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Relativistic BGK hydrodynamics
Pracheta Singha,1, ∗ Samapan Bhadury,1, 2, † Arghya Mukherjee,1, 3, ‡ and Amaresh Jaiswal1, §
1School of Physical Sciences, National Institute of Science Education and Research,
An OCC of Homi Bhabha National Institute, Jatni 752050, Odisha, India
2Institute of Theoretical Physics, Jagiellonian University,
ul. St. Lojasiewicza 11, 30-348 Krakow, Poland
3Department of Physics and Astronomy, Brandon University, Brandon, Manitoba R7A 6A9, Canada
(Dated: April 20, 2023)
Bhatnagar-Gross-Krook (BGK) collision kernel is employed in the Boltzmann equation to formu-
late relativistic dissipative hydrodynamics. 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 po-
tential and derive relativistic first-order dissipative hydrodynamics using it. We study the effects of
this new formulation on the coefficient of bulk viscosity.
I. INTRODUCTION
Relativistic Boltzmann equation governs the space-
time evolution of the single particle phase-space distribu-
tion function of a relativistic system. Moreover, suitable
moments of the Boltzmann equation are capable of de-
scribing the collective dynamics of the system. Therefore,
it has been extensively used to derive equations of rela-
tivistic dissipative hydrodynamics and obtain expressions
for the transport coefficients [1–15]. The collision 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 suggested to simplify the collision term in the
linearized regime [16–20].
Bhatnagar-Gross-Krook [16], and independently We-
lander [17], proposed a relaxation type model for the
collision term, which is commonly known as the BGK
model. This model was further simplified by Marle [18]
and Anderson-Witting [19] to calculate the transport co-
efficients. In the non-relativistic limit, Marle’s formula-
tion leads to the same transport coefficient 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 relativistic limit. The RTA has been employed exten-
sively in several areas of physics with considerable suc-
cess and has been widely employed in the formulation of
relativistic dissipative hydrodynamics [8–11, 21–33].
The RTA Boltzmann equation has provided remark-
able insights into the causal theory of relativistic hydro-
dynamics as well as a simple yet meaningful picture of
the collision mechanism in a non-equilibrium system. On
the other hand, the BGK collision term ensures conser-
vation of net particle four-current by construction, and
∗ pracheta.singha@gmail.com
† samapan.bhadury@niser.ac.in
‡ arbp.phy@gmail.com
§ a.jaiswal@niser.ac.in
is the precursor to RTA. While the RTA has been em-
ployed extensively, a consistent formulation of relativistic
dissipative hydrodynamics with the BGK collision term
is relatively less explored. This may be attributed to
the fact that the BGK collision kernel is ill defined for
relativistic 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 [34–44].
In this article, we take the first step towards formu-
lating a consistent framework of relativistic dissipative
hydrodynamics using the BGK collision kernel. Fur-
thermore, we propose a modified BGK collisions kernel
(MBGK), which is well defined even in the absence of
conserved particle four-current and is better suited for
the formulation of the relativistic dissipative hydrody-
namics. We find that there exists a free scalar parameter
arising from the freedom of matching condition. This
affects the scalar dissipation in the system, i.e., the co-
efficient of bulk viscosity. We study the effect on bulk
viscosity in several different scenarios.
II. RELATIVISTIC DISSIPATIVE
HYDRODYNAMICS
The conserved net particle four-current, Nµ, and the
energy-momentum tensor, Tµν , of a system can be ex-
pressed in terms of the single particle phase-space distri-
bution function and the hydrodynamic variables as [45],
Nµ =
∫
dP pµ
(
f − f̄
)
= nuµ + nµ, (1)
Tµν =
∫
dP pµpν
(
f + f̄
)
= � uµuν− (P0+δP )∆µν+ πµν,
(2)
where the Lorentz invariant momentum integral mea-
sure is defined as dP = g d3p/
[
(2π)3E
]
with g being
the degeneracy factor and E =
√
|p|2 +m2 being the
on-shell energy of the constituent particle of the medium
with three-momentum p and mass m. Here f ≡ f(x, p)
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mailto:pracheta.singha@gmail.com
mailto:samapan.bhadury@niser.ac.in
mailto:arbp.phy@gmail.com
mailto:a.jaiswal@niser.ac.in
2
and f̄ ≡ f̄(x, p) are the phase-space distribution func-
tions for particles and anti-particles, respectively. In the
above equations, n is the net particle number density,
� is the energy density, P0 is the equilibrium pressure,
nµ is the particle diffusion four-current, δP is the cor-
rection to the isotropic pressure, and πµν is the shear
stress tensor. We note that the fluid four-velocity uµ has
been defined in the Landau frame, uµT
µν = �uν . We
also define ∆µν ≡ gµν − uµuν as the projection opera-
tor orthogonal to uµ. In this article, we will be work-
ing 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, ∂µN
µ = 0,
and energy-momentum tensor, ∂µT
µν = 0. Using the
expressions of Nµ and Tµν from Eqs. (1) and (2), the
hydrodynamic equations can be obtained as,
ṅ+ nθ + ∂µn
µ = 0 (3)
�̇+ (�+ P0 + δP ) θ − πµνσµν = 0 (4)
(�+ P0 + δP ) u̇
α −∇α (P0 + δP ) + ∆αν ∂µπµν = 0 (5)
where we use the standard notation, Ȧ ≡ uµ∂µA for
the co-moving derivatives, ∇α ≡ ∆αβ∂β for the space-
like derivatives, θ = ∂µu
µ for the expansion scalar, and
σµν = 12 (∇
µuν +∇νuµ)− 13∆
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 using the so called matching
conditions. We relate these non-equilibrium quantities
with their equilibrium values as
n = n0 + δn, � = �0 + δ�, (6)
where n0 and �0 are the equilibrium net-number density
and the energy density, respectively, and, δn, δ� are the
corresponding non-equilibrium corrections. For a system
which is out-of-equilibrium, the distribution function can
be written as f = f0 + δf , where f0 is the equilibrium
distribution function and δf is the non-equilibrium cor-
rection. In the present work, we consider the equilib-
rium 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 equilib-
rium distribution for anti-particles is also taken to be of
the Maxwell-Juttner form with α→ −α.
We can now express the equilibrium hydrodynamic
quantities in terms of the equilibrium distribution func-
tion as,
n0 =
∫
dP (u · p)
(
f0 − f̄0
)
(7)
�0 =
∫
dP (u · p)2
(
f0 + f̄0
)
(8)
P0 = −
1
3
∆µν
∫
dP pµpν
(
f0 + f̄0
)
. (9)
Similarly, the non-equilibrium quantities can be ex-
pressed as
δn =
∫
dP (u · p)
(
δf − δf̄
)
(10)
δ� =
∫
dP (u · p)2
(
δf + δf̄
)
(11)
δP = −1
3
∆αβ
∫
dP pαpβ
(
δf + δf̄
)
, (12)
nµ = ∆µα
∫
dP pα
(
δf − δf̄
)
, (13)
πµν = ∆µναβ
∫
dP pαpβ
(
δf + δf̄
)
, (14)
where ∆µναβ ≡
1
2 (∆
µ
α∆
ν
β + ∆
µ
β∆
ν
α)−13∆
µν∆αβ is a trace-
less 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 δf̄ . To this end,
we consider the Boltzmann equation with BGK collision
kernel.
III. 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∂µf = C[f, f̄ ], p
µ∂µf̄ = C̄[f, f̄ ], (15)
for a single species of particles and its antiparticles. In
the above equation, C[f, f̄ ] and C̄[f, f̄ ] are the collision
kernels that contain the microscopic information of the
scattering processes. For the formulation of relativistic
hydrodynamics from the kinetic theory of unpolarized
particles, the collision kernel of the Boltzmann equa-
tion must satisfy certain properties. Firstly, the colli-
sion kernel must vanish for a system in equilibrium, i.e.,
C[f0, f̄0] = C̄[f0, f̄0] = 0. Further, in order to satisfy
the fundamental conservation equations in the micro-
scopic interactions, the zeroth and the first moments of
the collision kernel must vanish, i.e.,
∫
dPC = 0 and∫
dP pµ C = 0. Vanishing of the zeroth moment and
the first moment of the collision kernel follows from the
net particle four-current conservation and the energy-
momentum conservation, respectively.
In the present work, we consider the BGK collision ker-
nel which has the advantage that the particle four-current
3
is conserved by construction. The relativistic Boltzmann
equation with BGK collision kernel for particles can be
written as [40–44],
pµ∂µf = −
(u · p)
τR
(
f − n
n0
f0
)
, (16)
and similarly for anti-particles with f → f̄ and f0 → f̄0.
Here, τR is a relaxation time like parameter
1 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 first moment of the
above equations should lead to the conservation of the
energy-momentum tensor, defined in Eq. (2). However,
we find that the first moment of the Boltzmann equation,
Eq. (16), leads to,
∂µT
µν = − 1
τR
(
�− n
n0
�0
)
uν , (17)
which does not vanish automatically.
In order to have energy-momentum conservation ful-
filled by the Boltzmann equation with the BGK collision
kernel, Eq. (16), we require that
�n0 = �0n, (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∂µf =
− (u·p)τR (f − f0), is recovered from Eq. (16) if the second
matching condition is fixed as either � = �0 or equiva-
lently n = n0. For the RTA collision term, both match-
ing conditions � = �0 and n = n0, are necessary for
net particle four-current and energy-momentum conser-
vation. However, for the BGK collision kernel, both con-
servation equations are satisfied with only one matching
condition, Eq. (18), leaving the other condition free. We
shall see later that this scalar freedom affects the coeffi-
cient of bulk viscosity, which is the transport coefficient
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 poten-
tial. 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 collision kernel in order to extend
its regime of applicability. At this juncture, we are well
equipped to propose a modification to BGK collision ker-
nel that is well-defined for all values of chemical poten-
tial. To this end, we rewrite the condition necessary for
1 A more conventional notation is the collision frequency which is
defined as ν = 1/τR.
energy-momentum conservation from BGK collision ker-
nel, Eq. (18), in the form
n
n0
=
�
�0
. (19)
Substituting the above equation in Eq. (16), we obtain
Boltzmann equation for particles with a modified BGK
(MBGK) collision kernel,
pµ∂µf = −
(u · p)
τR
(
f − �
�0
f0
)
, (20)
and similarly for anti-particles with f → f̄ and f0 → f̄0.
The advantage of the above modification is that the colli-
sion kernel conserves energy-momentum by construction
and is applicable to systems even without any conserved
four-current, i.e., in the limit of vanishing chemical po-
tential. Additionally, MBGK is uniquely defined even for
a system with multiple conserved charges, as opposed to
the BGK collision kernel. In the case of finite chemi-
cal 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.
IV. NON-EQUILIBRIUM CORRECTION TO
THE DISTRIBUTION FUNCTION
In order to obtain the non-equilibrium correction to
the distribution function, we use Eq. (6) to rewrite the
MBGK Boltzmann equation, Eq. (20), as
pµ∂µf = −
(u · p)
τR
(
δf − δ�
�0
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 distribution function up to first-order in deriva-
tive, which we represent by δf1. However, obtaining the
expressions for δf1 from Eq. (21) is not straightforward
because it contains δ� which is defined in Eq. (11) as an
integral over δf . Therefore, to solve for δf1, 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∂µf0 = (AΠθ +Anp
µ∇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)
(�0 + P0)
, Aπ = −β. (24)
4
The coefficients χa and χb appearing in Eq. (23) are de-
fined via the relations
α̇ =χa θ, β̇ =χb θ, ∇µβ =
n0
�0+P0
∇µα− βu̇µ, (25)
χa =
I−20(�0+P0)− I
+
30n0
I+30I
+
10 − I
−
20I
−
20
, χb =
I+10(�0+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
(
∆αβp
αpβ
)q (
f0 ± f̄0
)
.
(27)
With the above definition, we identify n0 = I
−
10, �0 = I
+
20
and P0 = I
+
21.
We assume δf1 to have the same form as in Eq. (22),
δf1 = τR (BΠθ +Bnp
µ∇µα+Bπpµpνσµν) f0, (28)
and similarly for anti-particles. In the above expression,
the coefficients 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 δf1 in Eq. (11) to
obtain
δ� = τR
∫
dP (u · p)2
(
BΠf0 + B̄Πf̄0
)
θ. (29)
Using Eqs. (22), (28) and (29) into Eq. (21) and compar-
ing both sides, we get
− AΠ
(u · p)
= BΠ −
1
�0
∫
dP (u · p)2
(
BΠf0 + B̄Πf̄0
)
,
(30)
Bn = −
An
(u · p)
, Bπ = −
Aπ
(u · p)
. (31)
Another set of equations in terms of ĀΠ, Ān and Āπ can
be obtained by considering MBGK equation, analogous
to Eq. (21), for anti-particles. Here, coefficients Bn, B̄n,
Bπ and B̄π are easily determined but BΠ and B̄Π require
further investigation.
To obtain their expressions, we consider BΠ to be of
the general form, BΠ =
∑+∞
k=−∞ bk (u · p)
k
and B̄Π =∑+∞
k=−∞ b̄k (u · p)
k
. Substituting these in Eq. (30) and
its corresponding equation for anti-particles, we can con-
clude that the only non-zero bk and b̄k are the ones with
k = −1, 0, 1. We obtain
BΠ =
1∑
k=−1
bk (u · p)k , B̄Π =
1∑
k=−1
b̄k (u · p)k . (32)
Substituting Eqs. (23) and (32) in Eq. (30), we find
b1 = b̄1 =χb −
β
3
and, b−1 = b̄−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
b̄0, we find two coupled equations which are identical and
leads to the relation
b̄0 = b0 + 2χa. (34)
Hence, we see that a unique solution for b0 and b̄0 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 b̄0 separately.
Nevertheless, at this stage, we can determine δf1 and
δf̄1 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
(�0 + P0)
}
pµ (∇µα) +
βpµpµσµν
(u · p)
]
,
(35)
δf̄1 = τR f̄0
[{
m2β
3 (u·p)
+ b0 + 2χa + (u·p)
(
χb −
β
3
)}
θ
+
{
1
(u · p)
+
n0
(�0 + 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)
(
c2s−
1
3
)}
θ
+
pµpµσµν
(u·p)
]
, (37)
where we have used χb = βc
2
s, with c
2
s being the squared
of the speed of sound, given by,
c2s =
(�0 + P0)
3�0 + (3 + z2)P0
. (38)
Here z ≡ m/T is the ratio of particle mass to tempera-
ture.
V. FIRST ORDER DISSIPATIVE
HYDRODYNAMICS
The first-order correction to the phase-space distribu-
tion 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 hydrodynamic quantities as
δn = νθ, δ� = eθ, δP = ρθ,
nµ = κ∇µα, πµν = 2ησµν , (39)
5
where,
ν = τR (χa + b0)n0, e = τR (χa + b0) �0, (40)
ρ = τR
[
(χa+b0)P0 + χb
(�0+P0)
β
− 5
3
βI+32 −
χan0
β
]
,
(41)
κ = τR
[
I+11 −
n20
β(�0 + P0)
]
, 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 corresponds 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 analyze entropy produc-
tion 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
Sµ = −
∫
dPpµ
[
f (ln f − 1) + f̄
(
ln f̄ − 1
) ]
. (43)
The entropy production is determined by taking four-
divergence of the above equation,
∂µS
µ = −
∫
dPpµ
[
(∂µf) ln f +
(
∂µf̄
)
ln f̄
]
. (44)
Using the MBGK Boltzmann equation, i.e., Eq. (21),
and keeping terms till quadratic order in deviation-from-
equilibrium, we obtain
∂µS
µ =
1
τR
∫
dP (u·p)
[(
δf − δ�
�0
f0
)
φ+
(
δf̄ − δ�
�0
f̄0
)
φ̄
]
.
(45)
where we have defined φ ≡ δf/f0 and φ̄ ≡ δf̄/f̄0.
Using Eqs. (35) and (36) in Eq. (45), we obtain,
∂µS
µ = −βΠ θ − nµ∇µα+ βπµνσµν , (46)
where,
Π = δP − χb
β
δ�+
χa
β
δn. (47)
It is important to note that the right-hand-side of
Eq. (46) represents entropy production due to dissipa-
tion in the system. Here the shear stress tensor πµν is the
tensor dissipation, the particle diffusion four-current nµ
is the vector dissipation and Π is the scalar dissipation,
referred to as the bulk viscous pressure2. From Eq. (47),
2 We can further identify that,
(
∂P
∂�
)
n
= χb
β
and,
(
∂P
∂n
)
�
= −χα
β
.
we observe that δP , δ�, and δn, all contribute to the bulk
viscous pressure. Comparing with the Navier-Stokes re-
lation of bulk viscous pressure, i.e., Π = −ζ θ, we obtain
the coefficient of bulk viscosity as,
ζ = −τR
[
χb
β
(�0 + P0)−
5βI+32
3
− χan0
β
+
(χa + b0)
β
(βP0 − χb�0 + χan0)
]
. (48)
Demanding that Eq. (46) does not violate the second law
of thermodynamics, i.e., ∂µS
µ ≥ 0, leads to the following
constraints [46],
ζ ≥ 0, κ ≥ 0, η ≥ 0. (49)
These three transport coefficients represent the three dis-
sipative 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 ex-
pected because the matching conditions are scalar condi-
tions 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 hydrody-
namic equations obtained using the MBGK Boltzmann
equation forms a class of hydrodynamic theories. A spe-
cific hydrodynamic theory is determined by a specific b0
parameter. We can access different hydrodynamic the-
ories by varying the b0 parameter, which is solely con-
trolled 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 func-
tion A±r as [20, 47],
A±r =
∫
dP (u · p)r
(
δf ± δf̄
)
. (50)
The second matching condition then amounts to assign-
ing a value for a given A±r . For instance, the RTA match-
ing conditions can be recovered by setting A−1 = A
+
2 = 0.
It is apparent that the choice of a second matching con-
dition is vast, and determination 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 the homogeneous part of δf vanishes3 [48] 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)
3 It must be noted that this is not the only class of matching con-
ditions that guarantee the zero value of the homogeneous part.
6
0.01 0.10 1 10 100
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
FIG. 1. Dependence of the parameter b0 on z for different
matching conditions. The red region corresponds to negative
values of ζ. The plot is for zero chemical potential.
In the next section, we explore the effect of different b0
on the coefficient of bulk viscosity.
VI. 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 man-
ifests in the parameter b0 which affects only the scalar
dissipation, namely bulk viscous pressure. On the other
hand, the vector (net particle diffusion) and tensor (shear
stress tensor) dissipation remain unaffected. Therefore,
we study only the properties of bulk viscous coefficient
in this section.
Before we proceed to quantify the effect of varying the
second matching condition on the coefficient of bulk vis-
cosity, we must establish the allowed values for the pa-
rameter 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 define the second
matching condition in Eq. (51), at zero chemical poten-
tial. 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
b0 = −χa +
[
χb (�0 + P0)− χan0 − (5/3)β2I+32
χb�0 − χan0 − βP0
]
. (52)
0.001 0.010 0.100 1 10 100
0.00
0.01
0.02
0.03
0.04
FIG. 2. Dependence of ζ/ (s0τRT ) on the T/m for various
α = µ/T values. The curves labelled RTA corresponds to
r = 2 and thoselabelled MBGK corresponds to r = 0.
We find the b0 parameter with non-negative values of r
respects the requirement of the second law of thermody-
namics 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 corre-
sponds 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 equiva-
lently, the allowed values of b0, we will restrict ourselves
to b0 corresponding to r ≥ 0 values. In Fig. 2 we plot
the dimensionless quantity ζ/ (s0τRT ) for MBGK with
r = 0, and RTA (r = 2) against T/m for different values
of chemical potential, where s0 ≡ (�0 +P0−µn0)/T . We
observe that ζ/ (s0τRT ) is a non-monotonous function of
temperature, having a maximum for each r for MBGK
case, similar to the behavior known from RTA [20, 47, 49].
We also note that the dependence of ζ/ (s0τRT ) on α is
also non-monotonous, which can be realized by observ-
ing 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
conditions on the behavior of the bulk viscosity for the
MBGK collision kernel, we focus on the zero chemical
potential limit. In this limit, we study the scaling be-
havior of the ratio of the coefficient of bulk viscosity to
shear viscosity, ζ/η, with conformality measure 1/3− c2s.
In Fig. 3, we plot the ratio (ζ/η)/(1/3 − c2s)2 as a func-
tion 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 mea-
sure, characteristic 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 con-
verge. In order to better understand the behavior of ζ/η
7
0.001 0.010 0.100 1 10 100
0
20
40
60
80
0.0 0.5 1.0 1.5 2.0 2.5 3.0
40
50
60
70
80
FIG. 3. Variation of the dimensionless quantity (ζ/η)/(1/3−
c2s)
2 with respect to z for various matching conditions de-
termined by r. Inset: 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.
in these regimes, we separately analyze the small-z and
large-z limits.
A. 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 behav-
ior of the conformality measure is given by
(
1
3 − c
2
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
− c2s
)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 depen-
dence in the same small-z limit, the coefficient Γ depends
on the matching condition through b0, and equivalently
r, as is evident from Eq. (54). In the inset of Fig. 3, 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.
B. 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
0.4 2 4 6 8 10
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
0.4 2 4 6 8 10
0.2
0.3
0.4
0.5
 
FIG. 4. Proper time evolution of bulk viscous pressure, scaled
by equilibrium pressure. Inset: Proper time evolution of tem-
perature. Boost-invariant Bjorken expansion is considered to
generate curves for different r values.
of 1/z and is given by, 13 − c
2
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)
for all r. Considering only the leading terms in this ex-
pansion, we find (ζ/η)/(1/3−c2s)2 = 6, as is evident from
Fig. 3. Considering terms up to 1/z in the expansion, we
get,
ζ
η
= 2
√
3
(
1
3
− c2s
)3/2
, (56)
which is independent of r and hence the second match-
ing condition. The above equation is the scaling rela-
tion we obtain in the non-relativistic limit. In this limit,
the MBGK and RTA results coincide implying that the
properties of the fluid are independent of the nature of
collision with BGK collision kernel.
C. Bjorken expansion
In order to study the effect of present hydrody-
namic formulation on evolution of rapidly expanding
medium, we consider the case of transversely homoge-
neous and purely longitudinal boost-invariant expansion,
vz = z/t [50]. It is convenient to work in the Milne co-
ordinate system, (τ, x, y, ηs), where τ =
√
t2 − z2 is the
longitudinal proper time and ηs = tanh
−1 (z/t) is the
space time rapidity and the metric tensor is given by
gµν =
(
1,−1,−1,−τ2
)
. In this case, the fluid four ve-
locity becomes uµ = (1, 0, 0, 0) and all functions of space
and time depend only on τ . For zero chemical potential,
within MBGK framework, Eq. (4) can be written as,
�̇0 + δ�̇+ (�0 + P0) θ + (δ�+ δP ) θ − πµνσµν = 0, (57)
8
where, θ = 1/τ , �̇ = d�/dτ and πµνσµν = 4η/3τ
2 where
η is given in Eq. (42). The free parameter b0 enters in
the evolution equation through the scalar deviations δ�
and δP given in Eqs. (39)-(41). Moreover, δ�̇ is given by,
δ�̇ =
τR
τ
[(
�0ḃ0 + b0�̇0
)
− b0�0
τ
]
, (58)
where,
ḃ0 =β̇
[
b0
(
1
β
+
Ir+1,0
Ir,0
)
− β
{
Ir+1,0
Ir,0
(
c2sI4,0 − I4,1
I3,0
)
−
(
c2sIr+2,0 − Ir+2,1
Ir,0
)}]
, (59)
is obtained from our choice of b0 in Eq. (51).
The energy evolution equation then takes the form,
�̇0 +
τR
τ
(
�0ḃ0 + b0�̇0
)
+
(�0 + P0)
τ
[
1 +
(
b0 + c
2
s
) τR
τ
]
− τR
τ2
(b0�0 + 3βI32) = 0 . (60)
Implementing the ḃ0 obtained in Eq. (59), we next solve
Eq. (60) for the proper-time evolution of temperature.
The corresponding proper-time evolution of the bulk
viscous pressure is then obtained from Eqs. (47) and
(48). The evolution equations are solved numerically
with a set of initial conditions corresponding to rel-
ativistic heavy-ion collisions, namely, the initial tem-
perature is considered to be T0 = 0.5 GeV at initial
proper-time τ0 = 0.5 fm, the relaxation time is taken
as τR = 0.5 fm and the mass of the medium constituents
is assumed to be temperature independent with a fixed
value m = 0.3 GeV. With these given initial conditions,
the proper time evolution of the bulk viscous pressure
and temperature are obtained considering a fixed set of
r values as shown in Fig. 4. From the inset of Fig. 4,
it can be observed that the temperature evolution is not
sensitive to the choices of r. On the other hand, the bulk
viscous pressure, scaled by the equilibrium pressure, de-
pends significantly on the choices of r.
VII. SUMMARY AND OUTLOOK
In this work, we have provided the first formulation
of relativistic dissipative hydrodynamics from BGK col-
lision kernel, which represents a generalization of RTA
collision kernel. We found that relativistic BGK hydro-
dynamics is controlled by a free parameter related to the
freedom of a matching condition, which modifies the co-
efficient of bulk viscous pressure. On the other hand,
the BGK kernel is ill defined for vanishing chemical po-
tential as well as for a system with multiple conserved
charges. We thus proposed a modified BGK collision ker-
nel, which is free from such issues, and advocate it to be
better suited for derivation of hydrodynamic equations.
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.
Moreover, at finitechemical potential, two descriptions
become identical. We identified a class of matching con-
ditions for which the homogeneous part of the solution
to the relativistic Boltzmann equation vanishes, and RTA
turns out to be a special case of that. We examined the
effect of choice of matching condition on dissipative coef-
ficients and also studied scaling properties of the ratio of
coefficients of bulk viscosity to shear viscosity on the con-
formality measure. The importance of the bulk viscosity
in the hydrodynamic evolution of quark gluon plasma
has been emphasized in Refs. [51–53]. Our framework
provides a direct control over this first-order transport
coefficient through a choice of matching condition via b0.
The present formulation of hydrodynamics with a
modified BGK collision kernel opens up several possibil-
ities for future investigations. This MBGK collision ker-
nel may also find potential applications in non-relativistic
physics domain where BGK collision is widely used. The
formulation of causal hydrodynamics with MBGK col-
lision kernel is an immediate possible extension. For-
mulation 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 the-
ories also exhibit interesting features like fixed points and
attractors [54, 55], which could also be studied within the
MBGK hydrodynamics framework. The present article
forms the basis for all these studies which we leave for
future explorations.
ACKNOWLEDGEMENTS
The authors acknowledge Sunil Jaiswal for several
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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	Relativistic BGK hydrodynamics
	Abstract
	I Introduction
	II Relativistic dissipative hydrodynamics
	III The Boltzmann equation and conservation laws
	IV Non-equilibrium correction to the distribution function
	V First order dissipative hydrodynamics
	VI Results and discussions
	A Small-z behaviour
	B Large-z behaviour
	C Bjorken expansion
	VII Summary and outlook
	 Acknowledgements
	 References