1. Introduction
Rational extended thermodynamics (RET) is a theory applicable to nonequilibrium phenomena out of local equilibrium. It is expressed by a hyperbolic system of field equations with local constitutive equations and is strictly related to the kinetic theory with the closure method of the hierarchies of moment equations in both classical and relativistic frameworks [
1,
2].
The first relativistic version of the modern RET was given by Liu, Müller, and Ruggeri (LMR) [
3] considering the Boltzmann–Chernikov relativistic equation [
4,
5,
6]:
in which the distribution function
f depends on
, where
are the space-time coordinates,
is the four-momentum,
,
Q is the collisional term, and
. For monatomic gases, the relativistic moment equations associated with (
1), truncated at tensorial index
are:
with
where
c denotes the light velocity,
m is the particle mass in the rest frame, and
If
, the tensor reduces to
; moreover, the production tensor in the right-side of (
2) is zero for
, because the first 5 equations represent the conservation laws of the particle number and of the energy-momentum, respectively.
When
, we have the relativistic Euler system
where, also in the following,
and
have the physical meaning, respectively, of the particle number vector and the energy-momentum tensor. Instead, when
, we have the LMR theory of a relativistic gas with 14 fields:
Recently, Pennisi and Ruggeri first constructed a relativistic RET theory for polyatomic gases with (
2) in the case of
[
7] (see also [
8,
9]) whose moments are given by
where the distribution function
depends on the extra variable
, similar to the classical one (see [
2] and references therein), that has the physical meaning of the
molecular internal energy of internal modes in order to take into account the exchange of energy due to the rotation and vibration of a molecule, and
is the state density of the internal mode.
In [
7], by taking the traceless part of the third order tensor (i.e.,
) as a field instead of
in (
5)
, the relativistic theory with 14 fields (RET
) was proposed. It was also shown that its classical limit coincides with the classical RET
based on the binary hierarchy [
2,
10,
11]. The beauty of the relativistic counterpart is that there exists a single hierarchy of moments, but, as was noticed by the authors, to obtain the classical theory of RET
, it was necessary to put the factor 2 in front of
in the last equation of (
6)! This was also more evident in the theory with any number of moments, where Pennisi and Ruggeri generalized (
6) considering the following moments [
12]:
In this case, we need a factor
in (
7) to obtain, in the classical limit, the binary hierarchy.
To avoid this unphysical situation, Pennisi first noticed that
appearing in (
7) are the first two terms of the Newton binomial formula for
. Therefore he proposed in [
13] to modify, in the relativistic case, the definition of the moments by using the substitution:
that is, instead of (
7), the following moments are proposed:
Such definitions are more physical because now the full energy (the sum of the rest frame energy and the energy of internal modes)
appears in the moments.
The aim of this paper is to consider the system (
5) with moments given by (
8). In this way, for the case with
also, by taking the trace part of
as a field, we have 15 field equations, and to close the system, we adopt the molecular procedure of RET based on the maximum entropy principle.
The paper is organized as follows. In
Section 2, the values of generic moments in an equilibrium state are estimated in the general case. In
Section 3, the RET theory for 15 fields (RET
) is proposed, and the constitutive quantities are closed near the equilibrium state. By adopting a variant of the BGK model appropriate for polyatomic gases proposed by Pennisi and Ruggeri [
14], the production tensor is derived. In
Section 4, the four-dimensional entropy flux and the entropy production are deduced within the second order with respect to the nonequilibrium variables. Then, we show the condition of convexity of the entropy density and the positivity of the entropy production, which ensure the well-posedness of the Cauchy problem and the entropy principle as a result. We also discuss in
Section 5 the case of the diatomic gases for which all coefficients are expressed in closed form in terms of the ratio of two Bessel functions, similar to the case of monatomic gases. In
Section 6, we study the ultra-relativistic limit. In
Section 7, the principal subsystems of RET
are studied. First, we obtain RET
in which all field variables have physical meaning. Then, at the same level as RET
in the sense of the principal subsystem, there also exists the subsystem with 6 fields in which the dissipation is only due to the dynamical pressure. This system is important in the case that the bulk viscosity is dominant compared to the shear viscosity and heat conductivity, and it must be particularly interesting in cosmological problems. The simplest subsystem is the Euler non-dissipative case with 5 fields. In
Section 8, we use the Maxwellian iteration and, as a result, the phenomenological coefficients of the Eckart theory, that is, the heat conductivity, shear viscosity, and bulk viscosity are determined with the present model. Finally, in
Section 9, we show that the classic limit of the present model coincides with the classical RET
studied in [
15].
2. Distribution Function and Moments at Equilibrium
The equilibrium distribution function
of polyatomic gas that generalizes the Jüttner one of monatomic gas was evaluated in [
7] with the variational procedure of the maximum entropy principle (MEP) [
1,
16,
17,
18]. Considering the first 5 balance equations of (
5) in equilibrium state:
MEP requires that the appropriate distribution function
is the one which maximizes the entropy density
under the constraints that the temporal parts
and
are prescribed. Here,
are, respectively, the Boltzmann constant, the particle number, the mass density, the four-velocity (
), the projector tensor
, the pressure, the energy, and the entropy density, and
is the metric tensor.
The equilibrium distribution function for a rarefied polyatomic gas that maximizes the entropy has the following expression [
7]:
with
T being the absolute temperature,
and
subjected to the following recurrence relations [
3,
7]:
The pressure and the energy compatible with the equilibrium distribution function (
9) are [
7]:
Taking into account that
where
is the internal energy, we deduce from (
12):
Therefore, the internal energy is a function only of
or, it is the same, of
T as in the classical case for rarefied gases.
The moments in equilibrium state
for
were deduced in [
13]:
where
are dimensionless functions depending only on
. Taking into account (
12) and (
15), we obtain
, and using the recurrence Formula (
10) and (
11), in [
13], the following recurrence relations hold:
It is interesting to see that all the scalar coefficients can be expressed in terms of the function
and of its derivatives with respect to
(or with respect to the temperature
T), and
is strictly related to the internal energy
by (
13). A similar situation is studied in the article [
15] for the non-relativistic case.
The values of
can be determined, by using the recurrence Formula (
16), according to the following diagram:
We see that all the
can be obtained from
by using Equation
, and the other
with
can be obtained from Equations
. In particular, we can evaluate the following ones that need to be known for the model with 15 fields in the subsequent sections:
5. Diatomic Gases
The system (
47) is very complex, in particular, because it is not simple to evaluate the function
, which involves two integrals (
12)
that cannot have analytical expression for a generic polyatomic gas. Taking into account the relations [
7]
where
denotes the modified Bessel function, we can rewrite
given in (
12)
in terms of the modified Bessel functions [
7]:
Moreover, to calculate the integrals, we need to prescribe the measure
. In [
7], the measure
was assumed as
because it is the one for which the macroscopic internal energy in the classical limit, when
, it converges with that of a classical polyatomic gas, where
D indicates the degree of freedom of a molecule. As was observed by Ruggeri, Xiao, and Zhao [
28] in the case of
(i.e.,
corresponding to diatomic gas), the energy
e has an explicit expression similar to monatomic gas:
Therefore, from (
12), we have
Using the following recurrence formulas of the Bessel functions
we can express
in terms of
In fact, we can obtain immediately the following expression:
which is a simple function similar to the one of monatomic gas, for which we have [
3]:
Taking into account that the derivatives of the Bessel function are known, all coefficients appearing in the differential system (
47) can be written explicitly in terms of
, by using (
59) and the recurrence Formula (
58). This is simple by using a symbolic calculus like Mathematica
®.
8. Maxwellian Iteration and Phenomenological Coefficients
In order to find the parabolic limit of a system (
47) and to obtain the corresponding Eckart equations, we adopt the Maxwellian iteration [
31] on (
47), in which only the first order terms with respect to the relaxation time are retained. The phenomenological coefficients, that is, the heat conductivity
, the shear viscosity
, and the bulk viscosity
, are identified with the relaxation time.
The method of the Maxwellian iteration is based on putting to zero the nonequilibrium variables on the left side of Equation (
47):
From the first three equations of (
83) and taking into account
(see (
12)), we can deduce
Putting (
84) in the remaining Equation (
83)
, we obtain the solution
with
and
where
are explicitly given by (
44) with the relaxation time
.
As the first three equations in (
85) are the Eckart equations, we deduce that
are the heat conductivity, the bulk viscosity, and the shear viscosity, respectively. In addition, we have a new phenomenological coefficient
, but as
doesn’t appear in either
or
(see Equation (
22) or the first three equations in (
47)), we arrive at the conclusion that the present theory converges to the Eckart one formed in the first three block equations of (
47) with constitutive Equation (
85), in which the heat conductivity, bulk viscosity, and shear viscosity are explicitly given by (
86)
.
We introduce, as in [
9], the dimensionless variables, as follows:
which are functions only of
.
8.1. Ultra-Relativistic and Classical Limit of the Phenomenological Coefficients
Taking into account Equations (
62) and (
63), it is simple to obtain the limit of (
87) when
:
In particular, in the most significant case in which
for which
, we have
Instead, in the classical limit for which
, it was proved in [
7] that the internal energy
converges to the classical internal energy of polytropic gas:
. Therefore, from (
13),
converges to
In the present case, using (
89), it is not difficult to find
deduced in (
17) in the limit
, as follows:
Therefore, in the classical limit, we have
and we find from (
87)
which are in perfect agreement with the phenomenological coefficients of the classical RET theory [
2].
8.2. Phenomenological Coefficients in RET14 and RET6
By conducting the Maxwellian iteration to RET
as a principal subsystem of RET
, we may expect that a different bulk viscosity appears. This is because
is related to
by (
64), and it affects the balance laws corresponding to
in RET
. In fact, from (
66) and (
67), we can obtain the closed field equations for
, and then, through the Maxwellian iteration, as has been done in [
9], we obtain the bulk viscosity for RET
as follows:
We remark that the heat conductivity and the shear viscosity is the same between RET
and RET
.
Similarly, from (
79)
, we obtain the bulk viscosity estimated by RET
as follows:
It should be noted that, in the classical case studied in [
15], the bulk viscosities of RET
, RET
, and RET
are the same. In fact, in the classical limit,
and
coincide with
. However, due to the mathematical structure of the relativity (i.e., the scalar fields
and
appear together in the triple tensor), the method of the principal subsystem dictates the difference of the subsystems.
8.3. Heat Conductivity, Bulk Viscosity, and Shear Viscosity in Diatomic Gases
Inserting (
60), after cumbersome calculations (easy with Mathematica
®), we can obtain the phenomenological coefficients in the diatomic case:
with
Let us compare the phenomenological coefficients with the ones for the monatomic case obtained in [
9]. In
Figure 1, we plot the dependence of the dimensionless heat conductivity and shear viscosity on
for both diatomic and monatomic cases. Concerning
, we also plot the dimensionless bulk viscosity of RET
derived in (
93) in
Figure 2. We observe that in the ultra-relativistic limit and the classical limit, the figures are in perfect agreement with the limits (
88) and (
92) (for
). We remark, as is evidently shown in
Figure 2, how small the bulk viscosity in monatomic gas is with respect to that of the diatomic case.
It is also remarkable that the value of the bulk viscosity of RET
given by (
94) is quite near to the one of RET
. For this reason, we omit the plot of
in the figure. This indicates that RET
captures the effect of the dynamic pressure in consistency with RET
.
9. Classic Limit of the Relativistic Theory
We want to perform the classical limit
of the closed relativistic system (
47) now. For this purpose, we recall the limits of the coefficients given in (
90) and (
91). Moreover, taking into account the decomposition
, where
is the Lorentz factor, we have
, whose limit is
because
has zero limit, and a similar evaluation applies to
. Then,
Concerning the projection operator in the limit, it is necessary to remember that, with our choice of the metric,
, then
While from
The last two relations also hold without taking the non-relativistic limit. As a consequence, we have that
and
.
The relativistic material derivative (
46) of a function
f converges to the classical material derivative where we continue to indicate it with a dot. Then, the system (
47) becomes in the classical limit:
where
. The system (
95) coincides perfectly with the classical one obtained recently in [
15].
We remark that, as has been studied in [
15], for classical polytropic gases, RET
is derived as a principal subsystem of RET
by setting
. Moreover, RET
is derived from RET
as a principal subsystem of RET
by setting
and
. This corresponds to the fact that, in the classical limit, both
defined in (
64) and
defined in (
69) become zero.