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+Relativistic BGK hydrodynamics
+Pracheta Singhaa, Samapan Bhadurya,c, Arghya Mukherjeeb, Amaresh Jaiswala
+aSchool of Physical Sciences, National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Jatni 752050, Odisha, India
+bDepartment of Physics and Astronomy, Brandon University, Brandon, Manitoba R7A 6A9, Canada
+cInstitute of Theoretical Physics, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Krakow, Poland
+Abstract
+Bhatnagar-Gross-Krook (BGK) collision kernel is employed in the Boltzmann equation to formulate relativistic dissipative hydro-
+dynamics. In this formulation, we find that there remains freedom of choosing a matching condition that affects the scalar transport
+in the system. We also propose a new collision kernel which, unlike BGK collision kernel, is valid in the limit of zero chemical
+potential and derive relativistic first-order dissipative hydrodynamics using it. We study the effects of this new formulation on the
+coefficient of bulk viscosity.
+1. Introduction
+Relativistic Boltzmann equation governs the space-time evo-
+lution of the single particle phase-space distribution function of
+a relativistic system. Moreover, suitable moments of the Boltz-
+mann equation are capable of describing the collective dynam-
+ics of the system. Therefore, it has been extensively used to de-
+rive equations of relativistic dissipative hydrodynamics and ob-
+tain expressions for the transport coefficients [1–14]. The col-
+lision term in the Boltzmann equation, which describes change
+in the phase-space distribution due to the collisions of particles,
+makes it a complicated integro-differential equation. In order
+to circumvent this issue, several approximations have been sug-
+gested to simplify the collision term in the linearized regime
+[15–19].
+Bhatnagar-Gross-Krook [15], and independently Welander
+[16], proposed a relaxation type model for the collision term,
+which is commonly known as the BGK model. This model
+was further simplified by Marle [17] and Anderson-Witting [18]
+to calculate the transport coefficients. In the non-relativistic
+limit, Marle’s formulation leads to the same transport coeffi-
+cient as the BGK model but fails in the relativistic limit. On
+the other hand, the Anderson-Witting model, also known as the
+relaxation-time approximation (RTA), is better suited in the rel-
+ativistic limit. The RTA has been employed extensively in sev-
+eral areas of physics with considerable success and has been
+widely employed in the formulation of relativistic dissipative
+hydrodynamics [8–11, 20–32].
+The RTA Boltzmann equation has provided remarkable in-
+sights into the causal theory of relativistic hydrodynamics as
+well as a simple yet meaningful picture of the collision mech-
+anism in a non-equilibrium system. On the other hand, the
+Email addresses: pracheta.singha@gmail.com (Pracheta Singha),
+samapan.bhadury@niser.ac.in (Samapan Bhadury),
+arbp.phy@gmail.com (Arghya Mukherjee), a.jaiswal@niser.ac.in
+(Amaresh Jaiswal)
+BGK collision term ensures conservation of net particle four-
+current by construction, and is the precursor to RTA. While the
+RTA has been employed extensively, a consistent formulation
+of relativistic dissipative hydrodynamics with the BGK colli-
+sion term is relatively less explored. This may be attributed to
+the fact that the BGK collision kernel is ill defined for relativis-
+tic systems without a conserved net particle four-current. This
+has limited the use of the BGK collision kernel to the studies
+related to flow of particle number and/or charge [33–43].
+In this article, we take the first step towards formulating a
+consistent framework of relativistic dissipative hydrodynamics
+using the BGK collision kernel. Furthermore, we propose a
+modified BGK collisions kernel (MBGK), which is well de-
+fined even in the absence of conserved particle four-current and
+is better suited for the formulation of the relativistic dissipative
+hydrodynamics. We find that there exists a free scalar parame-
+ter arising from the freedom of matching condition. This affects
+the scalar dissipation in the system, i.e., the coefficient of bulk
+viscosity. We study the effect on bulk viscosity in several dif-
+ferent scenarios.
+2. Relativistic dissipative hydrodynamics
+The conserved net particle four-current, Nµ, and the energy-
+momentum tensor, T µν, of a system can be expressed in terms
+of the single particle phase-space distribution function and the
+hydrodynamic variables as [44],
+Nµ =
+�
+dP pµ �
+f − ¯f
+�
+= n uµ + nµ,
+(1)
+T µν =
+�
+dP pµpν �
+f + ¯f
+�
+= ϵ uµuν − (P0 + δP) ∆µν + πµν, (2)
+where the Lorentz invariant momentum integral measure is de-
+fined as dP = g d3p/
+�
+(2π)3E
+�
+with g being the degeneracy fac-
+tor and E =
+�
+|p|2 + m2 being the on-shell energy of the con-
+stituent particle of the medium with three-momentum p and
+Preprint submitted to Physics Letters B
+January 3, 2023
+arXiv:2301.00544v1  [nucl-th]  2 Jan 2023
+
+mass m. Here f ≡ f(x, p) and ¯f ≡
+¯f(x, p) are the phase-
+space distribution functions for particles and anti-particles, re-
+spectively. In the above equations, n is the net particle number
+density, ϵ is the energy density, P0 is the equilibrium pressure,
+nµ is the particle diffusion four-current, δP is the correction to
+the isotropic pressure, and πµν is the shear stress tensor. We
+note that the fluid four-velocity uµ has been defined in the Lan-
+dau frame, uµT µν = ϵuν. We also define ∆µν ≡ gµν − uµuν as
+the projection operator orthogonal to uµ. In this article, we will
+be working in a flat space-time with metric tensor defined as,
+gµν = diag(1, −1, −1, −1).
+Hydrodynamic equations are essentially the equations for
+conservation of net particle four current, ∂µNµ = 0, and energy-
+momentum tensor, ∂µT µν = 0. Using the expressions of Nµ and
+T µν from Eqs. (1) and (2), the hydrodynamic equations can be
+obtained as,
+˙n + nθ + ∂µnµ = 0
+(3)
+˙ϵ + (ϵ + P0 + δP) θ − πµνσµν = 0
+(4)
+(ϵ + P0 + δP) ˙uα − ∇α (P0 + δP) + ∆α
+ν∂µπµν = 0
+(5)
+where we use the standard notation, ˙A ≡ uµ∂µA for the co-
+moving derivatives, ∇α ≡ ∆αβ∂β for the space-like derivatives,
+θ = ∂µuµ for the expansion scalar, and σµν = 1
+2 (∇µuν + ∇νuµ) −
+1
+3∆µνθ for the velocity stress-tensor.
+To express the conserved net particle four-current and the
+energy-momentum tensor in terms of hydrodynamic variables
+in Eqs. (1) and (2), we chose Landau frame to define the fluid
+four-velocity. Additionally, the net-number density and energy
+density of a non-equilibrium system needs to be defined us-
+ing the so called matching conditions. We relate these non-
+equilibrium quantities with their equilibrium values as
+n = n0 + δn,
+ϵ = ϵ0 + δϵ,
+(6)
+where n0 and ϵ0 are the equilibrium net-number density and
+the energy density, respectively, and, δn, δϵ are the corre-
+sponding non-equilibrium corrections. For a system which is
+out-of-equilibrium, the distribution function can be written as
+f = f0 + δ f, where f0 is the equilibrium distribution function
+and δf is the non-equilibrium correction. In the present work,
+we consider the equilibrium distribution function to be of the
+classical Maxwell-Juttner form, f0 = exp(−β u · p + α), where
+β ≡ 1/T is the inverse temperature, α ≡ µ/T is the ratio of
+chemical potential to temperature and u · p ≡ uµpµ. The equi-
+librium distribution for anti-particles is also taken to be of the
+Maxwell-Juttner form with α → −α.
+We can now express the equilibrium hydrodynamic quanti-
+ties in terms of the equilibrium distribution function as,
+n0 =
+�
+dP (u · p)
+�
+f0 − ¯f0
+�
+(7)
+ϵ0 =
+�
+dP (u · p)2 �
+f0 + ¯f0
+�
+(8)
+P0 = −1
+3∆µν
+�
+dP pµpν �
+f0 + ¯f0
+�
+.
+(9)
+Similarly, the non-equilibrium quantities can be expressed as
+δn =
+�
+dP (u · p)
+�
+δ f − δ ¯f
+�
+(10)
+δϵ =
+�
+dP (u · p)2 �
+δ f + δ ¯f
+�
+(11)
+δP = −1
+3∆αβ
+�
+dP pαpβ �
+δ f + δ ¯f
+�
+,
+(12)
+nµ = ∆µ
+α
+�
+dP pα �
+δ f − δ ¯f
+�
+,
+(13)
+πµν = ∆µν
+αβ
+�
+dP pαpβ �
+δ f + δ ¯f
+�
+,
+(14)
+where ∆µν
+αβ ≡ 1
+2(∆µ
+α∆ν
+β+∆µ
+β∆ν
+α)− 1
+3∆µν∆αβ is a traceless symmetric
+projection operator orthogonal to uµ as well as ∆µν. In order to
+calculate these non-equilibrium quantities, we require the out-
+of-equilibrium correction to the distribution function, δf and
+δ ¯f. To this end, we consider the Boltzmann equation with BGK
+collision kernel.
+3. The Boltzmann equation and conservation laws
+The covariant Boltzmann equation, in absence of any force
+term or mean-field interaction term, is given by,
+pµ∂µ f = C[ f, ¯f],
+pµ∂µ ¯f = ¯C[ f, ¯f],
+(15)
+for a single species of particles and its antiparticles. In the
+above equation, C[ f, ¯f] and ¯C[ f, ¯f] are the collision kernels
+that contain the microscopic information of the scattering pro-
+cesses. For the formulation of relativistic hydrodynamics from
+the kinetic theory of unpolarized particles, the collision ker-
+nel of the Boltzmann equation must satisfy certain properties.
+Firstly, the collision kernel must vanish for a system in equilib-
+rium, i.e., C[ f0, ¯f0] = ¯C[ f0, ¯f0] = 0. Further, in order to satisfy
+the fundamental conservation equations in the microscopic in-
+teractions, the zeroth and the first moments of the collision ker-
+nel must vanish, i.e.,
+�
+dPC = 0 and
+�
+dP pµ C = 0. Vanishing
+of the zeroth moment and the first moment of the collision ker-
+nel follows from the net particle four-current conservation and
+the energy-momentum conservation, respectively.
+In the present work, we consider the BGK collision kernel
+which has the advantage that the particle four-current is con-
+served by construction. The relativistic Boltzmann equation
+with BGK collision kernel for particles can be written as [15],
+pµ∂µ f = −(u · p)
+τR
+�
+f − n
+n0
+f0
+�
+,
+(16)
+and similarly for anti-particles with f → ¯f and f0 → ¯f0. Here,
+τR is a relaxation time like parameter1 which we assume to be
+the same for particles and anti-particles. It is easy to verify that
+the conservation of net particle four-current, defined in Eq. (1),
+follows from the zeroth moment of the above equations. The
+1A more conventional notation is the collision frequency which is defined
+as ν = 1/τR.
+2
+
+first moment of the above equations should lead to the con-
+servation of the energy-momentum tensor, defined in Eq. (2).
+However, we find that the first moment of the Boltzmann equa-
+tion, Eq. (16), leads to,
+∂µT µν = − 1
+τR
+�
+ϵ − n
+n0
+ϵ0
+�
+,
+(17)
+which does not vanish automatically.
+In order to have energy-momentum conservation fulfilled
+by the Boltzmann equation with the BGK collision kernel,
+Eq. (16), we require that
+ϵn0 = ϵ0n,
+(18)
+which we identify as one matching condition. Note that two
+matching conditions are required to define the non-equilibrium
+net number density and the energy density. Along with the
+above equation, we are left with the freedom of one matching
+condition. It is important to observe that the RTA Boltzmann
+equation, pµ∂µ f = − (u·p)
+τR ( f − f0), is recovered from Eq. (16)
+if the second matching condition is fixed as either ϵ = ϵ0 or
+equivalently n = n0. For the RTA collision term, both matching
+conditions ϵ = ϵ0 and n = n0, are necessary for net particle four-
+current and energy-momentum conservation. However, for the
+BGK collision kernel, both conservation equations are satisfied
+with only one matching condition, Eq. (18), leaving the other
+condition free. We shall see later that this scalar freedom af-
+fects the coefficient of bulk viscosity, which is the transport co-
+efficient corresponding to scalar dissipation in the system.
+Note that the equilibrium net number density, defined in
+Eq. (7), vanishes in the limit of zero chemical potential. This
+implies that the BGK collision term in Eq. (16) is ill defined
+in this limit, which is relevant for ultra-relativistic heavy-ion
+collisions. Therefore, it is desirable to modify the BGK col-
+lision kernel in order to extend its regime of applicability. At
+this juncture, we are well equipped to propose a modification
+to BGK collision kernel that is well-defined for all values of
+chemical potential. To this end, we rewrite the condition nec-
+essary for energy-momentum conservation from BGK collision
+kernel, Eq. (18), in the form
+n
+n0
+= ϵ
+ϵ0
+.
+(19)
+Substituting the above equation in Eq. (16), we obtain Boltz-
+mann equation for particles with a modified BGK (MBGK) col-
+lision kernel,
+pµ∂µ f = −(u · p)
+τR
+�
+f − ϵ
+ϵ0
+f0
+�
+,
+(20)
+and similarly for anti-particles with f → ¯f and f0 → ¯f0. The
+advantage of the above modification is that the collision ker-
+nel conserves energy-momentum by construction and is appli-
+cable to systems even without any conserved four-current, i.e.,
+in the limit of vanishing chemical potential. In the case of finite
+chemical potential, the matching condition, Eq. (18), ensures
+net particle four-current conservation. It is important to note
+that BGK and MBGK are completely equivalent for the purpose
+of the derivation of hydrodynamic equations at finite chemical
+potential. In the following, we consider the MBGK Boltzmann
+equation, Eq. (20), to obtain non-equilibrium correction to the
+distribution function.
+4. Non-equilibrium correction to the distribution function
+In order to obtain the non-equilibrium correction to the dis-
+tribution function, we use Eq. (6) to rewrite the MBGK Boltz-
+mann equation, Eq. (20), as
+pµ∂µ f = −(u · p)
+τR
+�
+δ f − δϵ
+ϵ0
+f0
+�
+,
+(21)
+and similarly for anti-particles. The next step is to solve the
+above equation, order-by-order in gradients. In this work, we
+intend to obtain the non-equilibrium correction to the distribu-
+tion function up to first-order in derivative, which we repre-
+sent by δ f1. However, obtaining the expressions for δf1 from
+Eq. (21) is not straightforward because it contains δϵ which is
+defined in Eq. (11) as an integral over δ f. Therefore, to solve
+for δ f1, we examine each term individually. Up to first-order
+in gradients, the structure of the term on the left-hand side of
+Eq. (21) has the form,
+pµ∂µ f0 =
+�
+AΠθ + Anpµ∇µα + Aπpµpνσµν
+�
+f0,
+(22)
+and similarly for anti-particles. Here,
+AΠ = −
+�
+(u · p)2 �
+χb − β
+3
+�
+− (u · p) χa + βm2
+3
+�
+,
+(23)
+An = 1 − n0 (u · p)
+(ϵ0 + P0),
+Aπ = − β.
+(24)
+The coefficients χa and χb appearing in Eq. (23) are defined via
+the relations
+˙α = χa θ,
+˙β = χb θ,
+∇µβ =
+n0
+ϵ0 + p0
+∇µα − β˙uµ
+(25)
+χa = I−
+20(ϵ0 + P0) − I+
+30n0
+I+
+30I+
+10 − I−
+20I−
+20
+,
+χb = I+
+10(ϵ0 + P0) − I−
+20n0
+I+
+30I+
+10 − I−
+20I−
+20
+, (26)
+where, the thermodynamic integrals are given by,
+I±
+nq =
+(−1)q
+(2q + 1)!!
+�
+dP (u · p)n−2q �
+∆αβpαpβ�q �
+f0 ± ¯f0
+�
+.
+(27)
+With the above definition, we identify n0 = I−
+10, ϵ0 = I+
+20 and
+P0 = I+
+21.
+We assume δ f1 to have the same form as in Eq. (22),
+δ f1 = τR
+�
+BΠθ + Bnpµ∇µα + Bπpµpνσµν
+�
+f0,
+(28)
+and similarly for anti-particles. In the above expression, the co-
+efficients BΠ, Bn and Bπ needs to be determined using Eq. (21),
+up to first order in derivatives. To that end, we substitute the
+expression for δ f1 in Eq. (11) to obtain
+δϵ = τR
+�
+dP (u · p)2 �
+BΠ f0 + ¯BΠ ¯f0
+�
+θ
+(29)
+3
+
+Using Eqs. (22), (28) and (29) into Eq. (21) and comparing both
+sides, we get
+−
+AΠ
+(u · p) = BΠ − 1
+ϵ0
+�
+dP (u · p)2 �
+BΠ f0 + ¯BΠ ¯f0
+�
+(30)
+Bn = −
+An
+(u · p) ,
+Bπ = −
+Aπ
+(u · p).
+(31)
+Another set of equations in terms of ¯AΠ, ¯An and ¯Aπ can be
+obtained by considering the MBGK equation, analogous to
+Eq. (21), for anti-particles. Note that the coefficients Bn, ¯Bn,
+Bπ and ¯Bπ are easily determined but BΠ and ¯BΠ require further
+investigation.
+To obtain their expressions, we consider BΠ to be of the gen-
+eral form, BΠ = �+∞
+k=−∞ bk (u · p)k and ¯BΠ = �+∞
+k=−∞ ¯bk (u · p)k.
+Substituting these in Eq. (30) and its corresponding equation for
+anti-particles, we can conclude that the only non-zero bk and ¯bk
+are the ones with k = −1, 0, 1. We obtain
+BΠ =
+1
+�
+k=−1
+bk (u · p)k ,
+¯BΠ =
+1
+�
+k=−1
+¯bk (u · p)k .
+(32)
+Substituting Eqs. (23) and (32) in Eq. (30), we find
+b1 = ¯b1 = χb − β
+3
+and,
+b−1 = ¯b−1 = m2β
+3 ,
+(33)
+where we have also used the relation analogous to Eq. (30) for
+anti-particles. On the other hand, for b0 and ¯b0 we find two
+coupled equations, which are identical and can be simplified to
+the relation,
+¯b0 = b0 + 2 χa.
+(34)
+Hence, we see that a unique solution for b0 and ¯b0 can not be
+obtained but they are constrained by the above relation. We
+need to provide one more condition, which we recognize as the
+second matching condition, to fix b0 and ¯b0 separately.
+Nevertheless, at this stage, we can determine δf1 and δ ¯f1 up
+to a free parameter, b0, by using Eqs. (30)-(34) into Eq. (28),
+and similarly for anti-particles. We obtain,
+δf1 = τR f0
+� �
+m2β
+3 (u · p) + b0 + (u · p)
+�
+χb − β
+3
+��
+θ
+−
+�
+1
+(u · p) −
+n0
+(ϵ0 + P0)
+�
+pµ �
+∇µα
+�
++ βpµpµσµν
+(u · p)
+�
+,
+(35)
+δ ¯f1 = τR ¯f0
+� �
+m2β
+3 (u · p) + b0 + 2 χa + (u · p)
+�
+χb − β
+3
+��
+θ
++
+�
+1
+(u · p) +
+n0
+(ϵ0 + P0)
+�
+pµ �
+∇µα
+�
++ βpµpµσµν
+(u · p)
+�
+.
+(36)
+Note that for vanishing chemical potential, we have α = χa = 0.
+In this case, Eqs. (35) and (36) coincide to give
+δf1
+����µ=0 = τR βf0
+�� m2
+3 (u·p) + b0
+β +(u·p)
+�
+c2
+s− 1
+3
+��
+θ+ pµpµσµν
+(u·p)
+�
+,
+(37)
+where we have used χb = βc2
+s, with c2
+s being the squared of the
+speed of sound, given by,
+c2
+s =
+(ϵ0 + P0)
+3ϵ0 + �3 + z2� P0
+.
+(38)
+Here z ≡ m/T is the ratio of particle mass to temperature.
+5. First order dissipative hydrodynamics
+The first-order correction to the phase-space distribution
+functions of the particles and anti-particles at finite µ are given
+by Eqs. (35) and (36). Substituting them in Eqs. (10)-(14), we
+obtain the first-order expressions for non-equilibrium hydrody-
+namic quantities as
+δn = νθ,
+δϵ = eθ,
+δP = ρθ,
+nµ = κ∇µα,
+πµν = 2ησµν,
+(39)
+where,
+ν = τR (χa + b0) n0,
+e = τR (χa + b0) ϵ0,
+(40)
+ρ = τR
+�
+(χa + b0)P0 + χb
+(ϵ0 + P)
+β
+− 5
+3βI+
+32 − χan0
+β
+�
+,
+(41)
+κ = τR
+������I+
+11 −
+n2
+0
+β(ϵ0 + P)
+������ ,
+η = τR β I+
+32.
+(42)
+Note that the parameter b0 appears in the expressions of ν, e
+and ρ. Of these, ν and e vanishes for b0 = −χa which cor-
+responds to the Landau matching condition and RTA collision
+kernel. Conductivity κ and the coefficient of shear viscosity
+η does not contain the parameter b0, and the expressions for
+these two transport coefficients, given in Eq. (42), matches with
+those derived using RTA collision kernel [11]. Next, we ana-
+lyze entropy production in the MBGK setup in order to identify
+dissipative transport coefficients in Eqs. (40)-(42).
+To study entropy production, we start from the kinetic theory
+definition of entropy four-current, given by the Boltzmann’s H-
+theorem, for a classical system
+S µ = −
+�
+dPpµ�
+f (ln f − 1) + ¯f
+�
+ln ¯f − 1
+� �
+.
+(43)
+The
+entropy
+production
+is
+determined
+by
+taking
+four-
+divergence of the above equation,
+∂µS µ = −
+�
+dPpµ� �
+∂µ f
+�
+ln f +
+�
+∂µ ¯f
+�
+ln ¯f
+�
+.
+(44)
+Using the MBGK Boltzmann equation, i.e., Eq. (21), and keep-
+ing terms till quadratic order in deviation-from-equilibrium, we
+obtain
+∂µS µ = 1
+τR
+�
+dP (u·p)
+��
+δ f − δϵ
+ϵ0
+f0
+�
+φ +
+�
+δ ¯f − δϵ
+ϵ0
+¯f0
+�
+¯φ
+�
+. (45)
+where we have defined φ ≡ δ f/ f0 and ¯φ ≡ δ ¯f/ ¯f0.
+Using Eqs. (35) and (36) in Eq. (45), we obtain,
+∂µS µ = −β Π θ − nµ∇µα + βπµνσµν,
+(46)
+4
+
+where,
+Π = δP − χb
+β δϵ + χa
+β δn.
+(47)
+It is important to note that the right-hand-side of Eq. (46) repre-
+sents entropy production due to dissipation in the system. Here
+the shear stress tensor πµν is the tensor dissipation, the parti-
+cle diffusion four-current nµ is the vector dissipation and Π is
+the scalar dissipation, referred to as the bulk viscous pressure2.
+From Eq. (47), we observe that δP, δϵ, and δn, all contribute to
+the bulk viscous pressure. Comparing with the Navier-Stokes
+relation of bulk viscous pressure, i.e., Π = −ζ θ, we obtain the
+coefficient of bulk viscosity as,
+ζ = −τR
+�χb
+β (ϵ0 + P0) − 5βI+
+32
+3
+− χan0
+β
++ (χa + b0)
+β
+( βP0 − χbϵ0 + χan0)
+�
+.
+(48)
+Demanding that Eq. (46) does not violate the second law of
+thermodynamics, i.e., ∂µS µ ≥ 0, leads to the following con-
+straints [45],
+ζ ≥ 0,
+κ ≥ 0,
+η ≥ 0.
+(49)
+These three transport coefficients represent the three dissipative
+transport phenomena of the system related to the transport of
+momentum and charge. We see that out of the three transport
+coefficients, only ζ depends on the parameter b0 and the second
+matching condition is necessary to uniquely determine ζ. This
+is to be expected because the matching conditions are scalar
+conditions and should only affect the scalar dissipation in the
+system, i.e., bulk viscosity. In the following, we specify the
+second matching condition.
+With the parameter, b0 still not specified, the hydrodynamic
+equations obtained using the MBGK Boltzmann equation forms
+a class of hydrodynamic theories. A specific hydrodynamic the-
+ory is determined by a specific b0 parameter. We can access
+different hydrodynamic theories by varying the b0 parameter,
+which is solely controlled by the second matching condition.
+Thus, picking a specific second matching condition will fix b0
+and hence the hydrodynamic theory. To this end, we define a
+function A±
+r as [19, 46],
+A±
+r =
+�
+dP (u · p)r �
+δf ± δ ¯f
+�
+.
+(50)
+The second matching condition then amounts to assigning a
+value for a given A±
+r . For instance, the RTA matching condi-
+tions can be recovered by setting A−
+1 = A+
+2 = 0. It is apparent
+that the choice of a second matching condition is vast, and de-
+termination of the full list of the allowed ones is a non-trivial
+task that goes beyond the scope of the present work. Presently,
+for the second matching condition, we shall restrict our analysis
+to a special set A+
+r = 0. These matching conditions ensures that
+2We can further identify that,
+� ∂P
+∂ϵ
+�
+n = χb
+β and,
+� ∂P
+∂n
+�
+ϵ = − χα
+β .
+0.01
+0.10
+1
+10
+100
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+Figure 1: Dependence of the parameter b0 on z for different matching condi-
+tions. The red region corresponds to negative values of ζ. The plot is for zero
+chemical potential.
+the homogeneous part of δ f vanishes3 [47] and are also valid in
+the zero chemical potential limit. Using Eqs. (35) and (36) in
+our proposed matching condition A+
+r = 0, we obtain
+b0 = −
+�
+1/I+
+r,0
+� �
+χbI+
+r+1,0 − βI+
+r+1,1 + χa
+�
+I+
+r,0 − I−
+r,0
+��
+.
+(51)
+In the next section, we explore the effect of different b0 on the
+coefficient of bulk viscosity.
+6. Results and discussions
+In this section, we study the effect of MBGK collision kernel
+on transport coefficients. In the previous Section, we found that
+the effect of MBGK collision kernel manifests in the parameter
+b0 which affects only the scalar dissipation, namely bulk vis-
+cous pressure. On the other hand, the vector (net particle dif-
+fusion) and tensor (shear stress tensor) dissipation remain unaf-
+fected. Therefore, we study only the properties of bulk viscous
+coefficient in this section.
+Before we proceed to quantify the effect of varying the sec-
+ond matching condition on the coefficient of bulk viscosity, we
+must establish the allowed values for the parameter b0. To this
+end, we note that the second law of thermodynamics demands
+that the coefficient of bulk viscosity must be positive, Eq. (49).
+In Fig. 1, we plot b0 vs z for different values of r required to
+3It must be noted that this is not the only class of matching conditions that
+guarantee the zero value of the homogeneous part.
+5
+
+0.001
+0.010
+0.100
+1
+10
+100
+0.00
+0.01
+0.02
+0.03
+0.04
+Figure 2: Dependence of ζ/ (s0τRT) on the T/m for various α = µ/T values.
+The curves labelled RTA corresponds to r = 2 and those labelled MBGK cor-
+responds to r = 0.
+define the second matching condition in Eq. (51), at zero chem-
+ical potential. The red region in Fig. 1 corresponds to the part
+of b0-z plane where the coefficient of bulk viscosity becomes
+negative. Therefore all values of r for which the curves for b0
+lies in the red zone are not physical and must be discarded. The
+boundary of the red region corresponds to the ζ = 0 line and is
+given by
+b0 = −χa +
+������
+χb (ϵ0 + P0) − χan0 − (5/3) β2I+
+32
+χbϵ0 − χan0 − βP0
+������ .
+(52)
+We find the b0 parameter with non-negative values of r respects
+the requirement of the second law of thermodynamics Eq. (49).
+The black line with r = 2 represents the b0 for which the
+MBGK reduces to the RTA, where b0 vanishes for all z. From
+numerical analysis, we find that large negative values of r leads
+to b0 which corresponds to negative ζ. In Fig. 1, we see that the
+curve for b0, which corresponds to r = −4, passes through the
+physically forbidden region.
+Having determined the allowed range of r and equivalently,
+the allowed values of b0, we will restrict ourselves to b0 cor-
+responding to r ≥ 0 values.
+In Fig. 2 we plot the dimen-
+sionless quantity ζ/ (s0τRT) for MBGK with r = 0, and RTA
+(r = 2) against T/m for different values of chemical potential,
+where s0 ≡ (ϵ0 + P0 − µ n0)/T. We observe that ζ/ (s0τRT)
+is a non-monotonous function of temperature, having a maxi-
+mum for each r for MBGK case, similar to the behavior known
+from RTA [19, 46, 48]. We also note that the dependence of
+ζ/ (s0τRT) on α is also non-monotonous, which can be realized
+by observing that not only the position of the peak for α = 1 is
+at higher T/m values than for α = 0 and α = 2.5, but the peak
+value is also higher for α = 1 compared to α = 0 and α = 2.5.
+To better understand the effect of changing matching condi-
+tions on the behavior of the bulk viscosity for the MBGK col-
+lision kernel, we focus on the zero chemical potential limit. In
+this limit, we study the scaling behavior of the ratio of the coef-
+ficient of bulk viscosity to shear viscosity, ζ/η, with conformal-
+ity measure 1/3−c2
+s. In Fig. 3, we plot the ratio (ζ/η)/(1/3−c2
+s)2
+0.001
+0.010
+0.100
+1
+10
+100
+0
+20
+40
+60
+80
+Figure 3: Variation of the dimensionless quantity (ζ/η)/(1/3−c2
+s)2 with respect
+to z for various matching conditions determined by r.
+as a function of z for different r values. We observe that this
+ratio saturates in both small-z and large-z limits indicating a
+squared dependence of ζ/η on the conformality measure, char-
+acteristic to weakly coupled systems. We also observe that in
+the small-z limit, this ratio saturates to different values whereas
+in the large-z limit, they all converge. In order to better un-
+derstand the behavior of ζ/η in these regimes, we separately
+analyze the small-z and large-z limits.
+Small-z behaviour : The small-z limit, i.e., m/T ≪ 1, is
+the ultra-relativistic limit where the mass of the particles can
+be ignored compared to the temperature of the system. At zero
+chemical potential, the small-z limiting behavior of the confor-
+mality measure is given by
+� 1
+3 − c2
+s
+�
+= z2
+36 + O
+�
+z3�
+. On the other
+hand, the small-z behavior of the ratio ζ/η is found to be
+ζ
+η = Γ(r)
+�1
+3 − c2
+s
+�2
++ O
+�
+z5�
+,
+(53)
+for all r. We find the r-dependence of the coefficient to be,
+Γ(r) ≡ lim
+z→0
+ζ/η
+� 1
+3 − c2s
+�2 = 15(r2 + 23r + 10)
+4(r + 1)
+,
+(54)
+for r ≥ 0. Thus, while the ratio ζ/η shows a z4 dependence in
+the same small-z limit, the coefficient Γ depends on the match-
+ing condition through b0, and equivalently r, as is evident from
+Eq. (54). In Fig. 4, we show the variation of the coefficient Γ as
+a function of r. We observe that for r = 2, we recover the RTA
+value, Γ = 75, marked with a red dot in Fig. 4.
+Large-z behaviour : On the opposite end, i.e., at the large-z
+limit where m/T ≫ 1, we have the non-relativistic limit. In this
+limit, the conformality measure is expanded in powers of 1/z
+and is given by, 1
+3 − c2
+s = 1
+3 − 1
+z + O
+� 1
+z2
+�
+. The behaviour of the
+ratio ζ/η in the same limit is given by,
+ζ
+η = 2
+3 − 3
+z + O
+� 1
+z2
+�
+,
+(55)
+6
+
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+40
+50
+60
+70
+80
+Figure 4: Variation of the scaling coefficient Γ, defined in Eqs. (53) and (54),
+with respect to parameter r. The red dot represents the RTA value of Γ = 75.
+for all r. Considering the leading terms in this expansion, we
+find,
+ζ
+η = 6
+�1
+3 − c2
+s
+�2
+,
+(56)
+which is independent of r and hence the second matching con-
+dition, as is evident from Fig. 3. In this limit, the MBGK and
+RTA results coincide implying that in the non-relativistic limit,
+the properties of the fluid are independent of the nature of col-
+lision with BGK collision kernel.
+7. Summary and outlook
+In this work, we have provided the first formulation of rela-
+tivistic dissipative hydrodynamics from BGK collision kernel.
+We first propose a modified BGK collision kernel which we ad-
+vocate to be better suited for derivation of hydrodynamic equa-
+tions. We show that at finite chemical potential, where BGK
+collision kernel is defined, the formulation of relativistic hy-
+drodynamics with MBGK collision kernel becomes identical to
+that obtained using BGK collision. The advantage of MBGK
+is that it is well defined even in the limit of vanishing chemi-
+cal potential, and represents a generalization of RTA collision
+kernel. In the formulation of relativistic BGK hydrodynamics,
+we found the theory is controlled by a free parameter related
+to freedom of a matching condition, which affects the coeffi-
+cient of bulk viscous pressure. It is important to note that the
+BGK or MBGK collision kernels are affected by the matching
+conditions, which in turn affects the dissipative processes in the
+system. We have identified a class of matching conditions for
+which the homogeneous part of the solution to the relativistic
+Boltzmann equation vanishes. We examined the effect of choice
+of matching condition on dissipative coefficients and also stud-
+ied some scaling properties of the ratio of coefficients of bulk
+viscosity to shear viscosity on the conformality measure.
+The present formulation of hydrodynamics with a modified
+BGK collision kernel opens up several possibilities for future
+investigations. This MBGK collision kernel may also find po-
+tential applications in non-relativistic physics domain where
+BGK collision is widely used. The formulation of causal hydro-
+dynamics with MBGK collision kernel is an immediate possible
+extension. Formulation of higher-order hydrodynamic theories
+may be affected more significantly as the evolution equations
+of scalar, vector, and tensor dissipative quantities contain cross-
+terms giving rise to the possibility of them being controlled by
+the matching conditions. Higher-order theories also exhibit in-
+teresting features like fixed points and attractors [49, 50], which
+could also be studied within the MBGK hydrodynamics frame-
+work. The present article forms the basis for all these studies
+which we leave for future explorations.
+Acknowledgements
+The
+authors
+acknowledge
+Sunil
+Jaiswal
+for
+sev-
+eral
+useful
+discussions.
+A.J.
+was
+supported
+in
+part
+by
+the
+DST-INSPIRE
+faculty
+award
+under
+Grant
+No.
+DST/INSPIRE/04/2017/000038.
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