Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 60

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And all productions vanish in equilibrium.In order to obtain field equations for the densities uĮ, the balanceequations must be supplemented by constitutive equations. Such constitutive equations relate the fluxes FĮa and the productions ȆĮ to the densities in a materially dependent manner. In extended thermodynamics theconstitutive relations have the forms(D C(ˆD C (W E )and3D3ˆ D (W E )so that the fluxes FĮa and the productions ȆĮ at a point and a time dependonly on the densities uĮ at that point and time.

We may say that theconstitutive equations are local in space-time.59CIf the constitutive functions (ˆD and 3̂ D were known explicitly, wecould eliminate FĮa and ȆĮ from the equations of balance and obtain explicitfield equations for the uĮ’s. They form a quasi-linear system of partialdifferential equations of first order. Every solution of this system is called athermodynamic process.59Thus no gradients or time derivatives do occur among the variables in the constitutiveequations. In particular, there is no temperature gradient, and yet heat conduction isaccounted for, because the heat flux is counted among the fields .2568 Thermodynamics of Irreversible ProcessesSymmetric Hyperbolic SystemsIn reality, of course, the constitutive functions are not known and it is thetask of the constitutive theory to determine those functions or, at least, toreduce their generality.

The tools of the constitutive theory are certainuniversal physical principles which represent expectations based on longexperience. The main principles arex entropy inequality, x requirement of concavity, x principle of relativity.The first two of these represent the entropy principle and, in particular,the second one guarantees thermodynamic stability and hyperbolicity of thefield equations.The entropy inequality is an additional balance law. We writewh wh awt wxa∑ t0for all thermodynamic processes.h is the entropy density, and ha is the entropy flux.

∑ is the entropyproduction. All three are constitutive quantities so that in extendedthermodynamics we havehhˆ(uE ), h ahˆ a (uE ) and ∑∑ (uE ).The requirement of concavity demands that h be a concave functionof uĮ:w 2JwW D wW Enegative definite.The principle of relativity states that the field equations and the entropyinequality have the same form in all Galilei frames.60The key to the exploitation of the entropy inequality lies in theobservation that the inequality must hold for thermodynamic processes, i.e.solutions of the field equations.

In a manner of speaking the field equationsprovide constraints for the fields that must satisfy the entropy inequality. A60In relativistic thermodynamics we require invariance of the equations under Lorentztransfomations, but this is not a subject of this book, although relativistic thermodynamicsis an interesting application of extended thermodynamics.

See: I-Shih Liu, I. Müller, T.Ruggeri: “Relativistic thermodynamics of gases.” Annals of Physics 100 (1986). Also: I.Müller, T. Ruggeri: “Rational Extended Thermodynamics.” Springer, New York (1998)2nd edition.Extended Thermodynamics257lemma by Liu61 proves that it is possible to use Lagrange multipliers ȁĮ –functions of uĮ – to eliminate such constraints. Indeed, the new inequalityȘuα ˜Fα aؘh ˜ h a Λα É ȆαÙ • 0 must hold for all fields uα (xi ,t).˜t ˜xa˜xaÊ ˜tÚThis impliesdh/D d uD ,d hC/D dFD C ,and/D 3 D t 0 ,so that in equilibrium all but the first five Lagrange multipliers vanish. Theresidual inequality ȁĮȆĮ t 0 represents the entropy source or dissipation.In order to appreciate the mathematical structure of the system of fieldequations we change variables from the densities uĮ to the Lagrange multipliers ȁĮ and obtain for the scalar and vector potentials hƍ = – h + ȁĮ uĮ andhaƍ = – ha + ȁĮ FĮawh cw/DuD ,w h cCw/D(D Cso that the field equations readw 2 h c w/Ew 2 hcC w/Ew/D w/E wVw/D w/E wZ C3D(D1,2,...P ).All four matrices in this system are symmetric and the first one isnegative definite.62 Therefore the system of field equations – written interms of the Lagrange multipliers – is a symmetric hyperbolic system.Hyperbolicity guarantees finite speed of propagation and symmetrichyperbolic systems have convenient and desirable mathematical properties,namely well-posedness of Cauchy problems, i.e.

existence, uniqueness andcontinuous dependence on the data. The desire for finite speeds ofpropagation was the primary original incentive for the formulation of extended thermodynamics, see below. There are n speeds of propagation andthey may be calculated from the characteristic equation of the system offield equations, viz.61I-Shih Liu: “Method of Lagrange multipliers for the exploitation of the entropy principle.”Archive for Rational Mechanics and Analysis 46 (1972).62This follows from the concavity of the entropy density in terms of the densitiesuĮ , since hƍ= – h + ȁĮ uĮ defines a Legendre transformation associated with the mapȁĮ ļ uĮ..2588 Thermodynamics of Irreversible ProcessesÈ ˜2h„ؘ2h„aVna Ùdet ɘ ȁα ˜ ȁ β ÚÊ ˜ ȁα ˜ ȁ β0.na and V denote direction and speed of propagation.

Obviously, before anyvalues for wave speeds can actually be calculated, the synthetic character ofthe equations of this section must be replaced by more concrete relations sothat hƍ(ȁȕ) and hƍ a(ȁȕ) can be identified. The most immediate concretizationof the present formal framework is provided by extended thermodynamicsof moments, see below.Growth and Decay of WavesNon-linear hyperbolic equations tend to evolve discontinuities in the fields,even if the initial data are smooth. On the other hand, steep gradientsinvolve strong dissipation with a tendency to smooth out the solution. Thusthere exists a competition between non-linearity and dissipation which maylead to smooth solutions for all times.

This is important for a system of fieldequations to be realistic, since most phenomena that occur in the real world,are smooth. After all: Natura non fecit saltus.63An instructive example for the competition of non-linearity anddissipation is the growth and decay of acceleration waves, i.e. movingsingular surfaces along which uĮ(x,t) (Į = 1,2...n) are continuous, but theirgradients are not. As one moves with the wave, its amplitude A –representing the jumps of the gradients – obeys a Bernoulli equation,provided that the wave moves into an area of a homogeneous and timeindependent equilibrium 64w3 DG# w8F E # 2 NDFJ #GV wW EwW EC630.

DAccording to Aristoteles: “Historia animalium.” Aristoteles said it in Greek, of course, andin quite a different context. The familiar quotation is often used in connection with thesteep, but smooth structure of shock waves.64 An excellent review of waves – in particular acceleration waves – is given by P. Chen:“Growth and decay of waves in solids. Mechanics of Solids III” Handbuch der Physik6A/3 Springer, Heidelberg (1973).I believe that the first person to calculate the rate of change of the amplitude A(t) of anacceleration wave was W.A.

Green: “The growth of plane discontinuities propagating intoa homogeneous deformed material.” Archive for Rational Mechanics and Analysis 16(1964).The present compact form of the Bernoulli equation – with right and left eigenvectors – isdue to G. Boillat: “La propagation des ondes.” Gauthier-Villars, Paris (1965).Extended Thermodynamics259V is a characteristic speed and lĮ and dĮ are the left and right eigenvaluesof the matrix˜Fα1˜uβin the one-dimensional field equations1wuD wFDwtwx1ȆD( D 1,2,...n ) .The solution of the Bernoulli equation reads# (0)G DV1 # (0) DC (G DV 1)# (V )so that A(t) remains finite unless the initial amplitude A(0) is large.In general – for arbitrary solutions instead of merely acceleration waves –the condition for smooth solutions is not decisively known.

There exists asufficient condition for smoothness65 which, however, is not necessary.Characteristic Speeds in Monatomic GasesWe recall the generic equations of transfer in the kinetic theory of gases, cf.Chap. 4, and apply this to a polynomial in velocity components by setting\ PEK1EK2 ...EKN . In this manner we obtain equations of balance for³ µc cmoments ui1i2 ...ili1 i2...cil fdc of the distribution function f whichreadwW K1K2 ...KNwVwW K1K2 ...KNC3 K K ...KNwZ C1 2(N0,1,2...0 ) .Since each index may assume the values 1,2,3, there aren = 1/6 (N + 1)(N + 2)(N + 3)equations. These equations fit into the formal framework of extendedthermodynamics, see above, but they are simpler. Indeed, on the left handside there is only one flux, namely WK K ...K C – the last one – which is not1 2Nexplicitly related to the fields WK K ...K (l = 1,..N).1 265NS.

Kawashima: “Large-time behaviour of solutions to hyperbolic-parabolic systems ofconservation laws and applications.” Proceedings of the Royal Society of Edinburgh A106 (1987).2608 Thermodynamics of Irreversible ProcessesTherefore the results of the previous sections may be carried over to thepresent case, in particular the exploitation of the entropy inequality. Thatinequality reads according to the kinetic theory of gases, cf. Chap. 4˜Èf˜ ÈfØØdc Ù dc Ù • 0 .ÉÊ k Ô f lnÉÊ k Ô ca f lnÚeYeY ژt˜xaThe exploitation makes use of the Lagrange multipliers /K1K2 ...KN(l = 1,2,…N ) and the moment character of the densities and fluxes impliesthat the distribution function has the formfNY exp 1k Ç l 0 Λ i1i2 ...il µci1 ci2 ...cilso that the scalar and vector potentials may be written asNh„ kY Ô exp 1k Ç l 0 Λ i1i2 ...il µci1 ci2 ...cil dc andh „a kY Ô ca exp k1 Ç l 0 Λ i1i2 ...il µci1 ci2 ...cil dc .NInsertion into the characteristic equation for the calculation of wavespeeds givesdet Ô (c na a V ) ci1 ..cil c j1 ..c jn f equ dc0provided that the wave propagates into a region of equilibrium.

fequ is theMaxwell distribution, cf. Chap. 4.Thus the calculation of characteristic speeds and, in particular, themaximal one, the pulse speed requires no more than simple quadratures andthe solution of an nth order algebraic equation. It is true that the dimensionof the determinant increases rapidly with N: For N = 10 we have 286columns and rows, while for N = 43 we have 15180 of them. But then, thecalculation of the elements of the determinant and the determination of Vmaxmay be programmed into the computer and Wolf Weiss (1956– ) has thevalues ready for any reasonable N at the touch of a button, see Fig. 8.6. Werecognize that the pulse speed goes up with increasing N and it neverExtended Thermodynamics261Fig.

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