Monatomic gas as a singular limit of polyatomic gas in molecular extended thermodynamics with many moments

Annals of Physics 372 (2016) 83–109

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Monatomic gas as a singular limit of polyatomic gas in molecular extended thermodynamics with many moments Takashi Arima a , Tommaso Ruggeri b,∗ , Masaru Sugiyama c , Shigeru Taniguchi d a

Department of Mechanical Engineering, Faculty of Engineering, Kanagawa University, Japan

b

Department of Mathematics and Alma Mater Research Center on Applied Mathematics AM2 , University of Bologna, Bologna, Italy c

Graduate School of Engineering, Nagoya Institute of Technology, Japan

d

Department of Creative Engineering, National Institute of Technology, Kitakyushu College, Japan

highlights • • • • •

Molecular extended thermodynamics of monatomic gas and of polyatomic gas are compared. Monatomic gas as a singular limit of polyatomic gas is studied. In the singular limit it is proved that the characteristic variables of polyatomic gas vanish. In one-dimensional case the constitutive equations of typical systems are derived. Asymptotic behaviors of linear wave and of shock wave in the singular limit are studied.

article

abstract

info

Article history: Received 21 January 2016 Accepted 19 April 2016 Available online 26 April 2016 Keywords: Extended thermodynamics Rarefied polyatomic gases Rarefied monatomic gases Singular limit Kinetic theory Moment equations

The difference in the theoretical structure between monatomic and polyatomic gases in highly nonequilibrium states is discussed from the viewpoint of molecular extended thermodynamics (MET) of rarefied gases, which is free from the local equilibrium assumption. The MET theories of these two types of gases are based on the moment balance equations with different hierarchy structures due to whether the internal degrees of freedom of a molecule are incorporated in their distribution functions or not. In particular, the number of balance equations in the MET theory of polyatomic gases is greater than the number in the corresponding theory of monatomic gases. The closure procedure for the system of balance

∗

Corresponding author. E-mail addresses: [email protected] (T. Arima), [email protected] (T. Ruggeri), [email protected] (M. Sugiyama), [email protected] (S. Taniguchi). http://dx.doi.org/10.1016/j.aop.2016.04.015 0003-4916/© 2016 Elsevier Inc. All rights reserved.

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equations of polyatomic gases obtained in a recent paper (Arima et al., 2014) is adopted. We prove that the solutions for polyatomic gases converge, in the limit where the degrees of freedom of a molecule D tend to 3, to the ones for monatomic gases provided that we impose appropriate initial conditions compatible with monatomic gases. Thus a MET theory of rarefied monatomic gases can be identified as a singular limit of the corresponding MET theory of rarefied polyatomic gases. As illustrative examples, the asymptotic behaviors when D → 3 in the dispersion relation of ultrasonic waves and in the shock wave structure are shown. © 2016 Elsevier Inc. All rights reserved.

1. Introduction In a thermodynamic theory of rarefied gases composed of molecules, it is sometimes convenient to classify them into two groups, monatomic and polyatomic gases. A polyatomic molecule has both the translational degrees of freedom and the internal – rotational and vibrational – degrees of freedom, while a monatomic molecule has only the translational degrees of freedom. In an appropriate temperature range, we can assume that the translational degrees of freedom are fully excited and satisfy the equipartition law of energy, and also assume that some of the internal degrees of freedom, say rotational and vibrational degrees of freedom, are excited but the others are not. Total excited degrees of freedom will be referred to as effective degrees of freedom. We will confine our study within such a temperature range throughout the present paper. If the internal modes are totally frozen, a polyatomic gas behaves monatomic like. There are characteristic quantities expressing the difference between monatomic and polyatomic gases. In the case of perfect fluids described by the Euler system of equations, the difference shows up only in the caloric equation of state (see (7)) through the effective degrees of freedom of a molecule D. As mentioned above, D = 3 for monatomic gases and D > 3 for polyatomic gases. For the description of viscous heat-conducting fluids, the Navier–Stokes–Fourier (NSF) theory [1,2] has frequently been adopted under the assumption of local equilibrium. In this case the caloric equation of state alone is not sufficient to distinguish monatomic and polyatomic gases, because the dynamic pressure (nonequilibrium pressure) Π associated with the bulk viscosity ν exists in polyatomic gases. Therefore polyatomic gases are characterized by D > 3, ν ̸= 0 and Π ̸= 0, while the behavior of monatomic gases is described in the limit to D = 3, ν = 0 and Π = 0 (the Stokes assumption). Rational extended thermodynamics (RET, or simply ET) was proposed and developed to obtain hyperbolic systems, which, in the limit of small relaxation times, reduce to the parabolic system of NSF both in monatomic gases (13-field theory) [3] and polyatomic gases (14-field theory) [4]. The spirit of the RET theory is to close phenomenologically the system of balance equations with local-type constitutive equations by imposing the universal physical principles; entropy, causality, and objectivity principles. In principle this phenomenological approach can be done also when the number of field is large but there are technical mathematical difficulties. To overcome these difficulties an alternative closure procedure was proposed. In this procedure, we assume that the independent variables are expressed in terms of the moments of a distribution function and according with the proposal of Müller and Ruggeri [5], we call the theory molecular extended thermodynamics (MET). In MET we can adopt the maximum entropy principle (MEP), which says that the most suitable distribution function maximizes the entropy density under some constraints. MEP was developed in [6,5] for monatomic gases and in [7,8] for polyatomic gases. In the framework of MET, the equivalence between the closure procedures based on the entropy principle and on the MEP was proved by Boillat and Ruggeri [9] in the case of monatomic gases and recently by Arima, Mentrelli and Ruggeri [8,10] for polyatomic gases (see also [4]).

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The balance equations of the ET theory of monatomic gases have a single tensorial hierarchy structure, starting from the conservation laws of mass, momentum and energy, in which the flux in one equation becomes the density in the next equation [5]. A representative theory is the phenomenological theory with 13 independent fields: mass density, velocity, temperature, shear stress and heat flux [3,11]. This theory is fully consistent with the closed system of moment equations obtained by the Grad method in the kinetic theory of gases [12]. Since the ET theory with the single hierarchy structure is motivated by the moment equations based on the Boltzmann equation of monatomic gases, its applicability range is strictly limited only to rarefied monatomic gases. This limitation is caused, in particular, by the absence of the dynamic pressure, i.e., the nonequilibrium part of the trace part of the stress. If it existed, it would disconnect the equivalence between the energy density and the trace part of the momentum flux. Therefore the single hierarchy structure is no longer valid for dense gases nor for rarefied polyatomic gases. Recently, the ET theory for rarefied polyatomic gases and also for moderately dense gases with 14 independent fields was proposed by taking the dynamic pressure into account [4,13–16]. In this theory, the balance equations have a binary hierarchy structure; one hierarchy consists of the balance equations for mass density, momentum density and momentum flux (momentum-like hierarchy), the other one consists of the balance equations for energy density and energy flux (energy-like hierarchy). This theory has shown its superiority compared to the NSF theory in the studies of sound waves [17,18], light scattering [19], shock waves [20,21], and heat conduction [22,23]. As far as the kinetic counterpart is concerned, a crucial step towards the development of a theory of rarefied polyatomic gases is the work by Borgnakke and Larsen [24] in which the distribution function is assumed to depend on an additional continuous variable representing the internal energy of a molecule, thus allowing to take into account the exchange of energy (other than translational) in binary collisions. This model was initially used for Monte Carlo simulations of polyatomic gases, and later it has been applied to the derivation of appropriate Boltzmann equation by Bourgat, Desvillettes, Le Tallec and Perthame [25]. Using this distribution function it was shown by Pavić, Ruggeri and Simić [7], that the 14-field theory is in full agreement with the kinetic approach using the method of MEP. This approach with the binary hierarchy and a distribution function that depends on an extra variable was generalized recently to the case with any number of moments by Arima, Mentrelli and Ruggeri [8]. They showed that it is possible to close the system by using the procedure of MEP and to prove the equivalence of MET with the entropy principle. This is valid not only for classical gases but also for degenerate gases [10]. The theories with more independent fields are expected to describe more details of nonequilibrium phenomena. Now we have grasped an important point that, from the viewpoint of ET, the theoretical difference between polyatomic and monatomic gases resides in the difference in the hierarchy structure of balance equations in addition to the caloric equation of state. Therefore, one of the essential problems that have not yet been solved completely is to clarify the relationship between the MET theories of monatomic and polyatomic gases when D tends to 3, where D is assumed to be a continuous variable. It was proved by the present authors [26] that, in the limit D → 3, the system of 14-field equations for polyatomic gases has the same solutions as the system of 13-field equations for monatomic gases with null dynamic pressure Π in the following sense: The limit should be regarded as a singular limit with an appropriate initial condition compatible with a property of monatomic gases, i.e., in this case, Π = 0 at an initial time. In this sense, we can say that the binary hierarchy of polyatomic gases collapses into the single hierarchy of monatomic gases in the limit D → 3. What happens if the MET theory has larger number of independent fields than 14? In this case, there appear many new nonequilibrium variables and equations which characterize polyatomic gases in addition to the dynamic pressure. The aim of this paper is to obtain the relationship between the MET theories of monatomic and polyatomic gases when D tends to 3 in such general cases. In particular, we will study the singular limit of the ET theory with many independent fields, where such nonequilibrium variables vanish. We will prove that the solutions of the system of polyatomic gases coincide with those of monatomic gases in this singular limit for appropriate initial data. In this way, we will have a unified and general molecular extended thermodynamics theory that is valid for both monatomic and polyatomic gases.

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Concerning the studies of polyatomic gases taking into account the importance of the internal modes of a molecule, there exist some other interesting works. For example, see [27–32]. These works are typically based on the kinetic theory, and their continuum limit is usually related to the classical NSF-type theory. While our theory is developed based on ET with the hyperbolic system of field equations, which, in the parabolic limit, includes the NSF theory. The paper is organized as follows: In Sections 2 and 3, the MET closures for rarefied monatomic and polyatomic gases are briefly reviewed. In Section 4, we discuss the singular limit D → 3 and we prove, at least in the BGK approximation for the collision term, that a solution of the polyatomic binary hierarchy converges to the corresponding one of the monatomic single hierarchy provided that the initial conditions are compatible with monatomic gases. In Section 5, the closure procedure and their convergence to monatomic gas are shown in the case of one space dimension. In Section 6, we show explicitly the results for particular number of moments and in Section 7 the asymptotic behaviors of the solutions in the dispersion relation of linear sound waves and in the shock wave structure are presented. Finally in Section 8 the conclusions are written down. 2. Molecular extended thermodynamics of rarefied gases Before discussing the singular limit of the MET theory of polyatomic gases, a brief survey on the MET theories of rarefied monatomic and polyatomic gases is presented. 2.1. MET of monatomic gases The kinetic theory of monatomic gases adopts the assumption that a state of a gas can be described by the one-body distribution function f (x, c, t ), where x and c are, respectively, the position and the velocity of a molecule. f (x, c, t ) dx dc gives the number density of molecules at time t and in the volume element dx dc of the phase space centered at (x, c) ∈ R3 × R3 . The rate of change of the distribution function in the absence of external force is determined by the Boltzmann equation:

∂t f + ci ∂i f = Q (f ),

(1)

where the symbols ∂t and ∂i denote partial derivatives with respect to time t and space variables xi (i = 1, 2, 3), respectively,1 and the collision term Q (f ) represents the rate of change of the distribution function f due to molecular collisions. Our aim is to describe the evolution of macroscopic observable quantities under special conditions instead of finding the exact solution of (1). Therefore we define macroscopic quantities by taking moments of the distribution function f :

 F = R3

m f dc,

 Fi1 ···ik =

R3

m ci1 · · · cik f dc,

ik ∈ {1, 2, 3}, ∀k ∈ N+

where m is the mass of a molecule, and we determine appropriate evolution equations for these moments. It is easily seen from (1) that the moments of the distribution function constitute a hierarchy with infinite number of balance equations in which the flux in one equation becomes the density in the next one. 2.1.1. Truncated balance equations based on the Boltzmann equation We assume that nonequilibrium properties of the system can be described by the first N tensorial equations and then the hierarchy of infinite balance equations is truncated, where N is an arbitrary truncation order. The system can be written as

∂t FA + ∂i FiA = PA ,

(0 6 A 6 N )

(2)

1 Throughout the paper, summation with respect to repeated indexes is assumed, where the range of the sum is to be understood in the context: when the index represents a spatial coordinate, the range of the sum is from 1 to 3; in all the other cases the sum is intended over the variability region of the repeated indexes.

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where the following notations are introduced:

 FA = R3



mfcA dc,

FiA = R3



mfci cA dc,

PA = R3

mQ (f )cA dc

(3)

with

 FA =

F Fi1 ···iA

for A = 0 for 1 6 A 6 N ,

0 0 Pi1 ···iA

for A = 1 for 2 6 A 6 N ,

FiA =

Fi Fi i1 ···iA

for A = 0

 PA =



for A = 0 for 1 6 A 6 N ,

Pll = 0,

and

 cA =

1 ci1 · · · ciA

for A = 0 for 1 6 A 6 N .

The capital indexes, here and hereafter, run from 0 to N. System (2) will be denoted as ‘‘(N )-system’’. The balance equations obtained, in particular, for the collision invariants m, mci and mc 2 /2 (c 2 = cl cl ) turn out to be the conservation laws of mass, momentum, and energy since the corresponding production terms P , Pi , Pll are null. Therefore the first 13 moments are identified as the macroscopic variables commonly used in continuum mechanics: F = ρ,

Fi = ρvi ,

Fll = 2ρε + ρv , 2

Fij = −tij + ρvi vj , Flli = 2qi − 2til vl − tll vi + ρv 2 vi ,

(4)

where ρ is the mass density, ρvi is the momentum density with vi being the macroscopic gas velocity, ε is the specific internal energy, qi is the heat flux, and tij = −pδij + σij = −(p + Π )δij + σ⟨ij⟩

(5)

is the stress tensor with p, Π = −σll /3, and σ⟨ij⟩ being, respectively, the pressure, the dynamic pressure, and the deviatoric (traceless) part of the viscous stress σij . Due to the special structure of the hierarchy, we notice that the trace part of the momentum flux must be equal to the energy density. Comparison between the trace of (4)2 and (4)3 and taking into account (5), we obtain 3p = 2ρε,

Π = 0.

(6)

As the caloric equation of state for rarefied gases is given by

ε=

Dp 2ρ

,

(7)

where D is the effective degrees of freedom of a molecule, the condition (6) implies D = 3. Therefore, as is well known, the hierarchy structure (2) derived from the Boltzmann equation is valid only for monatomic gases. Moreover the dynamic pressure vanishes in this case. 2.1.2. (N − )-system and the closure The prototype of the ET theory [11] is the 13-field theory with independent variables F , Fi , Fij and Flli , which is in full agreement with Grad’s 13-moment theory [12]. However, the system of the 13field theory is not a particular case of the (N )-system because this system has only the trace part of the third order tensor in addition to the (2)-system. We will denote the system with (N − 1)-tensorial equations combined by the trace part with respect to indexes iN −1 and iN of the Nth order tensorial equation as ‘‘(N − )-system’’. From its definition (N − )-system is valid for N > 2. In this notation the 13-field theory is the ‘‘(3− )-system’’.

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In this subsection, as an example, we show the closure procedure of the ((N + 1)− )-system. For convenience, we introduce the symmetric traceless tensor F⟨i1 i2 ···iN ⟩ defined by [33]   N

2 

F⟨i1 i2 ···iN ⟩ =

aNk δ(i1 i2 · · · δi2k−1 i2k Fi2k+1 ···iN )j1 ···jk j1 ···jk

k=0

with N !(2N − 2k − 1)!!

aNk = (−1)k

,

(N − 2k)!(2N − 1)!!(2k)!!

where the following notations are adopted:

 N  

  N 2



if N is even.

2

=

N !! =

  N − 1 if N is odd.

N −1 2





(N − 2j).

j =0

2

Then the independent variables of the ((N +1)− )-system are F⟨A⟩ (0 6 A 6 N ) and FllA′ (0 6 A′ 6 N −1), and the system of balance equations is expressed as2

∂t F⟨A⟩ + ∂i Fi⟨A⟩ = P⟨A⟩ , (0 6 A 6 N )

∂t FllA′ + ∂i FlliA′ = PllA′ .   0 6 A′ 6 N − 1

(8)

We have put:

 FllA′ = R3

mfc 2 cA′ dc,

 FlliA′ = R3

mfc 2 ci cA′ dc,

 PllA′ = R3

mQ (f )c 2 cA′ dc.

(9)

The number of moments n((N +1)− ) of this system is given by 1

n((N +1)− ) =

6

(N + 1)(N 2 + 8N + 6).

In MET, we pick up the proper distribution function f by using the MEP: find f such that it maximizes the entropy density:

 h = −k

f log f dc R3

(k: the Boltzmann constant) under the constraints that the moments are expressed by f (see (3)). Then the proper distribution function is given by [5,3]



m 

χ , χ = Λ⟨A⟩ c⟨A⟩ + ΛllA′ c 2 cA′ , (10) k where Λ⟨A⟩ and ΛllA′ denote the Lagrange multipliers that coincide with the main fields [9] corresponding to the densities F⟨A⟩ and FllA′ . The main fields of the system with full moment such as (2) are also obtained from the linear combination of the main fields Λ⟨A⟩ and ΛllA′ . f = exp −1 −

2 Notations for the moments of the traceless tensors are defined as follows: F⟨A⟩ =

P⟨A⟩ =

 F

Fi

F ⟨i1 ···iA ⟩  0 0

P

⟨i1 ···iA ⟩

for A = 0 for A = 1 for 2 6 A 6 N , for A = 0 for A = 1 for 2 6 A 6 N .

Fi⟨A⟩ =

 Fi

Fii1

F

i ⟨i1 ···iA ⟩

for A = 0 for A = 1 for 2 6 A 6 N ,

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Remark. For simplicity reason, in (10), we have used the notation f as the distribution function compatible with the truncated system of the balance equations, but, in reality, it is more appropriate to use fN to indicate the number of moments that we prescribe. Changing N we have a different distribution function that maximizes the entropy density. Of course fN is not a solution of the Boltzmann equation, and we have the conjecture that the exact distribution function f , namely the solution of the Boltzmann equation, is obtained as the limit of fN when N tends to infinity. To prove this conjecture is, however, still a difficult and open problem. In (10)2 we omit also the summation symbols on the repeated indexes: for A from 0 to N and for A′ from 0 to N − 1. In equilibrium, all the coefficients in the polynomial χ vanish except for those corresponding to the first five moments (F , Fi and Fll ), which are identified as the hydrodynamic variables [34]. Therefore the distribution function f in an equilibrium state, f E , is, as is expected, the Maxwellian distribution f M: fM =

 m  ρ  m 3/2 exp − C2 , m 2π kT 2kT

where T is the temperature of a monatomic gas defined by the caloric equation of state

ε=

3 k

T,

2m

and Ci = ci − vi is the peculiar velocity (C 2 = Ci Ci ). All results explained above are valid also for a case far from equilibrium provided that the integrals in (3) with f given by (10) are convergent. The problem of the convergence of the moments is one of the main issues in a far-from-equilibrium case. In particular, the index of truncation N in (3) must be even. Moreover, if the above-mentioned conjecture is true, we need another convergence requirement for χ . These problems were studied by Boillat and Ruggeri [9] (see also [4]). For both mathematical (convergence) reason and physical importance of a process not far from equilibrium, which are carefully explained in [35], the distribution function (10) may be expanded around a local equilibrium state. Therefore, the distribution function is approximated as m



f ≈ fM 1−

k

˜ ⟨A⟩ c⟨A⟩ + Λ ˜ llA′ c 2 cA′ Λ

˜ ⟨A⟩ = Λ⟨A⟩ − ΛE⟨A⟩ , Λ



,

(11)

˜ llA′ = ΛllA′ − ΛEllA′ , Λ

where the superscript ‘‘E’’ means that all the quantities are evaluated at the local equilibrium state characterized by f M . Plugging (11) into (3)1 , we have a linear algebraic system by which we can ˜ ⟨A⟩ and Λ ˜ llA′ in terms of the densities F⟨A⟩ and FllA′ : evaluate Λ





J⟨M A⟩⟨B⟩

JllM⟨A⟩B′

M JllA ′ ⟨B ⟩

M JiijjA ′ B′

˜ ⟨B⟩ Λ ˜ ΛllB′



 =

F⟨A⟩ − F⟨EA⟩ E FllA′ − FllA ′

 ,

(12)

where3 M JAB =−

m2



k

R3

f M cA cB dc.

(13)

3 More specifically, J⟨M A⟩⟨B⟩ = − M JiijjA ′ B′ = −

m2 k m2 k

 R3

 R3

f M c⟨A⟩ c⟨B⟩ dc, f M c 4 cA′ cB′ dc.

M JllA ′ ⟨ B⟩ = −

m2 k

 R3

f M c 2 cA′ c⟨B⟩ dc,

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˜ ⟨A⟩ and Λ ˜ llA′ into (11), the explicit dependence of the truncated fluxes (3)2 and By inserting Λ productions (3)3 on the densities F⟨A⟩ and FllA′ is obtained, and the truncated system (8) becomes a closed system. The generalization of this procedure to higher order expansions of the distribution function was made in [36]. 2.2. Molecular extended thermodynamics for rarefied polyatomic gases Recently the procedure of MET has been extended to the theory of rarefied polyatomic gases with D > 3 on the basis of the kinetic model of polyatomic gases that was proposed by Bourgat et al. [25] in the framework of the Borgnakke–Larsen procedure [24]. This model was applied also to chemically reacting mixtures [37]. The basic feature of this model is the presence of a non-negative parameter I which reflects the internal degrees of freedom of a molecule. The velocity distribution function depends also on this parameter: f ≡ f (x, c, t , I ), and the collision term Q (f ) on the right-hand side of the Boltzmann equation (1) also takes into account the influence of the internal degrees of freedom through the collisional cross section. In this model, the hydrodynamic variables ρ , ρvi and ρε can be defined in terms of the collision invariants m, mci and mc 2 /2 + I as follows:

ρ=



∞

 R3

  ∞ ρvi = mfci ϕ (I ) dI dc,  2  R3 0  ∞ c I mf + ϕ (I ) dI dc,

mf ϕ (I ) dI dc, 0

1

ρε + ρv = 2

2

 R3

2

0

m

where the non-negative measure ϕ(I ) dI,

ϕ(I ) = I δ ,

δ=

D−5

(δ > −1),

2

is introduced in order to recover the caloric equation of state for polyatomic gases in equilibrium (7). Following the definition of the hydrodynamic variables, the two kinds of moments of the distribution function f are introduced4 [7]:



∞



FA = R3

∞

R3



 R3

δ

∞

∞



GlliA′ =

∞



 R3

2I



cA′ I δ dIdc,

2I



m

c2 +

0

m



ci cA′ I δ dIdc,

mQ (f ) c 2 +

QllA′ =

0

 mf

R3

mQ (f )cA I δ dIdc,

c2 +

0



0

PA =

 mf

R3

mfci cA I dIdc,

FiA =

∞



GllA′ =

0







mfcA I δ dIdc,

0

2I m



(14)

cA′ I δ dIdc.

It is possible to construct two hierarchies of balance equations that, after truncation, read as follows:

∂t FA + ∂i FiA = PA ,

∂t GllA′ + ∂i GlliA′ = QllA′ ,

(15)

′

where 0 6 A 6 N and 0 6 A 6 M, being N and M the truncation orders of the F -hierarchy (momentum-like hierarchy) and of the G-hierarchy (energy-like hierarchy), respectively. The system (15) is denoted as ‘‘(N , M )-system’’.

4 The notations for the moments are as follows:

 GllA′ =

Gll Glli1 ···iA′

 QllA′ =

0 Qlli1 ···iA′

for A′ = 0 for 1 6 A′ 6 M , for A′ = 0 for 1 6 A′ 6 M .

 GlliA′ =

Glli Gllii1 ···iA′

for A′ = 0 for 1 6 A′ 6 M ,

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In [8], it was proved that the requirement of the Galilean invariance of the system, implies: M 6 N − 1. It is worth noting that the first two equations of the F -hierarchy represent the conservation laws of mass and momentum (P = 0, Pi = 0),5 the first equation of the G-hierarchy represents the conservation law of energy (Qll = 0), and in each of the two hierarchies the flux in one equation appears as the density in the following one—a feature in common for the single hierarchy of monatomic gases. The first 14 fields are identified as the macroscopic variables commonly used: F = ρ, Fi = ρvi , Fij = −tij + ρvi vj , Gll = 2ρε + ρv 2 , Glli = 2qi − 2til vl + 2ρεvi + ρv 2 vi .

(16)

Although it is possible, in principle, to construct various systems depending on the various choice of N and M, the relation between N and M can be restricted by requiring the physically relevant feature to the system: the (N , M )-system has equilibrium characteristic velocities that depend on the degrees of freedom D. It was proved that this requirement with Galilean invariance is satisfied if and only if [8] M = N − 1. This case includes the Euler system ((1, 0)-system) with 5 moments: F , Fi , Gll , and the ET theory with 14 moments in (16) ((2, 1)-system). This result is also true for the (N , (N − 1)− ) and (N − , (N − 1)− )systems, but the (N − , (N − 1))-system is excluded because it does not assure the Galilean invariance. For an (N , N − 1)-system the number of moments is given by n(N ,N −1) =

1 6

(N + 1)(N + 2)(2N + 3),

for an (N , (N − 1)− )-system; 1

n(N ,(N −1)− ) =

6

(2N 3 + 9N 2 + N + 12),

and, for an (N − , (N − 1)− )-system; n(N − ,(N −1)− ) = 1 +

1 6

N (N − 1)(11 + 2N ).

The truncated system of moment equations is closed by the MEP with the entropy density h defined now as follows:



∞



h = −k R3

f log f I δ dIdc.

(17)

0

Details are shown in [7] in the case of 14 fields, and in [8,10] in the case of many independent fields. Remark. The (N − , N − 2)-system has a special feature that each moment of G-series has a one-to-one correspondence in the trace part of the F -series. Then, as the total energy can be decomposed into the translational and internal energy, each moment of GllA′ can be decomposed into two moments; one relating to the translational mode, i.e., FllA′ , and the other one relating to the internal mode, i.e., GllA′ − FllA′ . From this feature, although equilibrium characteristic velocities are independent of D, the (N − , N − 2)-system is physically and mathematically interesting. One of the examples is the ET theory with 6 fields, F , Fi , Fll , Gll , ((2− , 0)-system) [38–40], which is the simplest theory next to the Euler system. This theory can explain experimental data well in such a case that the effect of the dynamic pressure is enormously larger than that of the shear stress and the heat flux. And this theory has the correspondence relation with the well-known Meixner theory of relaxation processes with one internal parameter [41,42]. Recently the 6-field ET theory with nonlinear constitutive equations was

5 In the case of polyatomic gases, P ̸= 0. ll

92

T. Arima et al. / Annals of Physics 372 (2016) 83–109

proposed [43]. The kinetic theoretical basis of this nonlinear ET theory [44] and the application to the shock wave structure [45] have also been studied. The ET theory with 17 fields ((3− , 1)–system) that adopts F , Fi , Fij , Flli , Gll , Glli as independent fields is another example. This theory is in full agreement with the kinetic theory of polyatomic gases [46,47] that adopts 17 moments where an internal mode of the energy and the internal heat flux are introduced. Moreover, recently, Rahimi and Struchtrup [48] deduced Grad’s 36-moment equations based on the kinetic model with the two collision terms of BGK-type, and adopted the method of regularization. 3. Closure of the system of moment equations for polyatomic gases In the following sections, the characteristic features of the MET theory of polyatomic gases with an arbitrary number of moments will be discussed by introducing the new fields that express the difference between the momentum-like hierarchy and the energy-like hierarchy. Then the singular limit of MET of polyatomic gases will be studied in detail. 3.1. Characteristic variables of polyatomic gases We try to single out the variables that characterize polyatomic gases. In the ET theory with 14 fields, this problem was studied by recalling that a polyatomic gas is characterized by the following relation: if D > 3,

Gll ̸= Fll

and by the existence of the dynamic pressure, which is expressed by

Π=

1 3

(Fll − Gll ) −

 1 E Fll − GEll . 3

In a similar way, we pay attention to the difference between F and G-series as the characteristic nonequilibrium variables typical of polyatomic gases:

 E  E ΠA′′ = (FllA′′ − GllA′′ ) − FllA ′′ − GllA′′ , where 0 6 A′′ 6 N − 2 and we have put as in monatomic case:



∞



FllA′′ = R3

mfc 2 cA′′ I δ dIdc.

(18)

0

The dynamic pressure Π is the first component of ΠA′′ except for a factor 1/3. Therefore the multiindex tensors ΠA′′ (0 6 A′′ 6 N − 2) play the role like the dynamical pressure when the moments are more than 14, and we will prove in the following that ΠA′′ vanish in the limit of monatomic gas provided that these are zero at the initial time. We call for this reason ΠA′′ the dynamical multi-index pressure tensors. Let us introduce variables IA′′ defined by



∞



IA′′ = FllA′′ − GllA′′ = −2 R3

0





fcA′′ I δ+1 dIdc,

(19)

and let IiA′′ and RA′′ be given by ∞

IiA′′ = FlliA′′ − GlliA′′ = −2 R3



∞



RA′′ = PllA′′ − QllA′′ = −2 R3

fci cA′′ I δ+1 dIdc,

0

Q (f )cA′′ I δ+1 dIdc.

0

Then the dynamical multi-index pressure tensors ΠA′′ are the nonequilibrium part of IA′′ . If we adopt IA′′ , instead of FllA′′ , as independent variables for the closed (N , N − 1)-system, the system (15) can be rewritten in an equivalent way as follows:

∂t F⟨A⟩ + ∂i Fi⟨A⟩ = P⟨A⟩ ,

T. Arima et al. / Annals of Physics 372 (2016) 83–109

∂t IA′′ + ∂i IiA′′ = RA′′ ,   0 6 A′′ 6 N − 2

(0 6 A 6 N )

93

∂t GllA′ + ∂i GlliA′ = QllA′ .

(20)

(0 6 A 6 N − 1) ′

The unknown fields (independent variables) are {F⟨A⟩ , IA′′ , GllA′ }. In order to converge for the binary hierarchy to the single hierarchy in the limit D → 3, it is necessary that the hierarchy of the variables IA′′ vanishes. Therefore we will focus particularly on the evolution of the variables IA′′ . 3.2. Closure of the new system In MET, the system (20) is closed by using the MEP with the entropy density (17) and, as a consequence, the proper distribution function for the truncated system (20) is given by [8] m 



f = exp −1 −

χ ,

k

χ = λ⟨A⟩ c⟨A⟩ −

2I m

  2I νA′′ cA′′ + c 2 + µA′ cA′ ,

(21)

m

where λ⟨A⟩ , νA′′ and µA′ are the main field components (Lagrange multipliers). These main fields are the linear combination of the main fields of the system (15). By inserting (21) into (14) and (19), the main fields are evaluated, in principle, in terms of the densities F⟨A⟩ , IA′′ and GllA′ . For the same reasons of convergence stated before for monatomic gas, the distribution function (21) is expanded around a local equilibrium state: f ≈ fE





m

1−

k

λ˜ ⟨A⟩ c⟨A⟩ −

2I m

   2I ν˜ A′′ cA′′ + c 2 + µ ˜ A′ cA′

(22)

m

with

λ˜ ⟨A⟩ = λ⟨A⟩ − λE⟨A⟩ , ν˜ A′′ = νA′′ − νAE′′ , µ ˜ A′ = µA′ − µEA′ . The equilibrium distribution function f E obtained by Pavić, Ruggeri and Simić [7] (see also [37] and [4]) is given by fE =

   ρ  m 3/2 m 2I exp − C2 + , A(T ) m 2π kT 2kT m 1

(23)

where the normalization factor A(T ) is defined as A(T ) =

∞



 exp −

0

I



kT

I δ dI = (kT )1+δ Γ (1 + δ),

and Γ stands for the gamma function. All the main fields in equilibrium vanish, except for those corresponding to the conservation laws of mass, momentum and energy [34]. By inserting (22) into (14)  and (19), a linear algebraic system, from which we can evaluate the

˜ ⟨A⟩ , ν˜ A′′ , µ nonequilibrium main field λ ˜ A′ in terms of the densities F⟨A⟩ , IA′′ and GllA′ , is obtained: 

−K⟨A⟩B′′

J⟨A⟩⟨B⟩  −KA′′ ⟨B⟩ JiiA′ ⟨B⟩ + KA′ ⟨B⟩

LA′′ B′′

−KiiA′ B′′ − LA′ B′′









Jii⟨A⟩B′ + K⟨A⟩B′ F⟨A⟩ − F⟨EA⟩ λ˜ ⟨B⟩ −KiiA′′ B′ − LA′′ B′   ν˜ B′′  =  ΠA′′  , JiijjA′ B′ + 2KiiA′ B′ + LA′ B′ µ ˜ B′ GllA′ − GEllA′

(24)

where J⟨A⟩⟨B⟩ = −

m2 k

K⟨A⟩B′′ = − LA′′ B′′ = −



m2

R3



k

m2 k

∞



∞

 R3



0

∞

 R3

f E c⟨A⟩ c⟨B⟩ I δ dIdc,

0

0

2 m 4 m2

f E c⟨A⟩ cB′′ I δ+1 dIdc, f E cA′′ cB′′ I δ+2 dIdc.

(25)

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T. Arima et al. / Annals of Physics 372 (2016) 83–109

Inserting (23) into (25), we can obtain the following useful relations that express J⟨A⟩⟨B⟩ , K⟨A⟩B′′ and LA′′ B′′

M in terms of the corresponding quantities of monatomic gases JAB (see (13)) and the parameter δ :

J⟨A⟩⟨B⟩ = J⟨M A⟩⟨B⟩ , K⟨A⟩B′ = 2cs2 (1 + δ)J⟨M , A⟩B′

(26)

LA′′ B′ = 4cs4 (1 + δ)(2 + δ)JAM′′ B′ ,



where cs =

k T. m

By using these relations, we can order (24) and, in the matrix form, these are

expressed as J⟨M A⟩⟨B⟩

JiiM⟨A⟩B′ + 2cs2 (1 + δ)J⟨M A⟩B′

J⟨M A⟩B′′

 λ˜ ⟨B⟩ M  −2c 2 (1 + δ)˜νB′′   0 −2cs2 JAM′′ B′′ −4cs4 (1 + δ)JiiA ′′ B′ s  2 2 M 2 M M M M M µ ˜ B′ JiiA′ ⟨B⟩ JiiA′ B′′ + 2cs JA′ B′′ JiijjA′ B′ + 2cs (1 + δ) JiiA′ B′ + 2cs JA′ B′   F⟨A⟩ − F⟨EA⟩   2 (27) =  ΠA′′ − 2cs (1 + δ) FA′′− FAE′′  . GllA′ − GEllA′ − 2cs2 (1 + δ) FA′ − FAE′     ˜ ⟨A⟩ , ν˜ A′′ , µ From (27), the main field λ ˜ A′ can be calculated in terms of the densities F⟨A⟩ , ΠA′′ , GllA′ . 



Then, since the constitutive functions for fluxes satisfy the similar equation with (27), i.e., Fi⟨N ⟩ − FiE⟨N ⟩   2   Πi(N −2) − 2cs (1 + δ) Fi(N −2) − FiE(N −2)  Glli(N −1) − GElli(N −1) − 2cs2 (1 + δ) Fi(N −1) − FiE(N −1)



  =



M 2 M Jikk ⟨N ⟩B′ + 2cs (1 + δ)Ji⟨N ⟩B′

JiM ⟨N ⟩⟨B⟩

JiM ⟨N ⟩B′′

0

−2cs2 JiM (N −2)B′′

M Jikk (N −1)⟨B⟩

M 2 M Jikk (N −1)B′′ + 2cs Ji(N −1)B′′

M −4cs4 (1 +δ)Jikk (N −2)B′



M 2 M 2 M Jikkll (N −1)B′ + 2cs (1 + δ) Jikk(N −1)B′ + 2cs Ji(N −1)B′

 λ˜ ⟨B⟩ × −2cs2 (1 + δ)˜νB′′  , µ ˜ B′

 



(28)

the closed field equations are obtained except for the production terms. 3.3. Production terms in the BGK model In order to close the system, it is necessary to express the production terms by using the densities. For this purpose, we need the concrete expression of the collision term Q (f ). In the present study, however, we avoid entering into the precise modeling of the collision term (see also the remark in the Section 8). Therefore, let us introduce the simplest model, i.e., the BGK model: Q (f ) = −

f −fE

τ

,

where τ stands for a relaxation time. Although, in practical problems, there exist several kinds of the relaxation time and the difference of the order of magnitude of relaxation times plays an important role such as studied in [48], we assume here that all relaxation processes have the same relaxation time τ . Then the production terms are obtained as P ⟨A ⟩ = −

 1 F⟨A⟩ − F⟨EA⟩ ,

τ

1 RA′′ = − ΠA′′ ,

τ

QA′ = −

1

τ

GllA′ − GEllA′ .



T. Arima et al. / Annals of Physics 372 (2016) 83–109

95

4. Singular limit of a polyatomic gas to a monatomic gas Let us consider the singular limit: (i) we take the limit where D approaches 3 continuously, and (ii) we adopt an appropriate initial condition compatible with a property of monatomic gases. In this limit, we will show that the variables IA′′ converge to zero and the remaining fields F⟨A⟩ and GlliA′ approach the fields in the single hierarchy (8). 4.1. The limit D → 3 As explained above, the main fields are evaluated in terms of the densities by using the relation (27). It is convenient to introduce the new fields:

ν˜ B∗′′ ≡ −2cs2 (1 + δ)˜νB′′ , then the relation between the main fields and the densities (27) is rewritten as



J⟨M A⟩⟨B⟩

  0 M

JiiA′ ⟨B⟩

J⟨M A⟩B′′

−2cs2 JAM′′ B′′ JiiA′ B′′ + 2cs2 JAM′ B′′ M

JiiM⟨A⟩B′



    F⟨A⟩ − F⟨EA⟩ λ˜ ⟨B⟩  . ΠA′′ 0  lim  ν˜ B∗′′  = lim  D→3 D→3 E M ′ µ ˜ ′ GllA − GllA′ B JiijjA′ B′

(29)

From (19), (25)2 and (26)2 , lim IAE′′ = lim

D→3

D→3

k m

K0A′′ = 0,

E lim IiA ′′ = lim

D→ 3

D→3

k m

K1A′′ = 0.

Therefore, in the limit D → 3, IA′′ and IiA′′ have only the nonequilibrium part ΠA′′ and ΠiA′′ . From (29)2 the fields νB∗′′ are expressed explicitly by ΠA′′ 6 : 1  −1 lim ν˜ B∗′′ = − 2 J M A′′ B′′ ΠA′′ , 2cs

D→3

(30)

where the element of the matrix J M is JAM′′ B′′ . Concerning the nonequilibrium flux of the dynamical pressure tensors:

ΠiA′′ = IiA′′ − IiAE ′′ , evaluated in the limit of D → 3, from (28) and (30), we obtain

  −1 ΠiA′′ = −2cs2 JiAM′′ B′′ lim ν˜ B∗′′ = JiAM′′ B′′ J M B′′ C ′′ ΠC ′′ . D→ 3

Then, in the limit D → 3, the field equations of the dynamical pressure tensors (20)2 become

    −1  −1  1 ∂t ΠA′′ + JiAM′′ B′′ J M B′′ C ′′ ∂i ΠC ′′ = − δA′′ C ′′ + ∂i JiAM′′ B′′ J M B′′ C ′′ ΠC ′′ , τ

(31)

where δA′′ C ′′ is the Kronecker delta. It is remarkable that only ΠA′′ are involved in these equations, which are the first-order quasi-linear partial differential equations with respect to ΠA′′ . 4.2. Initial condition compatible with monatomic gases In the limit to monatomic gases, the initial condition should be compatible with a property of monatomic gases. Therefore we impose the condition:

ΠA′′ (x, 0) = 0.

6 We continue, for simplicity, using the same letter Π ′′ for the limit value. A

96

T. Arima et al. / Annals of Physics 372 (2016) 83–109

Under this condition, assuming the uniqueness of the solution of (31), we obtain

ΠA′′ (x, t ) = 0,

∀t .

Then we also find limD→3 ν˜ A∗′′ = 0, and ν˜ A′′ is undetermined. 4.3. Remaining field equations In the singular limit, the balance equations of non-vanishing fields are expressed as follows:

∂t F⟨A⟩ + ∂i Fi⟨A⟩ = P⟨A⟩ , (0 6 A 6 N )

∂t GllA′ + ∂i GlliA′ = QllA′ ,   0 6 A′ 6 N − 1

and, from (29), the main fields corresponding to F⟨A⟩ and GllA′ are calculated from



J⟨M A⟩⟨B⟩

JiiM⟨A⟩B′

M JiiA ′ ⟨B⟩

M JiijjA ′ B′





λ˜ ⟨B⟩ lim µ ˜ B′ D→3



 = lim

D→3

F⟨A⟩ − F⟨EA⟩ GllA′ − GEllA′

 .

(32)

Recalling (18), from (32), we notice that E lim (GllA′ − GEllA′ ) = lim (FllA′ − FllA ′ ). D→3

D→ 3

Taking into account the definition of GllA′ (14) and the relations (26), we notice E M lim GEllA′ = lim FllA ′ = FllA′ ,

D→ 3

D→3

M

where FllA′ is the moment defined in the case of monatomic gas (9) in an equilibrium state. The similar relations are satisfied for fluxes and productions. From these relations the G-hierarchy coincides with the F -hierarchy in the limit: lim GllA′ = lim FllA′ ,

D→ 3

D→3

lim GlliA′ = lim FlliA′ ,

D→ 3

D→3

lim QllA′ = lim PllA′ .

D→3

D→ 3

This concludes that the balance equations of non-vanishing fields have the same hierarchy structure as that of monatomic gases of the ((N + 1)− )-system. The relation between the main fields and the densities (32) coincides with the relation of the monatomic gases (12). Then the constitutive equations have the same form as those of monatomic gases, in other words, the equations of nonvanishing fields coincide with those of monatomic gases. 4.4. Singular limit of other systems The above procedure of the singular limit can be applied to the other systems. For example, the results for some typical cases are summarized as follows: the (N , (N − 1)− ) and (N − , (N − 1)− )systems converge to, respectively, the ((N + 1)2− ) and ((N + 1)2−− )-systems, where the ((N + 1)2− )system adopts an (N + 1)th order density with double trace in addition to the densities up until Nth order as independent fields,7 and the ((N + 1)2−− )-system adopts the densities of the ((N + 1)2− )system in which Nth order tensor has only a trace part. As an example of these systems, the densities of the (42− ) and (42−− )-systems are shown in Table 1. Up to now, we have shown the singular limit of the physically meaningful systems where equilibrium characteristic velocities depend on D. We can also prove the singular limit of the (N , M )system with M < N − 1 that adopts, as independent variables, F⟨A⟩ (0 6 A 6 N ), IA′ (0 6 A′ 6 M ), GllA′ (0 6 A′ 6 M ) and Fllα (M + 1 6 α 6 N − 2).8 In the singular limit, ΠA′ = 0 and this system

7 The (N + 1)2− -system is defined for N > 3. 8 If M = N − 2, F = 0. llα

T. Arima et al. / Annals of Physics 372 (2016) 83–109

97

Table 1 Correspondence relation between the systems of monatomic and polyatomic gases for particular N and M. Densities and number of moments are shown. λ∗ = λ∗ (D) indicates that the equilibrium characteristic velocities λ∗ depend on D or not. Monatomic gas

Polyatomic gas

(N)

Densities

n(N )

(N , M )

Densities

n(N ,M )

λ∗ = λ∗ (D)

(2− )

F , Fi , Fll

5

(3− )

F , Fi , Fij , Flli

13

F , Fi , Fij , Flli , Flljj

14

(42− )

F , Fi , Fij , Fijk , Flljj

21

(4− )

F , Fi , Fij , Fijk , Fllij

26

(1, 0) (2− , 0) (2, 1) (3− , 1) (3− , 2− ) (42−− , 2− ) (3 , 2 − ) (42− , 2− ) (3, 2) (4− , 2)

F , Fi , Gll F , Fi , Fll , Gll F , Fi , Fij , Gll , Glli F , Fi , Fij , Flli , Gll , Glli F , Fi , Fij , Flli , Gll , Glli , Glljj F , Fi , Fij , Flli , Flljj , Gll , Glli , Glljj F , Fi , Fij , Fijk , Gll , Glli , Glljj F , Fi , Fij , Fijk , Flljj , Gll , Glli , Glljj F , Fi , Fij , Fijk , Gll , Glli , Gllij F , Fi , Fij , Fijk , Fllij , Gll , Glli , Gllij

5 6 14 17 18 19 25 26 30 36

Yes No Yes No Yes No Yes No Yes No

(4

2−−

)

converges to the (N )-system of monatomic gas. Similarly the singular limit of the ((N +1)− , M ) system converges to the ((N + 1)− )-system. We summarize the correspondence relation between the systems of polyatomic and monatomic gases in the singular limit in Table 2. The correspondence is shown for some systems with particular N and M in Table 1. An interesting remark is that, from the view point of MET of polyatomic gases, appropriate models of monatomic gases have the trace in the highest tensorial equation, such as, the (N − ), (N 2− ) and (N 2−− )-systems. This fact implies that the (N )-system, which is usually utilized in the literature of the ET theory, is not appropriate as a model of monatomic gases. 4.5. Singular limit for the intrinsic (velocity-independent) fields In practical problems, instead of the system of moments (20), the system of field equations in terms of the velocity-independent variables, such as the mass density, internal energy, dynamic pressure, shear viscosity and heat flux, is useful. In general, the relation between the moments and the velocityindependent fields (intrinsic fields) is derived by imposing the Galilean invariance on the system of balance equations. In this section, we introduce the intrinsic fields and show the equations of these fields. In particular, the closed system of field equations of the velocity-independent part of ΠA′′ is obtained and the singular limit for these variables is also shown. The intrinsic fields (denoted with hat) are expressed, by using the peculiar velocity, as follows: FˆA =



 R3

∞

δ

mfCA I dIdC,

ˆ llA′ = G

 mf

R3

0

∞





0

2

C +

2I m



CA′ I δ dIdC.

By imposing the Galilean invariance on the system of balance equations, the moments FA , GllA′ are expressed by these intrinsic fields as follows: FA = XAB FˆB ,





ˆ llB′ + 2vl FˆlB′ + v 2 FˆB′ . GllA′ = XA′ B′ G

(33)

The coefficient XAB is a function only of the velocity, and it makes up an ((N + 1) × (N + 1))-matrix X . It should be noted that this matrix is the same as the one that dictates the velocity dependence of the monatomic (N )-system. Similarly XA′ B′ is an element of (N × N )-matrix. The details of the Galilean invariance and the features of this matrix are discussed in [49] in which, in particular, it was proved for the general system composed of the balance laws that the matrix is an exponential matrix: X = exp Ar vr





(34)

98

T. Arima et al. / Annals of Physics 372 (2016) 83–109 Table 2 Correspondence relation between the systems of monatomic and polyatomic gases. M < N − 1. Monatomic gas

Polyatomic gas

System

System

λ∗ = λ∗ (D)

(N , N − 1) ((N + 1)− , M ) (N , (N − 1)− ) ((N + 1)2− , (M + 1)− ) (N − , (N − 1)− ) ((N + 1)2−− , (M + 1)− ) (N , M )

Yes No Yes No Yes No No

((N + 1) ) −

((N + 1)2− ) ((N + 1)2−− ) (N )

with the constant matrices Ar , r = 1, 2, 3. In the case of ET, due to the particular order of the moments, the matrices Ar are nilpotent of order N and the matrix X becomes a polynomial matrix:



1

 vk1   vk vk 1 2  X = ..  .   vk1 · · · vkN

0

(h δk11)

h1 (k2 k1 )

2δ

.. .

v

  N h1 δ(k1 vk2 · · · vkN )

0

0

0

0

0

0

0

0

h1 h2 (k1 k2 )

δ δ .. .

···

1



.. . ···

h

h

h

δk11 δk22 · · · δkNN

    .   

According to the general result obtained in [49] (see also [4]), the system of field equations can be rewritten in terms of the intrinsic fields using the material derivative (indicated by a dot) as follows:

˙



Fˆ A + FˆA ∂i vi + ∂i FˆiA + ArAC FˆC v˙ r + FˆiC ∂i vr



=−

1



FˆA − FˆAE ,

τ ˙Gˆ ′ + Gˆ ′ ∂ v + ∂ Gˆ ′ + 2v˙ Fˆ ′ + 2Fˆ ′ ∂ v i lliA l lA llA llA i i ilA i l    1 ˆ llA′ − Gˆ EllA′ G + ArA′ C ′ Gˆ llC ′ v˙ r + Gˆ lliC ′ ∂i vr = − τ

(35)

with the conservation law of momentum:

  ρ v˙ i + ∂j (p + Π )δij − σ⟨ij⟩ = 0,   where Ar ≡ ArAC is given by (see (34))  ∂ XAC  r AAC = . ∂vr vi =0 Similarly, for the tensors IA′′ , the velocity-independent densities are introduced as

ˆIA′′ = FˆllA′′ − Gˆ llA′′ = −2



∞

 R3

fCA′′ I δ+1 dIdC,

0

and are related to the moments as follows: IA′′ = XA′′ B′′ ˆIB′′ . The velocity-independent part of ΠA′′ is also defined as

ˆ A′′ = ˆIA′′ − ˆIAE′′ . Π Then the field equations of ˆIA′′ are obtained as follows:

  ˙ˆI ′′ + ˆI ′′ ∂ v + ∂ ˆI ′′ + Ar ˆ ′′ ˙ r + ˆIiC ′′ ∂i vr = − 1 ΠA′′ . i iA A A i i A′′ C ′′ IC v τ

(36)

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99

The field equations of the velocity-independent variables corresponding to the system (20) with IA′′ , GllA′ and F⟨A⟩ are (36), (35)2 and the traceless part of the field equations (35)1 . For the traceless tensor F⟨A⟩ we have the following velocity dependence: tl ˆ tl ˆ F⟨A⟩ = XAB FB = XAB F⟨B⟩ , tl where XAB indicates the traceless counterpart of XAB . By imposing vi = 0 on (24), the velocity-

ˆ ⟨A⟩ , νˆ A′′ and µ independent main fields λ ˆ A′ are expressed by the velocity-independent densities with the coefficient matrix in (24) with quantities evaluated at vi = 0, i.e., quantities in (25) in terms of the peculiar velocity Ci instead of ci . In particular, in the limit D → 3, the closed system of field equations of ΠA′′ is obtained as  −1 ˙ˆ ′′ + JˆM ˆ C ′′ Π Jˆ M ∂i Π ′′ ′′ A iA B B′′ C ′′     −1  1 + ∂i vi δA′′ C ′′ + ∂i JˆiAM′′ B′′ Jˆ M ′′ ′′ =− B C τ    −1 ˆ C ′′ , + Ar ′′ ′′ v˙ r δB′′ C ′′ + JˆM′′ ′′ Jˆ M ∂ i vr Π A B

iB D

(37)

D′′ C ′′

where JˆAM′′ B′′ = −

m2



f M CA′′ CB′′ dC .

k

(38)

The singular limit is also achieved as before, that is, by adopting the initial condition compatible with ˆ A′′ (x, 0) = 0. Then the solution of (37), that is, the solution of the a property of monatomic gases: Π ˆ A′′ is obtained, for any time, as first-order quasi-linear partial differential equations of Π

ˆ A′′ (x, t ) = 0. Π 5. Closure and singular limit in the one-dimensional case In the one-dimensional case, it is easy to calculate the explicit expression of the constitutive equations and the field equations because the integral (38) can be replaced with the simple expression. For this reason, we study the one-dimensional case along the x(≡x1 )-axis, and introduce the following notations:

 Fp,q = Gp′ ,q′ =

∞



 q

mf (c1 )p c 2

R3

0





∞

 mf

R3

Ip′′ ,q′′ ≡ Fp′′ ,q′′ +1 − Gp′′ ,q′′



′  q′ (c1 )p c 2 I δ dIdc, m   ∞ ′′  q′′ δ+1 I dIdc = −2 f (c1 )p c 2

c2 +

0

2I

I δ dIdc,

R3

0

with the dynamical pressure tensors that become in the one dimensional case:

Πp′′ ,q′′ = Ip′′ ,q′′ − IpE′′ ,q′′ . The indexes are the non-negative integers satisfying 0 6 p + 2q 6 N ,

0 6 p′ + 2q′ 6 N − 1,

0 6 p′′ + 2q′′ 6 N − 2.

The balance equations (20) become, in the present case,

∂t Fp,0 + ∂x Fp+1,0 = Pp,0 , ∂t Ip′′ ,q′′ + ∂x Ip′′ +1,q′′ = Rp′′ ,q′′ ,

∂t Gp′ ,q′ + ∂x Gp′ +1,q′ = Qp′ ,q′ ,

(39)

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where ∂x denotes the partial derivative with respect to x. The production terms Pp,q , Rp′′ ,q′′ and Qp′ ,q′ are defined in a similar way as follows:

 Pp,q =

∞

 R3

 q

mQ (f )(c1 )p c 2

 Rp′′ ,q′′ = −2

 Qp′ ,q′ =

I δ dIdc,

0

R3

′′

Q (f ) (c1 )p

 2 q′′ c

I δ+1 dIdc,

0

∞

 R3

∞





mQ (f ) c 2 + 0

2I



m

′  q′ (c1 )p c 2 I δ dIdc.

Taking into account that the velocity-independent variables denoted with a hat are evaluated with the peculiar velocity C1 = c1 − v1 instead of the velocity of a molecule c1 , the relation (33) between the moments and the velocity-independent variables becomes as follows: Fp,q =

r /2] N [ 

Xp,q,r −2s,s Fˆr −2s,s ,

r =0 s =0 r /2] N −1 [  ′

Gp′ ,q′ =





ˆ r ′ −2s′ ,s′ + 2v Fˆr ′ −2s′ +1,s′ + v 2 Fˆr ′ −2s′ ,s′ , Xp′ ,q′ ,r ′ −2s′ ,s′ G

r ′ =0 s′ =0 r /2] N −2 [  ′′

Ip′′ ,q′′ =

Xp′′ ,q′′ ,r ′′ −2s′′ ,s′′ ˆIr ′′ −2s′′ ,s′′ .

r ′′ =0 s′′ =0

The coefficients are defined as follows:

Xp,q,r ,s =

min(p+q−r −s,q−s)      j=max(0,q−r −s)

  



p

p+q−r −s−j



q

q−s

q−s



j

2q−s−j v p+2q−(r +2s)

if q > s and p + q > r + s otherwise,

0

and also

Ap,q,r ,s =

 min(p+ q−r −s,q−s)     j=max(0,q−r −s)



p p+q−r −s−j



q

q−s

q−s

j



(p + 2q − (r + 2s)) 2q−s−j δp+2q,r +2s+1 if q > s and p + q > r + s otherwise,

  

0

where v ≡ v1 . The closure for the system of balance equations is obtained by the same procedure summarized in Section 3.2. In particular, we show the closure for the velocity-independent part of the constitutive fluxes. In this case, the velocity-independent and nonequilibrium parts of the main fields λ˜ r ,0 , ν˜ r ′′ ,s′′ , µ ˜ r ′ ,s′ are determined by the following equation corresponding to (24):   JˆpM+r ,0 −Kˆ p+r ′′ ,s′′ JˆpM+r ′ ,s′ +1 + Kˆ p+r ′ ,s′   −Kˆ p′′ +r ,q′′ Lˆ p′′ +r ′′ ,q′′ +s′′ −Kˆ p′′ +r ′ ,q′′ +s′ +1 − Lˆ p′′ +r ′ ,q′′ +s′   M M Jˆp′ +r ,q′ +1 + Kˆ p′ +r ,q′ −Kˆ p′ +r ′′ ,q′ +s′′ +1 − Lˆ p′ +r ′′ ,q′ +s′′ Jˆp′ +r ′ ,q′ +s′ +2 + 2Kˆ p′ +r ′ ,q′ +s′ +1 + Lˆ p′ +r ′ ,q′ +s′    ˜  Fp,0 λ˜ r ,0 ˆ p′′ ,q′′  , × ν˜ r ′′ ,s′′  = Π (40) ˜ p′ ,q′ µ ˜ r ′ ,s′ G where a quantity with a tilde indicates that the velocity-independent nonequilibrium part of the quantity, and the coefficients are expressed as follows [8]: 2

m JˆpM,q = −

k

 f R3

M

 2 q

(C1 ) C p

dC = −

m k

ρ



kT m

 2p +q

p

2 2 +q p+1

Γ



p+3 2

 +q

1 + (−1)p

√ π

,

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Kˆ p,q = 2cs2 (1 + δ)JˆpM,q , Lˆ p,q = 4cs4 (1 + δ)(2 + δ)JˆpM,q . In (40), the summations for repeated indexes are taken to be over all integers which satisfy (39). ˜ r ,0 , ν˜ r ′′ ,s′′ , µ ˆ p′′ ,q′′ , G˜ p′ ,q′ , the constitutive equations for the fluxes By using λ ˜ r ′ ,s′ evaluated by F˜p,0 , Π

ˆ t ′′ +1,u′′ , G˜ t ′ +1,u′ are obtained where the non-negative integers satisfy t = N, t ′′ + 2u′′ = N − 2 F˜t +1,0 , Π ′ and t + 2u′ = N − 1. Finally, after the constitutive equations are obtained, in the limit D → 3, the closed system of field equations for the dynamical pressure tensors corresponding to (37) becomes as follows: N −2 [t /2] ˙ˆ ′′ ′′ +   Hˆ (1) ˆ ′′ ′′ ′′ ∂Π Π p ,q p′′ ,q′′ ,t ′′ −2u′′ ,u′′ x t −2u ,u ′′

t ′′ =0 u′′ =0

N −2 [t ′′ /2]

=−

     1 ˆ t ′′ −2u′′ ,u′′ . + ∂x v δp′′ ,t ′′ −2u′′ δq′′ ,u′′ + ∂x Hˆ p(1′′),q′′ ,t ′′ −2u′′ ,u′′ + Hˆ p(2′′),q′′ ,t ′′ −2u′′ ,u′′ Π τ t ′′ =0 u′′ =0 (41)

Here we define (1)

ˆ p,q,t ,u = H

r /2] N −2 [ 



JˆpM+r −2s+1,q+s Jˆ M

 −1

r =0 s=0

(2)

ˆ p,q,t ,u = H

r /2] N −2 [ 

r −2s+t ,s+u

, (1)





ˆ r −2s,s,t ,u , Ap,q,r −2s,s v˙ δr −2s,t δs,u + (∂x v)H

r =0 s=0

where Jˆ M is a matrix of which element is JˆpM,q . 6. Examples of particular systems The closed systems of field equations and the singular limit for particular cases are studied in this section. By solving (40), we can obtain the velocity-independent nonequilibrium part of the main fields, and then the constitutive fluxes are closed. To avoid showing the long calculations and ˆ p′′ ,q′′ expressions of the field equations, we show only the closure equations and the field equations of Π (41) in the limit that D approaches 3. As particular examples, we choose the case of (1, 0), (2− , 0), (2, 1), (3− , 1), (3− , 2− ), (3, 2) and (4, 3)-systems. For simplicity, we study the one-dimensional case. 6.1. The 5-moment system (N = 1, M = 0) The independent variables F0,0 , F1,0 , G0,0 are F0,0 = ρ,

F1,0 = ρv,

G0,0 = 2ρε + ρv 2 ,

ˆ 0,0 = 2ρε. The field where the velocity-independent variables are denoted as Fˆ0,0 = ρ, Fˆ1,0 = 0, G equations are ∂t F0,0 + ∂x F1,0 = 0, ∂t F1,0 + ∂x F2,0 = 0, ∂t G0,0 + ∂x G1,0 = 0, where the constitutive functions F2,0 , G1,0 are determined as follows: F2,0 = p + ρv 2 ,

G1,0 = v(D + 2)p + ρv 3 .

In this case there is no quantity like ˆIA′′ , and therefore the number of field equations does not change in the singular limit.

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6.2. The 6-moment system (N = 2− , M = 0) The independent variables F0,0 , F1,0 , I0,0 , G0,0 are F0,0 = ρ,

F1,0 = ρv,

I0,0 = −(D − 3)p + 3Π ,

G0,0 = 2ρε + ρv 2 ,

ˆ 0,0 /3 is the dynamic pressure and the velocity-independent part of I0,0 is denoted as where Π = Π ˆI0,0 = −(D − 3)p + 3Π . The field equations of these variables are ∂t F0,0 + ∂x F1,0 = 0, ∂t F1,0 + ∂x F2,0 = 0, ∂t I0,0 + ∂x I1,0 = R0,0 ,

∂t G0,0 + ∂x G1,0 = 0,

and the constitutive functions F2,0 , I1,0 , G1,0 are determined as follows: F2,0 = p + Π + ρv 2 ,

I1,0 = −v {(D − 3)p − 3Π } ,

G1,0 = v {(D + 2)p + 2Π } + ρv 3 .

The characteristic variable of polyatomic gases is only Π . Its field equation when D approaches 3 is given by

˙ =− Π



1

τ

 + ∂x v Π .

(42)

In the singular limit with the null dynamical pressure at initial time, this system converges to the Euler system of monatomic gas [26]. 6.3. The 14-moment system (N = 2, M = 1) The independent variables F0,0 , F1,0 , F2,0 , I0,0 , G0,0 , G1,0 are F0,0 = ρ,

F1,0 = ρv,

G0,0 = 2ρε + ρv , 2

F2,0 = p + Π − σ + ρv 2 ,

I0,0 = −(D − 3)p + 3Π ,

G1,0 = 2q + v {(D + 2)p + 2Π − 2σ } + ρv 3 ,

(43)

where σ ≡ σ⟨11⟩ and q ≡ q1 , and the velocity-independent parts of F2,0 and G1,0 are denoted as ˆ 1,0 = 2q. The field equations of these variables are Fˆ2,0 = p + Π − σ and G

∂t F0,0 + ∂x F1,0 = 0, ∂t F1,0 + ∂x F2,0 = 0, ∂t F2,0 + ∂x F3,0 = P2,0 ,

∂t I0,0 + ∂x I1,0 = R0,0 ,

∂t G0,0 + ∂x G1,0 = 0, ∂t G1,0 + ∂x G2,0 = Q1,0 .

In the present case, the constitutive functions F3,0 , I1,0 , G2,0 are determined as follows: 6 q + 3v(p + Π − σ ) + ρv 3 , D+2 D−3 I1,0 = −2 q − v [(D − 3)p − 3Π ] , D+2 p 4(D + 5) G2,0 = {(D + 2)p + (D + 4)(Π − σ )} + qv + v 2 {(D + 5)p + 5Π − 5σ } + ρv 4 . ρ D+2 F3,0 =

As is the case with the 6-moment system, the characteristic variable of polyatomic gases is only the dynamic pressure Π . When D → 3, the field equation of Π is the same as (42), and therefore the 14-moment system converges to the Grad 13-moment system [26] in the singular limit. Hereafter, we will show the constitutive equations of the (3− , 1), (3− , 2− ), (3, 2) and (4, 3)systems, whose first 14 moments (F0,0 , F1,0 , F2,0 , I0,0 , G0,0 , G1,0 ) are in common with (43).

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6.4. The 17-moment system (N = 3− , M = 1) The independent variables are F0,0 , F1,0 , F2,0 , I0,0 , I1,0 , G0,0 , G1,0 , where I1,0 is

ˆ 1,0 − v [(D − 3)p + 3Π ] . I1,0 = Π The constitutive functions F3,0 , I2,0 , G2,0 are determined as follows: 3

F3,0 =

5 p

ˆ 1,0 − v 2 [(D − 3)p − 3Π ] , {(D − 3)p + (D − 6)Π − (D − 3)σ } + 2v Π   p 16 6 ˆ 1,0 + q = {(D + 2)p + (D + 4)(Π − σ )} + v Π ρ 5 3

I2,0 = − G2,0

 ˆ 1,0 + 2q + 3v(p + Π − σ ) + ρv 3 , Π

ρ

+ v 2 {(D + 5)p + 5Π − 5σ } + ρv 4 . The closed system of field equations with these constitutive equations completely coincides with that of Zhdanov’s theory [46] except for the production terms. In Zhdanov’s theory, the translational part of ˆ 1,0 . The field equations the heat flux Q = F˜1,1 /2 is adopted as the nonequilibrium variable instead of Π ˆ 1,0 (=2(Q − q)) in the limit D → 3 are of the characteristic variables of polyatomic gases Π and Π given as follows: 1

˙ + ∂x Π ˆ 1 ,0 = − Π





1

+ ∂x v Π ,     k ˙ˆ + 3 k T ∂ Π = − 1 + 2∂ v Π ˆ Π − 3 ∂ T + v ˙ Π. 1,0 x x 1 ,0 x m τ m 3

τ

(44)

As mentioned before, the 13-moment system of monatomic gases is achieved as the singular limit of the 14-moment system and also of the 17-moment system. 6.5. The 18-moment system (N = 3− , M = 2− ) The independent variables are F0,0 , F1,0 , F2,0 , I0,0 , I1,0 , G0,0 , G1,0 , G0,1 where I1,0 and G0,1 are

ˆ 1,0 − v [(D − 3)p − 3Π ] , I1,0 = Π ˜ 0,1 + 3(D + 2) G0,1 = G

p2

ρ

ˆ 1,0 + 4q) + v 2 {(D + 7)p + 7Π − 4σ } + ρv 4 , + 2v(Π

and the constitutive functions F3,0 , I2,0 , G2,0 , G1,1 are determined as follows: 3

 ˆ 1,0 + 2q + 3v(p + Π − σ ) + ρv 3 , Π 5   D−3 ˜ 0,1 − p (D − 3)p − 30 Π − (D − 3)σ I2,0 = − G 3(D + 7) ρ D+7 2 ˆ 1,0 − v [(D − 3)p − 3Π ] , + 2v Π   1 16 p ˜ 0 ,1 + 6 v Π ˆ 1,0 + q + {(D + 2)p − (D + 4)σ } G2,0 = G 3 5 3 ρ F3,0 =

G1,1

+ v 2 {(D + 5)p + 5Π − 5σ } + ρv 4 ,  p  ˆ 1,0 + 2(D + 11)q = (D + 6)Π ρ    5(D + 11) p 60 ˜ +v G0,1 + 5(D + 4)p + Π − 2(D + 11)σ 3(D + 7) ρ D+7

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+

27 5



ˆ 1,0 + v2 Π

28 9

 q

+ v 3 {(D + 11)p + 11Π − 8σ } + ρv 5 .

ˆ 1,0 in D → 3 are The field equations of the characteristic variables of polyatomic gases Π and Π the same as (44), and the system converges to the 14-moment system of monatomic gases ((42−− )system). 6.6. The 30-moment system (N = 3, M = 2) The independent variables are F0,0 , F1,0 , F2,0 , F3,0 , I0,0 , I1,0 , G0,0 , G1,0 , G2,0 , G0,1 , where F3,0 , I1,0 , G2,0 and G0,1 are F3,0 = F˜3,0 + 3v(p + Π − σ ) + ρv 3 ,

ˆ 1,0 − v [(D − 3)p − 3Π ] , I 1 ,0 = Π ˜ 2,0 + (D + 2) G2,0 = G

p2

ρ

˜ 0,1 + 3(D + 2) G0,1 = G

+ 2v(F˜3,0 + 2q) + v 2 {(D + 5)p + 5Π − 5σ } + ρv 4 ,

p2

ρ

ˆ 1,0 + 4q) + v 2 {(D + 7)p + 7Π − 4σ } + ρv 4 . + 2v(Π

The constitutive functions F4,0 , I2,0 , G3,0 , G1,1 are determined as follows:



F4,0 =



˜ 2,0 − G˜ 0,1 + 3 (D + 4) {(D + 7)p + 6Π } 6 (D + 7)G ρ p

(D + 4)(D + 7)

+ 4v F˜3,0 + 6v 2 (p + Π − σ ) + ρv 4 ,   D−3 D−3 ˜ 2,0 + ˜ 0,1 − p (D − 3)p − 30 Π I2,0 = − G G D+4 (D + 4)(D + 7) ρ D+7 2 ˆ 1,0 − v [(D − 3)p − 3Π ] , + 2v Π  p  G3,0 = (D + 6)F˜3,0 + 6q ρ (D + 7)(D + 8)G˜ 2,0 − 4G˜ 0,1 + ρp (D + 4) {(D + 4)(D + 7)p + 12Π } + 3v (D + 4)(D + 7) + v 2 (7F˜3,0 + 6q) + v 3 [(D + 9)p + 9Π − 9σ ] + ρv 5 ,  p  ˆ 1,0 + 2(D + 11)q G1,1 = (D + 6)Π ρ   (D + 11) (D + 2)G˜ 0,1 + 2(D + 7)G˜ 2,0 + 5 ρp (D + 4) {(D + 4)(D + 7)p + 12Π } +v (D + 4)(D + 7) 2 3 ˜ ˆ + v (3Π1,0 + 4F3,0 + 12q) + v [(D + 11)p + 11Π − 8σ ] + ρv 5 . ˆ 1,0 in the limit D → 3 The field equations of the characteristic variables of polyatomic gases Π and Π are the same as (44), and the system converges to the 26-moment system of monatomic gases ((4− )system). 6.7. The 55-moment system (N = 4, M = 3) The independent variables are F0,0 , F1,0 , F2,0 , F3,0 , F4,0 , I0,0 , I1,0 , I2,0 , I0,1 , G0,0 , G1,0 , G2,0 , G0,1 , G3,0 , G1,1 , and the constitutive functions are F5,0 , I3,0 , I1,1 , G4,0 , G2,1 . Because of the complexity, we omit to show

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105

the expressions of the densities and the constitutive equations, but we only show the field equations ˆ 1 ,0 , Π ˆ 2 ,0 , Π ˆ 0,1 in the limit D → 3 as follows: of the characteristic variables Π , Π

 + ∂x v Π , 3 τ   1 ˙ ˆ 1,0 − 3v˙ Π , ˆ ˆ + 2∂x v Π Π 1,0 + ∂x Π2,0 = − τ     1 k ˙ˆ + 3 k T ∂ Π ˆ ˆ ˆ 1,0 , Π = − + 3 ∂ v Π − 3 ∂ T + 2 v ˙ Π 2,0 x 1,0 x 2,0 x m τ m     1 k ˙ˆ + 5 k T ∂ Π ˆ ˆ 0,1 − 2(∂x v)Π ˆ 2,0 − 5 ∂x T + 2v˙ Π ˆ 1 ,0 . + ∂x v Π Π x 1,0 = − 0,1 m τ m ˙ + Π

1

ˆ 1 ,0 = − ∂x Π



1

The system converges for D → 3 to the 45-moment system of monatomic gases ((5− )-system). 7. Examples of the convergence of the solutions in the singular limit We study, as examples of the singular limit, asymptotic behaviors in the dispersion relation of sound waves and the shock wave structure when D approaches 3. We will focus, in particular, on the behaviors of the dynamical pressure tensors. 7.1. Dispersion relation of sound wave in the singular limit

¯ = (F˜p,q , G˜ p′ ,q′ , v)T superposed on the We study one-dimensional linear harmonic waves u ˆ 0 with the wave form: reference equilibrium state u ¯ = U¯ ei(ωt −kx) , u

(45)

where U¯ is the constant amplitude vector, ω is the frequency, and k is the complex wave number. By inserting the waveform (45) into the linearized system of field equations, we obtain the dispersion relation [17,50,51], from which the phase velocity vph and the attenuation factor α are derived as the functions of ω:

vph =

ω , ℜ(k)

α = −ℑ(k).

We now show the dependence of the dimensionless phase velocity vph /c0 and the attenuation − factor c0 τ α with D = 3.1 and 5 on the reduced √ frequency Ω (=τ ω) in the (2, 1) and (3 , 1)systems, where c0 is the sound velocity: c0 = (k/m)T0 (D + 2)/D with the temperature T0 at a reference equilibrium state. See Fig. 1. When D → 3, the dispersion relations of both systems approach uniformly the dispersion relation of (3− )-system of monatomic gases. Furthermore, we notice that ˆ 0,0 and Π ˆ 1,0 has no the mode of (3− , 1)-system that comes from the system of the field equations of Π counterpart in ET of monatomic gases. Since the solutions of such variables are zero in D → 3 because of the initial condition, the amplitude of this mode is zero. For higher-moment systems such as the (3, 2) and (4, 3)-systems, we can observe similar asymptotic behaviors. See Figs. 2 and 3. The dispersion relations of these systems converge to those of the (4− ) and (5− )-systems, respectively, and the modes that have no counterparts in monatomic gases have null solutions due to the initial condition. 7.2. Shock wave structure Next we study the asymptotic behavior of the structure of plane shock waves connecting two equilibrium states that satisfy the Rankine–Hugoniot conditions of the Euler equilibrium system.

106

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1.2

2.0 0.8

1.5 1.0

0.4 0.5 0.0

10–3

10–2

10–1

100

101

0.0

10–3

10–2

10–1

100

101

Fig. 1. Dependence of the dimensionless phase velocity (left) and attenuation factor (right) on the dimensionless frequency. The dashed line denotes the monatomic (3− )-system. The solid and dot-dashed lines with red (D = 3.1) and blue (D = 5) lines denote, respectively, the polyatomic (2, 1)–system and (3− , 1)-system. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.5

1.2

2.0 0.8

1.5 1.0

0.4 0.5 0.0

10–3

10–2

10–1

100

101

0.0

10–3

10–2

10–1

100

101

Fig. 2. Dependence of the dimensionless phase velocity (left) and attenuation factor (right) on the dimensionless frequency. The dashed line denotes the monatomic (4− )-system. The red and blue lines denote, respectively, the polyatomic (3, 2)-system with D = 3.1 and D = 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.5

1.2

2.0 0.8

1.5 1.0

0.4 0.5 0.0

10–3

10–2

10–1

100

101

0.0

10–3

10–2

10–1

100

101

Fig. 3. Dependence of the dimensionless phase velocity (left) and attenuation factor (right) on the dimensionless frequency. The dashed line denotes the monatomic (5− )-system. The red and blue lines denote, respectively, the polyatomic (4, 3)-system with D = 3.1 and D = 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Hereafter, we confine our study within the 17-field theory ((3− , 1)-system) in which the dynamical ˆ 1,0 (=2(Q − q)). For convenience, we introduce the dimensionless pressure tensors are Π and Π

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107

variables:

ρˆ ≡ ˆ ≡ Π

ρ , ρ0 Π kB m

ρ 0 T0

vˆ ≡ ,

v c0

,

qˆ ≡

Tˆ ≡

T T0

q kB m

ρ0 T0 c0

,

,

σˆ ≡ Qˆ ≡

σ kB m

ρ0 T0

,

Q kB m

ρ0 T0 c0

,

where a quantity with the subscript 0 represents the one at a reference constant equilibrium state before and far from a shock wave. As the system of balance equations is Galilean invariant, by using a coordinate system moving with a shock wave, we may assume, without loss of generality, that the shock wave is stationary. We adopt a method proposed by Weiss to solve the balance equations numerically [52] (see also [20,40] for details). For simplicity, we only show the results of the shock structure without the sub-shock [20,40]. In Fig. 4, we depict the wave profiles of the independent variables for D = 4, 3.5, 3.1, 3 at M0 = 1.5 where the Mach number in the unperturbed state M0 is defined as M0 = v0 /c0 . We notice from the figure that the wave profiles approach uniformly the ˆ 1,0 = 0 (Q = q) in the singular corresponding profiles of monatomic gases with Π = 0 and Π limit. 8. Summary and concluding remarks The singular limit of the polyatomic (N , N − 1)-system to the one of monatomic gases in MET has been studied. This limit is achieved by taking the effective degrees of freedom of a molecule D → 3 continuously under an appropriate initial condition compatible with a property of monatomic gases. In this limit, the characteristic variables of polyatomic gases, i.e., the dynamical pressure tensors defined by the difference between the momentum like-hierarchy and the energy like-hierarchy converge to zero for any time. The remaining field equations coincide with those of the monatomic ((N + 1)− )-system. Therefore, in this sense, the MET theory of polyatomic gases is valid not only for polyatomic gases but also for monatomic gases. Moreover this result indicates that, from the viewpoint of MET of polyatomic gases, the appropriate model of monatomic gases is the ((N + 1)− )system. These results are true also for the singular limit of other systems, e.g., the singular limit of the (N , (N − 1)− ) and (N − , (N − 1)− )-systems to the (N + 12− ) and (N + 12−− )-systems, respectively. Furthermore, for the systems with the D-independent equilibrium characteristic velocities, the ((N + 1)− , M ), ((N + 1)2− , (M + 1)− ), ((N + 1)2−− , (M + 1)− ) and (N , M )-systems with M 6 N − 2 converge to the ((N + 1)− ), ((N + 1)2− ), ((N + 1)2−− ) and (N )-systems, respectively. Several examples are given and the singular limit for the solutions are shown explicitly in the dispersion relations of sound waves and in the shock wave structure. In the present study, we have adopted the simplest BGK model as the collision term. If each velocity-independent nonequilibrium variable has its own relaxation time with no cross effect between the production terms, the present proof is also valid. The study of a general case by adopting more realistic collision model, which allows us to evaluate the order of magnitude of the relaxation times, is difficult to be carried out, but we may conjecture that quite similar results come out. We will try to study this problem in the future. Acknowledgments This work was partially supported by Japan Society of Promotion of Science (JSPS) No. 15K21452 (T.A.) and No. 25390150 (M.S.), and by National Group of Mathematical Physics GNFM-INdAM and by University of Bologna: FARB 2012 Project Extended Thermodynamics of Non-Equilibrium Processes from Macro- to Nano-Scale (T.R.). References [1] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, Dover, New York, 1984. [2] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1958.

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Fig. 4. Mass density profiles (left) and the profiles of the dynamic pressure (center, red) and the shear stress (center, black), and the heat flux (right, black) and Q (right, red) for D = 4, 3.5, 3.1, 3 at M0 = 1.5. Here xˆ is the dimensionless position coordinate defined by xˆ ≡ x/(c0 τ ). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) [3] I. Müller, T. Ruggeri, Rational Extended Thermodynamics, second ed., in: Springer Tracts in Natural Philosophy, vol. 37, Springer, New York, 1998. [4] T. Ruggeri, M. Sugiyama, Rational Extended Thermodynamics beyond the Monatomic Gas, Springer, Cham Heidelberg New York Dordrecht London, 2015, ISBN 978-3-319-13340-9. [5] I. Müller, T. Ruggeri, Extended Thermodynamics, first ed., in: Springer Tracts in Natural Philosophy, vol. 37, Springer, New York, 1993. [6] W. Dreyer, J. Phys. A: Math. Gen. 20 (1987) 6505–6517. [7] M. Pavić, T. Ruggeri, S. Simić, Physica A 392 (2013) 1302–1317. [8] T. Arima, A. Mentrelli, T. Ruggeri, Ann. Physics 345 (2014) 111–140. [9] G. Boillat, T. Ruggeri, Contin. Mech. Thermodyn. 9 (1997) 205–212. [10] T. Arima, A. Mentrelli, T. Ruggeri, Rend. Lincei Mat. Appl. 25 (2014) 275–291. [11] I.-S. Liu, I. Müller, T. Ruggeri, Ann. Physics 169 (1986) 191–219. [12] H. Grad, Comm. Pure Appl. Math. 2 (1949) 221–407. [13] T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Contin. Mech. Thermodyn. 24 (2011) 271–292. [14] T. Arima, M. Sugiyama, Atti Accad. Pelorit. Pericol. 91 (S1) (2013) A1–A15. [15] T. Ruggeri, M. Sugiyama, Acta Appl. Math. 132 (2014) 527–548. [16] M.C. Carrisi, S. Pennisi, T. Ruggeri, M. Sugiyama, Ricerche Mat. 64 (2015) 403–419. [17] T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Contin. Mech. Thermodyn. 25 (2013) 727–737.

T. Arima et al. / Annals of Physics 372 (2016) 83–109 [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]

109

T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Acta Appl. Math. 132 (2014) 15–25. T. Arima, S. Taniguchi, M. Sugiyama, Proceedings of The 34th Symposium on Ultrasonic Electronics, 2013, pp. 15–16. S. Taniguchi, T. Arima, T. Ruggeri, M. Sugiyama, Phys. Rev. E 89 (2014) 013025. S. Taniguchi, T. Arima, T. Ruggeri, M. Sugiyama, Acta Appl. Math. 132 (2014) 583–593. E. Barbera, F. Brini, M. Sugiyama, Acta Appl. Math. 132 (2014) 41–50. T. Arima, E. Barbera, F. Brini, M. Sugiyama, Phys. Lett. A 378 (2014) 2695–2700. C. Borgnakke, P.S. Larsen, J. Comput. Phys. 18 (1975) 405–420. J.-F. Bourgat, L. Desvillettes, P. Le Tallec, B. Perthame, Eur. J. Mech. B Fluids 13 (1994) 237–254. T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Phys. Lett. A 377 (2013) 2136–2140. Ya.B. Zel’dovich, Yu.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Dover, New York, 2002. E. Nagnibeda, E. Kustova, Non-equilibrium Reacting Gas Flow, Springer, Berlin, 2009. M. Groppi, G. Spiga, F. Zus, Phys. Fluids 18 (2006) 057105. M. Groppi, G. Spiga, J. Math. Chem. 26 (1999) 197–219. M. Bisi, G. Martaló, G. Spiga, Acta Appl. Math. 122 (2012) 37–51. G.M. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases, Springer, Berlin Heidelberg, 2010. H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin Heidelberg, 2005. G. Boillat, T. Ruggeri, Arch. Ration. Mech. Anal. 137 (1997) 305–320. M.N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969. F. Brini, T. Ruggeri, Contin. Mech. Thermodyn. 14 (2002) 165–189. L. Desvillettes, R. Monaco, F. Salvarani, Eur. J. Mech. B Fluids 24 (2005) 219–236. T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Phys. Lett. A 376 (2012) 2799–2803. T. Arima, T. Ruggeri, M. Sugiyama, S. Taniguchi, Meccanica 49 (2014) 2181–2187. S. Taniguchi, T. Arima, T. Ruggeri, M. Sugiyama, Phys. Fluids 26 (2014) 016103. J. Meixner, Ann. Phys. 43 (1943) 470–487. J. Meixner, Acoustica 2 (1952) 101–109. T. Arima, T. Ruggeri, M. Sugiyama, S. Taniguchi, Int. J. Non-Linear Mech. 72 (2015) 6–15. T. Ruggeri, Bull. Inst. Math. Acad. Sin. 11 (2016) 1–22. S. Taniguchi, T. Arima, T. Ruggeri, M. Sugiyama, Int. J. Non-Linear Mech. 79 (2016) 66–75. V.M. Zhdanov, Sov. Phys. JETP 26 (1968) 1187–1191. F.J. McCormack, Phys. Fluids 11 (12) (1968) 2533–2543. B. Rahimi, H. Struchtrup, Phys. Fluids 26 (2014) 052001. T. Ruggeri, Contin. Mech. Thermodyn. 1 (1989) 3–20. A. Muracchini, T. Ruggeri, L. Seccia, Wave Motion 15 (1992) 143–158. Z. Banach, W. Larecki, T. Ruggeri, Wave Motion 51 (2014) 955. W. Weiss, Phys. Rev. E 52 (1995) R5760.