Non-Linear Maximum Entropy Principle for a Polyatomic Gas subject to the Dynamic Pressure
aa r X i v : . [ m a t h - ph ] A p r Non-Linear Maximum Entropy Principle for aPolyatomic Gas subject to the Dynamic Pressure
Tommaso RuggeriDepartment of Mathematics and Alma Mater Research Center on AppliedMathematics AM , University of Bologna, Bologna, ItalyE-mail: [email protected] Dedicated to Tai-Ping Liu with esteem and affection . Abstract
We establish Extended Thermodynamics (ET) of rarefied polyatomic gaseswith six independent fields, i.e., the mass density, the velocity, the temperatureand the dynamic pressure, without adopting the near-equilibrium approximation.The closure is accomplished by the Maximum Entropy Principle (MEP) adoptinga distribution function that takes into account the internal degrees of freedom of amolecule. The distribution function is not necessarily near equilibrium. The resultis in perfect agreement with the phenomenological ET theory. To my knowledge,this is the first example of molecular extended thermodynamics with a non-linearclosure. The integrability condition of the moments requires that the dynamicalpressure should be bounded from below and from above. In this domain the systemis symmetric hyperbolic. Finally we verify the K-condition for this model and showthe existence of global smooth solutions.
1. Introduction
Rational extended thermodynamics [1] (hereafter referred to as ET) isa thermodynamic theory that is applicable to nonequilibrium phenomenawith steep gradients and rapid changes in space-time, which may be out oflocal equilibrium. It is expressed by a symmetric hyperbolic system of fieldequations with the convex entropy.As ET has been strictly related to the kinetic theory with the closure
AMS Subject Classification: 35L, 76A, 76P.Key words and phrases: Extended Thermodynamics, Non-Equilibrium Fluids, Symmetric Hyper-bolic systems, Maximum Entropy Principle. ommaso Ruggeri method of the hierarchy of moment equations, the applicability range of thetheory has been restricted within rarefied monatomic gases. Only recently,however, the ET theory of dense gases and of polyatomic rarefied gaseshas been successfully developed and has been obtained a 14-field theorythat, in the limit of small relaxation times (parabolic limit), reduces to theNavier-Stokes-Fourier classical theory [2]. This new approach to the caseof polyatomic rarefied gases, in particular, is in perfect agreement with thekinetic theory in which the distribution function depends on an extra variablethat takes into account the internal degrees of freedom of a molecule [3].Precise modeling of polyatomic gases and of dense gases in nonequilib-rium is an active and urgent issue nowadays with many important appli-cations like the study of shock wave structure [4, 5, 6], which is essentiallyimportant, for example, for the atmospheric reentry problem of a space ve-hicle.There are at least three different methods of closure of moment theoryassociated with the Boltzmann equation:1. A closure at the kinetic level proposed firstly by Grad in the case of 13moments, which is based on a perturbative procedure of the distributionfunction in terms of the Hermite polynomials [7];2. The phenomenological closure of ET by using the universal principlesof physics to select admissible constitutive equations [1, 8, 9] ;3. The kinetic closure of molecular ET by using the Maximum EntropyPrinciple (MEP) [10, 11].It is very suitable and in some sense surprising that the three different clo-sure methods give the same result in the case of the 13-moment theory formonatomic gases [1] and also in the 14-moment theory for rarefid polyatomicgases [3], provided that the thermodynamic processes are not far from equi-librium.We here want to focus mainly on MEP and therefore we want firstly tosummarize the principle and its limitation. The principle of maximum en-tropy has its root in statistical mechanics and is developed by E. T. Jaynesin the context of the theory of information related to the concept of theShannon entropy [12, 13]. MEP states that the probability distribution that on-Linear Maximum Entropy Principle represents the current state of knowledge in the best way is the one withthe largest entropy. Concerning the applicability of MEP in nonequilibriumthermodynamics, this was originally motivated by the similarity between thefield equations in ET and the moment equations, and later by the observa-tion made by Kogan [14] that Grad’s distribution function maximizes theentropy. The MEP was proposed in ET for the first time by Dreyer [10]. TheMEP procedure was then generalized by M¨uller and Ruggeri to the case ofany number of moments [11], and later proposed again and popularized byLevermore [15]. In the case of moments associtaed to the Boltzmann equa-tion the complete equivalence between the entropy principle and the MEPwas finally proved by Boillat and Ruggeri [16]. Later MEP was formulatedalso in a quantum-mechanical context [17, 18].As seen below, the truncated distribution function of MEP has the mean-ing also far from equilibrium provided that the integrals of the moments areconvergent. The problem of the convergence of the moments is one of themain questions in the far-from-equilibrium case. In particular, the index oftruncation of the moments N must be even. This implies that the Gradtheory with 13 moments is not allowed in a situation far from equilibrium!For this reason the truncated distribution function is formally expanded inthe neighborhood of equilibrium as a perturbation of the Maxwellian distri-bution. All closures by the MEP procedure are valid only near equilibrium.As a consequence, hyperbolicity exists only in some small domain of config-uration space near equilibrium [1, 19].The aim of this paper is to prove that, in the case of rarefied polyatomicgases, a theory can be established with the closure that is valid even far fromequilibrium. This is a theory with 6 independent fields, i.e., the mass density,the velocity, the temperature and the dynamic pressure. We will show thatthis non-linear closure matches completely the previous result obtained byusing only the macroscopic method [20].
2. Rarefied polyatomic Gas
A crucial step in the development of the kinetic theory of rarefied poly-atomic gases was made by Borgnakke and Larsen [21]. It is assumed thatthe distribution function depends on, in addition to the velocity of particles ommaso Ruggeri c , a continuous variable I representing the energy of the internal modes ofa molecule. This model was initially used for Monte Carlo simulations ofpolyatomic gases, and later it has been applied to the derivation of the gener-alized Boltzmann equation by Bourgat, Desvillettes, Le Tallec and Perthame[22]. The distribution function f ( t, x , c , I ) is defined on the extended domain[0 , ∞ ) × R × R × [0 , ∞ ). Its rate of change is determined by the Boltzmannequation which has the same form as in the case of monatomic gases: ∂ t f + c i ∂ i f = Q, (2.1)where the right-hand side, the collision term, describes the effect of collisionsbetween molecules. The collision term Q ( f ) now takes into account theexistence of the internal degrees of freedom through the collisional crosssection. Here ∂ t = ∂/∂t and ∂ i = ∂/∂x i .The idea, firstly proposed at the macroscopic level by Arima, Taniguchi,Ruggeri and Sugiyama [2] and successively in the kinetic framework in [3]and [23], is to consider, instead of the typical single hierarchy of moments, adouble hierarchy, i.e., an F -series at the index of truncation N and a G -seriesat the index M : ( N, M ) system given by ∂ t F + ∂ i F i = 0 ,∂ t F k + ∂ i F ik = 0 ,∂ t F k k + ∂ i F ik k = P k k , ∂ t G ll + ∂ i G ill = 0 , ... ∂ t G llj + ∂ i G llij = Q llj , ... ... ∂ t F k k ...k N + ∂ i F ik k ...k N = P k k ...k N , ... ∂ t G llj j ...j M + ∂ i G llij j ...j M = Q llj j ...j M . with F k k ··· k p = Z R Z ∞ mf ( t, x , c , I ) c k c k · · · c k p ϕ ( I ) dI d c , (2.2) G llk k ··· k q = Z R Z ∞ mf ( t, x , c , I ) (cid:18) c + 2 Im (cid:19) c k c k · · · c k q ϕ ( I ) dI d c , (2.3) on-Linear Maximum Entropy Principle and 0 ≤ p ≤ N, ≤ q ≤ M (when the index p = 0 we have F andwhen q = 0, G ll ). The double hierarchy is composed of the traditional velocity-moments F ’s and the energy-moments G ’s where the variable I ofthe internal modes plays a role. The connection between the index M and N is discussed in [23]. The non-negative measure ϕ ( I ) dI is introduced soas to recover the classical caloric equation of state for polyatomic gases inequilibrium. The functional form of ϕ will be given in the next section. Let us consider firstly the case of 5 moments corresponding to an Eulerfluid. In this case, N = 1 and M = 0. The collision invariants in this modelform a 5-vector: m (cid:18) , c i , c + 2 Im (cid:19) T , (2.4)which leads to hydrodynamic variables: FF i G ll = ρρv i ρv + 2 ρε = Z R Z ∞ m c i c + 2 I/m f ( t, x , c , I ) ϕ ( I ) dI d c , (2.5)The symbols are the usual ones: ρ, v i , ε are, respectively, the mass density,the i-th component of the velocity and the specific internal energy. Theentropy is defined by the relation: h = − k B Z R Z ∞ f log f ϕ ( I ) dI d c . (2.6)By introducing the peculiar velocity: C i = c i − v i , (2.7)we rewrite Eq. (2.5) as follows: ρ i ρε = Z R Z ∞ m C i C + 2 I/m f ( t, x , C , I ) ϕ ( I ) dI d C . (2.8) ommaso Ruggeri Note that the internal energy density can be divided into the translationalpart ρε T and the part of the internal degrees of freedom ρε I : ρε T = Z R Z ∞ mC f ( t, x , C , I ) ϕ ( I ) dI d C ,ρε I = Z R Z ∞ If ( t, x , C , I ) ϕ ( I ) dI d C . (2.9)The energy ρε T is related to the kinetic temperature T : ε T = 32 k B m T, (2.10)where k B and m are the Boltzmann constant and the atomic mass, respec-tively. The weighting function ϕ ( I ) is determined in such a way that itrecovers the caloric equation of state for polyatomic gases. If D is the de-grees of freedom of a molecule, it can be shown that the relation ϕ ( I ) = I α leads to the appropriate caloric equation of state: ε = D k B m T, α = D − . (2.11)The maximum entropy principle is expressed in terms of the follow-ing variational problem: determine the distribution function f ( t, x , C , I )such that h → max, under the constraints (2.5), or equivalently, due to theGalilean invariance, under the constraints (2.8). The result due to Pavic,Ruggeri and Simi´c [3] is summarized as follows: Theorem 1.
The distribution function that maximizes the entropy (2.6)under the constraints (2.8) has the form: f E = ρm ( k B T ) α Γ(1 + α ) (cid:18) m πk B T (cid:19) / exp (cid:26) − k B T (cid:18) mC + I (cid:19)(cid:27) . (2.12)This is the generalized Maxwell distribution function for polyatomicgases. In [3], the following theorem was also proved: Theorem 2.
If (2.12) is the local equilibrium distribution function with ρ ≡ ρ ( t, x ) , v ≡ v ( t, x ) and T ≡ T ( t, x ) , then the hydrodynamic variables ρ , on-Linear Maximum Entropy Principle v and T satisfy the Euler system: ∂ρ∂t + ∂∂x i ( ρv i ) = 0 ,∂∂t ( ρv j ) + ∂∂x i ( ρv i v j + pδ ij ) = 0 , (2.13) ∂∂t (cid:18) ρε + ρ v (cid:19) + ∂∂x i (cid:26)(cid:18) ρε + ρ v p (cid:19) v i (cid:27) = 0 with p = k B m ρ T, ε = D k B m T. (2.14)This is an important result because we can obtain the Euler equa-tions from the kinetic equation for any kind of polyatomic gases as wellas monatomic gases.
3. The 6 Moment-Equations for Polyatomic Gases
The 14-field theory, N = 2 and M = 1, gives us a complete phenomeno-logical model but its differential system is rather complex and the closureis in any way limited to a theory near equilibrium. Let us consider now asimplified theory with 6 fields (referred to as the ET6 theory): the massdensity ρ , the velocity v , the temperature T , and the dynamic (nonequi-librium) pressure Π. This simplified theory preserves the main physicalproperties of the more complex theory of 14 variables, in particular, whenthe bulk viscosity plays more important role than the shear viscosity andthe heat conductivity do. This situation is observed in many gases suchas rarefied hydrogen gases and carbon dioxide gases at some temperatureranges [24, 25, 5]. ET6 has another advantage to offer us a more affordablehyperbolic partial differential system. In fact, it is the simplest system thattakes into account a dissipation mechanism after the Euler system of perfect ommaso Ruggeri fluids. In the present case we have: ∂F∂t + ∂F k ∂x k = 0 ,∂F i ∂t + ∂F ik ∂x k = 0 ,∂F ll ∂t + ∂F llk ∂x k = P ll , ∂G ll ∂t + ∂G llk ∂x k = 0 (3.1)with FF i F ll = ρρv i ρv + 3( p + Π) = Z R Z ∞ m c i c f I α dI d c (3.2)and G ll = ρv + 2 ρε = Z R Z ∞ m ( c + 2 I/m ) f I α dI d c . (3.3) We want to prove the following theorem:
Theorem 3.
The distribution function that maximizes the entropy (2.6)under the constraints (3.2) (3.3) has the form: f = ρm ( k B T ) α Γ(1 + α ) m πk B T
11 + Π p ! / − α ) Π p ! α exp ( − k B T mC
11 + Π p ! + I − α ) Π p !!) . (3.4) All the moments are convergent provided that − < Π p <
23 (1 + α ) , α > − . (3.5) Proof : The proof of the theorem is accomplished with the use ofthe Lagrange multiplier method. Introducing the vector of the multipliers on-Linear Maximum Entropy Principle ( λ, λ i , λ ll , µ ll ), we define the functional: L = − Z R Z ∞ k B f log f I α dI d c + λ (cid:18) ρ − Z R Z ∞ mf I α dI d c (cid:19) ++ λ i (cid:18) ρv i − Z R Z ∞ mf c i I α dI d c (cid:19) + λ ll (cid:18) ρv + 3( p + Π) − Z R Z ∞ mc f I α dI d c (cid:19) ++ µ ll (cid:18) ρv + 2 ρε − Z R Z ∞ m (cid:18) c + 2 Im (cid:19) f I α dI d c (cid:19) . As this is a functional of the distribution function f and we want to maximizeit with respect to f with the given macroscopic quantities, this functionalcan be substituted by the following one: L = − Z R Z ∞ k B f log f I α dI d c − λ Z R Z ∞ mf I α dI d c −− λ i Z R Z ∞ mf c i I α dI d c − λ ll Z R Z ∞ mc f I α dI d c − (3.6) − µ ll Z R Z ∞ m (cid:18) c + 2 Im (cid:19) f I α dI d c . Since L is a scalar, it must retain the same value in the case of zero hydro-dynamic velocity v = due to the Galilean invariance. Therefore: L = − Z R Z ∞ k B f log f I α dI d C − ˆ λ Z R Z ∞ mf I α dI d C − ˆ λ i Z R Z ∞ mf C i I α dI d C − ˆ λ ll Z R Z ∞ mC f I α dI d C − (3.7)ˆ µ ll Z R Z ∞ m (cid:18) C + 2 Im (cid:19) f I α dI d C . Comparison between (3.6) and (3.7) yields the relations between the La-grange multipliers and the corresponding zero-velocity Lagrange multipliersindicated by hat: λ = ˆ λ − ˆ λ i v i + (ˆ λ ll + ˆ µ ll ) v ; λ i = ˆ λ i − λ ll + ˆ µ ll ) v i ; λ ll = ˆ λ ll µ ll = ˆ µ ll , (3.8)which dictate the velocity dependence of the Lagrange multipliers. We no-tice that these relations are in accordance with the general results of theGalilean invariance [26]. The Euler-Lagrange equation δ L /δf = 0 leads to ommaso Ruggeri the following form of the distribution function: f = exp − − mkB χ , (3.9)where χ = ˆ λ + ˆ λ i C i + ˆ λ ll C + ˆ µ ll (cid:18) C + 2 Im (cid:19) . By introducing the following variables: ξ = mk B (ˆ λ ll + ˆ µ ll ) , η i = mk B ˆ λ i , ζ = 2 k B ˆ µ ll , Ω = exp (cid:18) − − mk B ˆ λ (cid:19) , (3.10)the distribution function can be rewritten as f = Ωe − ζI e − ξC − η i C i . (3.11)Inserting (3.11) into the second equation of (3.2) evaluated at the zero ve-locity, we obtain immediately η i = 0. Then the remaining equations of (3.2)and (3.3) evaluated for v = 0 become ρ = Z R Z ∞ mf I α dI d C = mπ / Γ(1 + α ) Ω ξ / ζ α ,p + Π = 13 Z R Z ∞ mf C I α dI d C = mπ / Γ(1 + α ) Ω2 ξ / ζ α , (3.12) ρε = Z R Z ∞ mf (cid:18) C Im (cid:19) I α dI d C == mπ / Γ(1 + α ) Ω4 ξ / ζ α (cid:18) m (1 + α ) ξζ (cid:19) . From the integrability condition, we have ζ > , ξ > , α > − . (3.13) on-Linear Maximum Entropy Principle From (3.12) and (2.11), we obtain ε = 14 ξ (cid:26) m ( D − ξζ (cid:27) ,p = m D π / Γ (cid:18) D − (cid:19) Ω ξ / ζ D − (cid:26) m ( D − ξζ (cid:27) , Π = m π / Γ (cid:18) D − (cid:19) D − D − m ξζ ξ / ζ D − Ω . (3.14)We can invert these relations as follows: ξ = ρ p
11 + Π p ,ζ = ρm ( D − ρε − p + Π) = ρmp − D − p , Ω = ρmπ / Γ (cid:0) D − (cid:1) ρ p
11 + Π p ! ρmp − D − p ! D − . (3.15)The integrability conditions (3.13) imply that, for a bounded solution,the ratio Π /p must satisfy p + Π > D − p − > . (3.16)Inserting (3.15) into the distribution function (3.11), we obtain (3.4)and the proof is completed. When Π → Substituting (3.4) into the fluxes F llk , G llk and into the production term ommaso Ruggeri P ll of (3.1), we obtain after some calculations F ik = Z R Z ∞ mc i c k f I α dI d c = ρv i v k + ( p + Π) δ ik ,F llk = Z R Z ∞ mc c k f I α dI d c = (cid:0) p + Π) + ρv (cid:1) v k ,G llk = Z R Z ∞ m (cid:18) c + 2 Im (cid:19) c k f I α dI d c = ( ρv + 2 ρε + 2 p + 2Π) v k ,P ll = ˆ P ll = Z R Z ∞ mC Q ( f ) I α dI d C . (3.17)From the balance equations of momentum and of energy in continuummechanics, we know that F ik = ρv i v k − t ik , G llk = ( ρv + 2 ρε ) v k − t ik v i + 2 q k , where t ik = − pδ ik + σ ik is the stress tensor, σ ik = − Π δ ik + σ
D ερ δρ, δ
Π = 43 D ( D − εδρ. We notice that the sound velocity in (3.33) is independent of the degree offreedom D and coincide with the sound velocity of monatomic gas. Thiscurious fact was explained by a general theorem [23] in which was provedthat for particular choice of ( N, M ) systems in which belong the 6 momenttheory the characteristic velocities are independent on D .As only the last component of the production term f of the genericsystem (3.27) is non-zero (see (3.30)), the K-condition (3.28) is satisfied if δ Π = 0. This is true for contact wave and for sound waves. Therefore theK-condition is satisfied and, together with the convexity of the entropy, wecan conclude that, according to the general theorems, the 6-moment systemhas global smooth solutions for all time and the solution converges to theequilibrium one provided that the initial data are sufficiently smooth.
4. Conclusions
In the present paper, we deduced the system of equations for a dissi-pative fluid in which the dissipation is due only to the dynamical pressure.The closure was obtained by the method of the Maximum Entropy Principlewithout assuming that the processes are near equilibrium. This system isthe simplest example of non-linear dissipative fluid after the ideal case ofEuler. The system is symmetric hyperbolic with the convex entropy den-sity and the K-condition is satisfied. Therefore, in contrast with the Euler on-Linear Maximum Entropy Principle 19 case, there exist global smooth solutions provided that the initial data aresufficiently smooth. The result obtained here is in perfect agreement withthe one obtained by using only phenomenological theory of ET [20]. Thecomparison with experimental data in the case of shock waves is excellent[5]. Acknowledgments : This work was supported National Group of MathematicalPhysics GNFM-INdAM and by University of Bologna: FARB 2012 Project
Extended Ther-modynamics of Non-Equilibrium Processes from Macro- to Nano-Scale
The author thanksTakashi Arima for the interesting discussions on the contents of this paper.
References [1] I. M¨uller and T. Ruggeri,
Rational Extended Thermodynamics. 2nd edn.
Springer,New York, (1998).[2] T. Arima, S. Taniguchi, T. Ruggeri, and M. Sugiyama, Extended thermodynamicsof dense gases, Cont. Mech. Thermodyn. , 271 (2012).[3] M. Pavi´c, T. Ruggeri, and S. Simi´c, Maximum entropy principle for rarefied poly-atomic gases, Physica A, , 1302–1317, (2013).[4] S. Taniguchi, T. Arima, T. Ruggeri, and M. Sugiyama, Effect of dynamic pressureon the shock wave structure in a rarefied polyatomic gas, Phys. Fluids, , 016103,(2014).[5] S. Taniguchi, T. Arima, T. Ruggeri, M. Sugiyama, Thermodynamic theory of theshock wave structure in a rarefied polyatomic gas: Beyond the Bethe-Teller theory,Phys. Rev. E, , 013025, (2014).[6] S. Taniguchi, T. Arima, T. Ruggeri, M. Sugiyama, Shock Wave Structure in a Rar-efied Polyatomic Gas Based on Extended Thermodynamics, Acta Appl. Math. , pp.331-407, (1949).[8] I-S. Liu, I. M¨uller, Extended thermodynamics of classical and degenerate ideal gases,Arch. Rat. Mech. Anal. , 285–332, (1983).[9] I-S. Liu, I. M¨uller, T. Ruggeri, Relativistic thermodynamics of gases, Annals ofPhysics, , 191–219 (1986).[10] W. Dreyer, Maximization of the entropy in non-equilibrium, J. Phys. A: Math. Gen., , 6505–6517 (1987).[11] I. M¨uller, T. Ruggeri, Springer Tracts in Natural Philosophy (I edition), Springer-Verlag, New York, (1993).[12] E. T. Jaynes, Information Theory and Statistical Mechanics, Phys. Rev. , 620–630 (1957); Information Theory and Statistical Mechanics II, Phys. Rev., , 171–190, (1957).0 Tommaso Ruggeri [[13] J. N. Kapur, Maximum entropy models in science and engineering , John Wiley, NewYork, (1989).[14] M. N. Kogan, On the principle of maximum entropy, in
Rarefied Gas Dynamics (Vol.I), 359–368 Academic Press, New York, (1967).[15] C. D. Levermore, Moment Closure Hierarchies for Kinetic Theories, J. of StatisticalPhysics , 1021–1065, (1996).[16] G. Boillat, T. Ruggeri, Moment equations in the kinetic theory of gases and wavevelocities, Cont. Mech. Thermodyn. , 205–212, (1997).[17] P. Degond, C. Ringhofer, Quantum moment hydrodynamics and the entropy prin-ciple, J. Stat. Phys. , 587–628, (2003).[18] M. Trovato, L. Reggiani, Maximum entropy principle and hydrodynamic models instatistical mechanics, Riv. Nuovo Cimento Soc. Ital. Fis. , 99–266, (2012).[19] F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Con-tinuum Mech. Thermodyn. , 405–420, (1975).[22] J.-F. Bourgat, L. Desvillettes, P. Le Tallec, B. Perthame, Microreversible collisionsfor polyatomic gases Eur. J. Mech. B/Fluids, , 237–254, (1994).[23] T. Arima, A. Mentrelli, and T. Ruggeri, Molecular Extended Thermodynamics ofRarefied Polyatomic Gases and Wave Velocities for Increasing Number of Moments,Annals of Physics The Mathematical Theory of Non-Uniform Gases,Third Edition , Cambridge University Press, Cambridge, (1970).[25] T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Dispersion relation for soundin rarefied polyatomic gases based on extended thermodynamics, Continuum Mech.Thermodyn. , 727–737, 2013.[26] T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws,Continuum Mech. Thermodyn. EFERENCES Hokkaido Math. J. , 249-275 (1985).[40] B. Hanouzet and R. Natalini, Global existence of smooth solutions for partiallydissipative hyperbolic systems with a convex entropy. Arch. Rational Mech. Anal., , 89-117 (2003).[41] W. A. Yong, Entropy and global existence for hyperbolic balance laws. Arch. Ra-tional Mech. Anal. , 247-266, (2004).[42] J. Lou and T. Ruggeri, Acceleration Waves and Weak Shizuta-Kawashima Condition.Suppl. Rend. Circ. Mat. Palermo, Non Linear Hyperbolic Fields and Waves. A tributeto Guy Boillat,78