partially funded by
+the Swedish Research Council through grant agreement no.
+2020-05143. This research utilized the HPC facility sup-
+ported by the Technical Division at the Department of
+Physics, Stockholm University. This work was enabled by
+the research project grant ‘Understanding the Dynamic
+Universe’ funded by the Knut and Alice Wallenberg Foun-
+dation under Dnr KAW 2018.0067. JJ acknowledges sup-
+port by the Swedish Research Council (VR) under the
+project 2020-05143 – ”Deciphering the Dynamics of Cosmic
+Structure”. GL acknowledges support by the ANR BIG4
+project, grant ANR-16-CE23-0002 of the French Agence
+Nationale de la Recherche, and the grant GCEuclid from
+”Centre National d’Etudes Spatiales” (CNES). This work
+was supported by the Simons Collaboration on “Learning
+Article number, page 12 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+the Universe”. This work is conducted within the Aquila
+Consortium (https://aquila-consortium.org).
+A. Derivation of the conditional density posterior
+for real-space inference
+In this section, we describe how to arrive at the condi-
+tional density posterior of Equation (3) starting from Equa-
+tion (2). The former equation shows how we incorporate a
+gravitational structure growth model into our framework,
+whereas the latter represents the output of the Gibbs sam-
+pling process. Here, we discuss how we obtain constraints
+on the real-space late-time density field using galaxy obser-
+vations in redshift space.
+The present-day density field is conditionally indepen-
+dent of the observed galaxy redshifts given an ensemble of
+sampled redshifts. Therefore
+P(δ|z, zobs, θ)
+=
+P(δ|z, θ)
+=
+�
+Ng
+P(δ, Ng|z, θ)
+=
+�
+Ng
+P(δ|Ng)P(Ng|z, θ).
+(17)
+The last term in the sum indicates how we arrive from
+redshift-space observations to real-space galaxy counts. As
+elaborated on in (Jasche & Wandelt 2012, Eq. B4), this
+involves a transform from redshift- to real-space such that
+P(δ|z, θ) =
+�
+Ng
+P(δ|Ng)P(Ng|x(z, θ)),
+(18)
+x being the real-space galaxy positions. In what follows, we
+will omit dependence on (z, θ) for legibility. The latter term
+indicates how we bin galaxies onto a grid given their real-
+space positions. For this gridding operation, we use Nearest
+Grid Point projection such that
+P(Ng|x) =
+�
+i
+δD
+��������Ng
+i −
+�
+p
+WNGP(xi − xp)
+�������� ,
+(19)
+i being the voxel index, p being the galaxy index and
+WNGP(x) =
+3
+�
+n=1
+�1
+if |xn|Nn/Ln < 1
+0
+otherwise,
+(20)
+Substituting the above into Equation (18) we arrive at
+P(δ|z, θ) = P(δ|Ng).
+(21)
+B. Derivation of the conditional redshift posterior
+for real-space inference
+In this subsection we describe how we arrive at the condi-
+tional redshift posterior in Equation (9), while performing
+an inference of the density field in real space. Let us denote
+by θ the right ascension and declination of a galaxy and by
+u the redshifts of all other galaxies in the previous sampling
+step. We start by the redshift posterior of a given galaxy i
+and apply Bayes’ law
+P(zi|θ, u, δ, zobsi)
+=
+P(zi)P(θ, u, δ, zobsi, zi)
+P(θ, u, δ, zobsi)
+=
+P(zi)P(zobsi|θ, u, δ, zi)P(θ, u, δ|zi)
+P(zobsi|θ, u, δ)P(θ, u, δ)
+=
+P(θ, u, δ|zi)P(zi)
+P(u, θ, δ)
+P(zobsi|θ, u, δ, zi)
+P(zobsi|θ, u, δ)
+=
+P(zi|θ, u, δ)P(zobsi|θ, zi, δ, u)
+P(zobsi|u, δ, θ)
+=
+P(θ, zi|u)P(δ|θ, zi)
+P(δ|u)
+P(zobsi|θ, zi, δ, u)
+P(zobsi|u, δ, θ)
+(22)
+We assume zobsi is conditionally independent of u given zi
+and δ and therefore we arrive at
+P(zi|θ, u, δ, zobsi) = P(θ, zi|u)P(δ|θ, zi)
+P(δ|u)
+P(zobsi|θ, zi, δ)
+P(zobsi|δ, θ)
+(23)
+The first term on the right-hand side of the above equa-
+tion describes the distribution of galaxies in redshift space.
+However, as we perform a real-space inference, we trans-
+form this term from redshift- z, to comoving Cartesian, r,
+space
+P(θ, zi|u) =
+����r2(zi) sin (φ)∂r
+∂z
+����zi
+����P(x|u),
+(24)
+φ being the declination of the galaxy and x its real-space
+position. The real-space galaxy positions are used to build
+the gridded galaxy count field, Mg. Mg indicates the galaxy
+counts except for the galaxy under consideration. We the
+last term on the right-hand side of the above equation as a
+marginal over galaxy counts
+P(x|u) =
+�
+Mg
+P(Mg|u)P(x|Mg)
+(25)
+The first term in the sum is the Nearest Grid Point projec-
+tion that we use to grid galaxies at the field-level. We will
+indicate the Nearest Grid Point kernel with WNGP. Under
+the assumption that galaxies are Poisson-distributed, the
+number counts in each voxel are independent events. Since
+the Poisson intensity depends only on the density, our Pois-
+son model is homogeneous. Following these assumptions
+P(x|u)
+=
+�
+Mg
+P(x|Mg)
+×
+�
+i
+δD
+��������Mg
+i −
+�
+p
+WNGP(xi − xp)
+�������� ,
+(26)
+where i is the voxel index and p the galaxy index. We now
+want to rewrite the first term in the above sum with respect
+to the entire galaxy count field, Ng. It differs from Mg in that
+it includes the galaxy under consideration. The reason we
+rewrite the redshift posterior with respect to Ng is because
+the entire galaxy count field is used in the density inference.
+Below this point we will drop the dependence on u, because
+all quantities are conditionally independent of u given Mg.
+P(x|Mg) =
+1
+P(Mg)
+�
+Ng
+P(Ng)P(x|Ng)P(Mg|Ng, x)
+(27)
+Following our definition, the relationship between Mg and
+Ng is deterministic and yields
+P(x|Mg)
+=
+1
+P(Mg)
+�
+Ng
+P(Ng)P(x|Ng)
+Article number, page 13 of 16
+
+A&A proofs: manuscript no. aanda
+×
+�
+i
+δD �
+Mg
+i − [Ng
+i − WNGP(xi − x)]
+�
+=
+P(x|Ng),
+(28)
+because P(Ng) and P(Mg) are equal, as they differ by one
+(at the position of the galaxy under consideration). The
+above result is equal to finding a galaxy at position x given
+a number counts field.
+P(x|Ng) =
+�
+i
+WNGP(xi − x)
+δV
+Ng
+i
+Ntotal
+,
+(29)
+where δV is the volume of a voxel and Ntotal the total number
+of galaxies in the inference domain. As a result of the above,
+the second term on the right-hand side of Equation (23) is
+written as
+P(δ|θ, zi)
+P(δ|u)
+=
+�
+i
+Mi!
+Ni! λNi−Mi
+i
+=
+�
+i
+� λi
+Ni
+�WNGP(xi−x)
+,
+(30)
+as it represents the ratio between the Poisson distribution
+for all galaxies and all galaxies expect for the one we con-
+sider at each step. Substituting Equation (29) and Equa-
+tion (30) into Equation (23) we arrive at
+P(zi|θ, u, δ, zobsi)
+∝
+����r2(zi) sin (φ)∂r
+∂z
+����zi
+����
+×
+�
+i
+WNGP(xi − x)λiP(zobsi|θ, zi, δ).
+(31)
+As demonstrated in Jasche & Wandelt (2013) the above
+form suggests that each galaxy can be sampled indepen-
+dently from all others, as the dependence on u vanishes
+through the conditioning on galaxy counts and therefore,
+the Poisson intensity. Finally, since sin (φ) is a constant
+term, it serves as a proportionality constant in each galaxy’s
+target posterior and therefore vanishes. Finally, we drop the
+dependence on right ascension and declination for legibil-
+ity, since we only sample redshifts. Therefore, the process
+in Equation (1) is equivalent to drawing a sample from
+P(zi|δ, zobsi)
+∝
+����r2(zi)∂r
+∂z
+����zi
+����
+×
+�
+i
+WNGP(xi − x)λiP(zobsi|zi, δ).
+(32)
+C. Estimators of large-scale structure properties
+Accurate inferences of the large-scale structure with pho-
+tometric galaxy clustering are necessary because next-
+generation photometric surveys probe deeper redshifts and
+have a wider footprint than spectroscopic surveys. Below,
+we discuss the aspects of the cosmic large-scale structure
+which we demonstrate our method on, as well as the esti-
+mators we use.
+C.1. Peculiar velocities
+In order to derive the peculiar velocity field, we use the
+Simplex-In-Cell (SIC) estimator (e.g. Abel et al. 2012; Hahn
+et al. 2015; Leclercq et al. 2017). Contrary to kernel meth-
+ods, which sample the velocity field only at a discrete set
+of locations with poor resolution in low-density regions, the
+SIC estimator provides an estimate of the velocity field at
+any point in space. In this framework, the Lagrangian posi-
+tions of the particles in the inference domain are considered
+as vertices of unit cubes. The Delaunay tessellation of each
+unique cube defines six tetrahedra which are followed dur-
+ing the evolution. Subsequently, each tetrahedron deposits
+a value to the grid (Leclercq et al. 2017, Equation 37).
+We further explore kinematic properties of the peculiar
+velocity field. Here, we focus on the peculiar velocity diver-
+gence, defined as
+θ = −
+1
+f Ha∇ · v,
+(33)
+where f is the linear growth rate, H the Hubble parame-
+ter and a the cosmic scale factor. The divergence determines
+how the volume of a peculiar velocity flow changes. The pe-
+culiar velocity divergence can be used to constrain Ωm and
+structure formation (e.g. Bernardeau et al. 1995; Howlett
+et al. 2017). Further, the redshift-space distortion signal can
+be detected in the autocorrelation power spectrum of the
+peculiar velocity divergence and the cross-correlation power
+spectrum between the dark matter density and the peculiar
+velocity divergence (e.g. Bel et al. 2019). For ∇ · v > 0 the
+region is expanding, for ∇ · v < 0 the region is contracting,
+whereas for ∇ · v = 0 the volume remains invariant.
+C.2. Gravitational potential
+Probing gravitational tidal forces with photometric sur-
+veys is a challenging endeavor, but a necessary one (e.g.
+Troxel & Ishak 2015), as intrinsic galaxy alignments con-
+taminate weak lensing analyses and can be used as cos-
+mological probes. Further, as photometric surveys extend
+deeper in redshift, they allow us to probe the evolution of
+intrinsic alignments over time. This evolution can be used
+to constrain galaxy formation and evolution scenarios.
+For this purpose, we explore differing aspects of the dy-
+namics and kinematics of the large-scale structure. We de-
+rive the gravitational potential of the dark matter density
+field by solving Poisson’s equation in Fourier space
+Φ(k) = −4πGρ(k)
+|k|2
+,
+(34)
+where G is the gravitational constant, ρ the density and
+k the wavevector. We further derive the tidal shear of the
+gravitational potential, which is given by
+Ti j =
+∂2Φ
+∂xi∂x j
+,
+(35)
+where xi, ({i, j} = {1, 2, 3}) are comoving Cartesian coordi-
+nates. Our inference can further be used with cosmic web
+classification methods, particle- (e.g. Sousbie 2013) or cell-
+based (e.g. Libeskind et al. 2018; Buncher & Carrasco Kind
+2020) to infer cosmic structures constrained by photomet-
+ric redshifts. Such classifications have been used to study
+the environment of cosmological tracers (e.g. Leclercq et al.
+2016; Porqueres et al. 2018; Tsaprazi et al. 2022a). Kru-
+use et al. (2019) further showed that photometric redshifts
+are positively correlated with filaments detected in spectro-
+scopic galaxy observations.
+Article number, page 14 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
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