Molecular Extended Thermodynamics of Rarefied Polyatomic Gases with a new Hierarchy of Moments

Recently, Pennisi and Ruggeri [J Stat Phys 179, 231-246 (2020)] consider the classical limit of the relativistic theory of moments associated with the Boltzmann-Chernikov equation truncated at a tensorial index $N+1$ and they proved that there exists a unique possible choice of the moments in the classical case for a given $N$ both for monatomic and polyatomic gases. In particular, in polyatomic gases, there exists a new hierarchy of moments that is more general than the one considered in the recent literature. As consequence, when $N=2$, in the classical limit, there is a theory with $15$ fields. In this paper, we consider this system of moments, and we close the system using the maximum entropy principle. It is shown that the theory contains as a principal subsystem the previously polyatomic $14$ fields theory, and in the monatomic limit, in which the dynamical pressure vanishes, the differential system converges instead to Grad 13-moments system to the 14 moments theory proposed by Kremer [Annales de l'I.H.P. Physique th\'eorique, 45, 419-440 (1986)].


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It is well known that when the Knudsen number K n is very high, the appropriate theory of the monatomic gas is the Boltzmann equation where the state of the gas can be described by the distribution function f (x, ξ, t), being x ≡ (x i ), ξ ≡ (ξ i ), t the space coordinates, the microscopic velocity and the time, respectively, and Q denotes the collisional term. A huge literature exists on the Boltzmann equation in which very important mathematical contributions were given by Cercignani [1,2]. Associating to the distribution function, we can construct macroscopic observable quantities that are called moments (m is the atomic mass) 2 : F k 1 k 2 ...k n (x, t) = m 3 f (x, ξ, t) ξ k 1 ξ k 2 . . . ξ k n dξ, k 1 , k 2 , . . . , k n = 1, 2, 3 and n = 0, 1, 2, . . .
The surprising results was that the LMR theory converges, in the classical limit, to the monatomic ET 14 theory by 41 Kremer for monatomic gas not the Grad theory (ET 13 ) as was expected [3,18,19]. 42 For many years, the applicability range of RET was only limited to monatomic gases both in the classical and relativistic regime. For rarefied polyatomic gases, after some previous tentatives [20,21], Arima, Taniguchi, Ruggeri and Sugiyama [22] proposed a binary hierarchy of field equations with 14 fields (polyatomic ET 14 ) because now there is also, as a new field, the dynamical pressure relating to the relaxation of the molecular internal modes which is identically to zero in monatomic gases: ∂G lli ∂t + ∂G llik ∂x k = Q lli .
where F(= ρ) is the mass density, F i (= ρv i ) is the momentum density, G ll = ρv 2 + 2ρε is two times the energy density, F i j is the momentum flux, and G llk is the energy flux. As usual v i denotes the components of velocity and ε is the internal energy. F i jk and G llik are the fluxes of F i j and G lli , respectively, and P i j (P ll 0) and Q lli are the productions with respect to F i j and G lli , respectively. In the parabolic limit, this theory converges to the NSF theory, and in the monatomic singular limit, it converges to the Grad system [4,5,23]. This hierarchy was justified at kinetic level in [24][25][26] using the same form of Boltzmann equation (1) but with a distribution function f (x, ξ, t, I) that depends on a non-negative internal energy parameter I, that takes into account the influence of the internal degrees 3 When n = 0, the tensor reduces to A α . Moreover, the production tensor in the right-side of (5) is zero for n = 0, 1, because the first 5 equations represent the conservation laws of the particles number and the energy-momentum, respectively. 4 In the monatomic case, from (5), we have A αβ β = c 2 A α and therefore only 14 equations of (6) are independent.
of freedom of a molecule on energy transfer during collisions [27,28]. The theory with many moments was also developed in [24,26,29]: with the following definition of moments of polyatomic gases (ξ 2 = |ξ| 2 = ξ j ξ j ): where ϕ(I) is the state density corresponding to I, i.e., ϕ(I)dI represents the number of internal state between I 43 and I + dI. We need to remark that the two blocks of hierarchies in (8)   Recently, Pennisi and Ruggeri first constructed a relativistic version of polyatomic gas in the case of N = 2 [30]. Then, in [19], they studied the classical limit of generic moments equations (4) for a fixed N both in monatomic gas of which moments are (5) and in polyatomic gas of which moments are given by: with a distribution functions f (x α , p β , I) depends on the extra energy variable I similar to the classical one. They proved that there is a unique possible choice of classical moments for a prescribed truncation index N of (4). In particular, for N = 2, in the monatomic case, we have in the classical limit the monatomic 14-moment equations by Kremer [9] according with the old results of [18]. Instead, in the polyatomic case, for N = 2, we have, as classical limit, 15 moments in which, in addition to the previous polyatomic 14-moment equations (7), one equation for a mixed type moment H llmm defined by is involved. For many moments, the new hierarchy contains, in addition to the F s and G s hierarchies (8) with 50 n = 0, 1, . . . N and m = 0, 1, . . . N − 1, more complex N + 1 hierarchies for mixed type of moments (see [19]). For 51 more details on RET beyond the monatomic gas, see the new book of Ruggeri and Sugiyama [5].
The aim of the present paper is to study the closure of the most simple case of this new hierarchy, that is the system with 15 equations (ET 15 ): where H llmmk is the flux of H llmm given by (9), and R llmm is the production with respect to H llmm . In the following, 53 after presenting a equilibrium properties of the distribution function, we close the system (10) by means of MEP. As 54 the collisional term, we introduce the generalized BGK model for a relaxation processes of molecular internal modes 55 [25]. We show that the derived closed set of the moment equations involves the polyatomic ET 14 theory as a principal 56 subsystem, the monatomic ET 14 theory in the monatomic singular limit, and the NSF theory as its parabolic limit. First, we recall the equilibrium distribution function for polyatomic gas that was deduced first in the polytropic case (p, ε, ρ, T denote as usual the equilibrium pressure, the equilibrium specific internal energy, the mass density and the absolute temperature, while k B is the Boltzmann constant and the constant D = 3 + f i , where f i are the degree of freedom; in the monatomic gas D = 3) in [24,28] and in the present case of non polytropic gas in [25,31]: where f K E is the Maxwellian distribution function and f I E is the distribution function of the internal mode: with A(T ) is the normalization factor (partition function): and we have put with C i = ξ i − v i (C 2 = C j C j ) the peculiar velocity.
where the following relation is taking into account by (14) 2 and (15) The partition function is obtained by integrating (17) is the specific heat of the internal mode. We remark that the relation between the pressure and the translational internal energy is as follows: The specific entropy density in equilibrium is expressed by with its translational part s K and internal part s I which are given by

System of balance equations for 15 fields 64
The macroscopic quantities in (10) are defined as the moments of f as follows: and the production terms Since the intrinsic (velocity independent) variables are the moments in terms of the peculiar velocity C i instead of ξ i , the velocity dependence of the densities is obtained as follows: where a hat on a quantity indicates its velocity independent part. The conventional fields, i.e., mass density : specific internal energy density : specific internal energy density : total nonequilibrium pressure : shear stress : heat flux : are related to the intrinsic moments as follows: where the temperature of the system T is introduced through the caloric equation of state Let us decompose the intrinsic part of H llmm into the equilibrium part and the nonequilibrium part ∆ as follows: and ∆ is defined by Similarly, the velocity dependences of the fluxes and productions are obtained as follows: The velocity dependences in (22) and (27)

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The constitutive quantities are now the following momentŝ that is needed to be determined for the closure of the differential system together with the production terms P i j , Q lli 67 and R llmm . To close the system (10), we need the nonequilibrium distribution function f , which is derived from the MEP. According with the principle, the most suitable distribution function f of the truncated system (10) is the one that maximize the entropy under the constraints that the density moments F, F i , F i j , G ll , G lli , H llmm are prescribed as in (20). Therefore the best approximated distribution function f 15 is obtained as the solution of a variational problem of the following functional where λ, λ i , λ i j , µ, µ i , and ζ are the corresponding Lagrange multipliers of the constraints. As L is a scalar independent of frame proceeding as in [32], we can evaluate the right side of (28) in the rest frame of the fluid (v i = 0), and in this way we have the following velocity dependence of the Lagrange multipliers (according with the general theorem given in [32]): The distribution function f , which satisfies δL/δ f = 0, is Taking into account that, in equilibrium, f 15 coincides with the equilibrium distribution function (13), we can easily see that the equilibrium components of the Lagrange multipliers are given by multipliers of the Euler system, and those are the main field symmetrize the Euler system as was proved first by 71 Godunov (see [4,33]). 72 We observe that the highest power of peculiar velocity in χ in (30) 2 is even, i.e., C 4 . The highest power is same with the highest tensorial order of the system, and it is revealed in [19] that the highest tensorial order of the system obtained in the classical limit is always even, i.e., 2N. This fact indicates that, in principle, the moments can be integrable with the distribution function f 15 (concerning the integrability of moments see [13]). Nevertheless, for the non-linear moment closure, there is the problematic that was noticed first by Junk [34] that the domain of definition of the flux in the last moment equation is not convex, the flux has a singularity, and the equilibrium state lies on the border of the domain of definition of the flux. To avoid this difficulties in the molecular extended thermodynamics approach, we consider, as usual, the processes near equilibrium. Then, we expand (30) around an equilibrium state in the following form: where a tilde on a quantity indicates its nonequilibrium part. In the following, for simplicity, we use the notation Inserting (32) into (23) and (26), we obtain the following algebraic relation for Lagrange multipliers: where E with a quantity indicates the moment evaluated by the equilibrium distribution function, and Taking into account the moments of f I E , i.e., (18), (17) and (19), we obtain the following relation: whereF M k 1 k 2 ...k r is the equilibrium moments for monatomic gas defined bŷ Since f K E is the Maxwell distribution (14) 1 ,F M k 1 k 2 ...k r are easily obtained, e.g., From (33), the intrinsic nonequilibrium Lagrange multipliers are evaluated as functions of (ρ, T , Π, σ i j , q i , ∆) up to the first order with respect to the nonequilibrium fields, Π, σ i j , q i and ∆. Instead of ∆, it may be useful to introduce the following nonequilibrium field Then, we obtain as solution of (33):λ Inserting (31) and (36) into (29), we can write down the explicit form of the Lagrange multipliers. As is well known, the multipliers coincide with the main field by which the system (10) becomes symmetric hyperbolic. Therefore we heave the well-posed Cauchy problem (local  By using the distribution function (32) with (36), we obtain the constitutive equations for the fluxes up to the first order with respect to the nonequilibrium variables as follows: We recall that ε K and ε I given in (23) are not equilibrium variables since these are the moments of the nonequilibrium distribution function (instead, the sum of the two is an equilibrium value). Then, we can define two nonequilibrium temperatures (θ K , θ I ) such that, by inserting in the equilibrium state function instead of T , we obtain the non equilibrium internal energies (ε K , ε I ), i.e.: Recalling 2ρε K = 3P and (17) with (39) 1 , the total nonequilibrium pressure is expressed with θ K from (12) 1 as follows: Since P = p + Π, we have the following relations between the nonequilibrium temperature θ K and the dynamical pressure Π: Moreover, we have the relation among three temperatures from (16) and (25) as follows:

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In polyatomic gases, we may introduce two characteristic times corresponding to two relaxation processes we have assumed the condition: τ > τ K .

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To describe the above two separated relaxation processes, We adopt the generalized BGK collision term [40,41] (see also [25,42,43]) which treats the translational relaxation and internal relaxation separately is as follows: where the distribution functions f K is

Production terms 94
From the generalized BGK model (40), the production terms given by (21) are evaluated as follows: Since we consider linear constitutive equations, we neglect the quadratic term in the last expression of (41):

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Using the constitutive equations above, we obtain the closed system of field equations for the 15 independent fields (ρ, v i , T , Π, σ i j , q i , ∆) : 2ĉ I v + 2y I + 7 q k + 5 4y I + 7 where, from (35), In conclusion: The system (42) formed by 15 equations in the 15 unknown is closed with the provided equilibrium 96 state function (12) and relaxation times τ and τ K . 97 We remark that the field equations of ρ, v i , T , Π and σ i j are same with the ones of polyatomic 14 field theory 98 and the presence of ∆ involves only the last two equations of (42) .

Entropy density, flux and production 100
The entropy density h satisfies the entropy balance equation: where ϕ i is the non-convective entropy flux and Σ is the entropy production which are defined below.

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By adopting (32) with (36), we obtain the entropy density within second order with respect to the nonequilibrium variables This means that the entropy density is convex (in the limit of the approximation), it reaches the maximum at equilibrium and the system (42) provides the symmetric form in the main field components. Similarly, the entropy flux is obtained as follows: The entropy production Σ according with the symmetrization theorem [4,5,39] is obtained as scalar product between the main field given by (37) and the production vector given by (21). By taking into account (41) and (36), we have It is noteworthy that the entropy production is positive provided the relaxation times are together withĉ I v ≥ 0. 102 2.6. Characteristic velocities 103 The differential system (42) is particular case of a generic balance law system: and it is well known that the characteristic velocity V associated with a hyperbolic system of equations can be obtained by using the operator chain rule (see [4]): where n i denotes the i-component of the unit normal to the wave front, f is the production terms and δ is a differential 104 operator.

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Let us consider only one dimensional space-variable, and the system (42) reduces to only 7 scalar equations for the 7 unknown (ρ, v = v 1 , T , Π, σ = σ 11 , q = q 1 , ∆). After some cumbersome calculations, it is possible to prove that the system has the following 7 characteristic velocities evaluated in equilibrium: with whereĉ v = 3/2 +ĉ I v is the dimensionless specific heat. It is easy to prove that U 1st E given by (46) and U 2nd E by (47) this is the value of the characteristic velocity of monatomic ET theory with 10 moments (ET 10 ) in which (F, E and U 2nd E of ET 15 coincide with the monatomic ET 14 . In the limit that D → ∞, both of U 1st E of ET 15 and ET 14 approach to the one of monatomic ET 10 , and U 2nd E of ET 15 and ET 14 approach, respectively, to monatomic Euler and ET 4 .

Maxwellian iteration and phenomenological coefficients 121
The NSF theory is obtained by carrying out the Maxwellian iteration [45] on (42) in which only the first order terms with respect to the relaxation times are retained. Then we obtain and where Recalling the definition of the bulk viscosity ν, shear viscosity µ, and heat conductivity κ in the NFS theory: we have from (48) We note that ∆ andΠ is not present in the conservation laws of mass, momentum and energy. In particular, (49) indicates with (43)Π = 0.
This result seems similar to the case of monatomic ET 14 in which the nonequilibrium scalar field is equal to 0 in the 122 Maxwellian iteration [9].

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As usual in the BGK model, the Prandtl number predicted by the present model is not satisfactory. To avoid this 124 difficulty, one possibility is to regard the relaxation times τ, τ σ and τ q as functions of ρ and T , and estimate them by 125 using the experimental data on ν, µ and κ. On the other hand, τ ∆ and τ K are not related to such phenomenological 126 coefficients and the kinetic theory is needed for their estimation, or we may determine these relaxation times as 127 parameters to have a better agreement with some experimental data as it has been usually done for the bulk viscosity.  The monatomic gases are described in the limit ε I → 0 (y I → 0) and thereforeĉ I v → 0. In the limit, the equation for Π obtained by subtracting (42) 3 from (42) 4 becomes This is the first-order quasi-linear partial differential equation with respect to Π. As it has been studied in [23], the initial condition for (52) must be compatible with the case of monatomic gas, i.e., Π(0, x) = 0, and, assuming the uniqueness of the solution, the possible solution of Eq. (52) is given by If we insert the solution (53) into (24) and (38) with y I = 0 andĉ I v = 0, the velocity independent moments are expressed by the velocity independent moments of monatomic gasF M i 1 i 2 ···i n which are given in (34) as follows:

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The dependences of the phase velocity and the attenuation per wavelength on the frequency are studied. Let us confine our study within one-dimensional problem, that is, a plane longitudinal wave propagating in x-direction. Therefore, considering the symmetry of the wave, we have the following form: Moreover, we study a harmonic wave for the fields u = (ρ, v, T , Π, σ, q, ∆) with the angular frequency ω and the complex wave number k such that where w is a constant amplitude vector. From Eq. (10) with (54) and (55), the dispersion relation k = k(ω) is obtained by the standard way [3]. The phase velocity v ph and the attenuation factor α are calculated as the functions of the frequency ω by using the following relations: In addition, it is useful to introduce the attenuation per wavelength α λ : where λ is the wavelength.

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Let us introduce the following dimensionless parameters: Then the dispersion relation depends on these parameters with dimensionless specific heat of internal modeĉ I v .

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We emphasize that k = k(ω) does not depend on ρ, and its temperature dependence is determined through the 154 dimensionless specific heat that can be determined from statistical mechanics or experimental data.

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As an example, we adoptτ K = 0.001 which indicates the existence of the slow relaxation of internal mode [42,46,47]. We show the dependence of the phase velocity normalized by the equilibrium sound velocity c 0 : and the attenuation per wavelength on the dimensionless frequency in Fig. 2 in the case withĉ I v = 2. Around 156 Ω ∼ 10 0 (ω ∼ τ −1 ), we can observe a change of v ph and a peak of α λ . Since this is due to the relaxation of internal 157 mode relating to Π, both of the predictions by ET 14 and ET 15 coincide each other. Around Ω ∼ 10 3 (=τ −1 K ), we can 158 observe a steep change of v ph and a large peak of α λ . Since this is due to the relaxation of σ, q and ∆, the difference 159 between two theories emerges.

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According to the general results of Pennisi and Ruggeri [19], the classical limit of the relativistic theory of 162 moments provides more complex hierarchy than the F' and G' binary hierarchy. In this paper, we have studied the 163 case of N = 2 in which the classic limit dictates 15 fields for a non-polytropic polyatomic gas. We have obtained the 164 closure using the MEP. The closed field equations include the classical NSF theory as its parabolic limit and converge 165 to the monatomic ET 14 theory obtained by Kremer [9] in the monatomic singular limit. Moreover, we proved that the 166 polyatomic ET 14 theory is a principal subsystem of the present one, and according to the general results, the spectrum 167 of characteristic velocities of ET 15 includes the spectrum of eigenvalues of ET 14 . Finally, we have evaluated the 168 dispersion relation proving that, in the low-frequency region, the predictions by ET 14 and ET 15 theories coincide with 169 each other, while, in the high-frequency region, the difference between two theories emerges due to the existence of 170 the additional higher order moment. 171 We finally remark that, in the present approach, we treat the internal modes as a whole; however, in principle,