Physica 101A (1980) 588-598 © North-Holland Publishing Co.

HYDRODYNAMICAL FLUCTUATIONS IN EXTENDED

IRREVERSIBLE THERMODYNAMICS

D. JOU, J.M. RUBI and J. CASAS-VAZQUEZ

Departament de Termologia, Universitat AutOnoma de Barcelona, Bellaterra, Spain

Received 18 June 1979 Revised 17 September 1979

Starting from the generalized Gibbs equation of extended irreversible thermodynamics, we define a thermodynamic potential that provides a suitable description of the fluctuations of the hydrodynamical dissipative fluxes when it is used in an expression analogous to the classical Einstein formula for the probability of fluctuations. In the limit of vanishing relaxation times, our results coincide with those of Landau-Lifshitz. The effect of the rapid normal modes is taken into account as a stochastic noise in the evolution equations of the dissipative fluxes, and their covariance matrix is found from a fluctuation-dissipation theorem.

I. Introduction

The limitations of the local equilibrium hypothesis have led some authors j- 11) to propose a generalized Gibbs equation which takes into account the

dependence of a non-equilibrium entropy on the dissipative fluxes, besides on the classical thermodynamic variables. For a simple, one-component , isotro- pic fluid, the basic variables are taken to be the velocity v, the specific internal

energy u, the specific volume v and also the heat flux vector q and the viscous pressure tensor pv, which, at difference with classical hydrodynamical

theories, are considered here as independent variablesg). The macroscopic values of these variables at the steady state give only a partial determination

of the state of the system, and it is necessary to consider their fluctuations around the average steady state values in order to have a complete descrip-

tion of the system. We undertake here the problem of describing the fluctua- tions of the dissipative fluxes q and P~ in the presence of an external

temperature gradient and a superimposed velocity gradient. A similar problem in hydrodynamical fluctuations was treated by several

authors, mainly Landau-Lifshi tz 12) and Fox-Uhlenbeck~3), on the basis of Onsager-Machlup hypotheses~4). The aim of these authors was to study the regression of the fluctuations of the classical hydrodynamical variables u, v and v, by assuming that this phenomenon obeys, on the average, the same phenomenological equations that describe the decay of the system towards

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HYDRODYNAMICAL FLUCTUATIONS IN EXTENDED THERMODYNAMICS 589

equilibrium when a constraint is suddenly removed. So, they added to the balance hydrodynamicai equations random terms whose covariance matrix was to be calculated. Since the resulting equations are non-linear, Landau- Lifshitz argued that these random terms were caused by random stresses and heat fluxes, and they obtained their correlations directly from the linear Fourier and Navier-Stokes equations by establishing an analogy between these equations and the typical form of a Langevin equation~2). However, the Fourier and Navier-Stokes laws are in fact not differential equations, and the dissipative fluxes are not the time derivatives of any quantity. Fox and Uhlenbeck~3), by linearizing the hydrodynamicai balance equations, pro- ceeded to calculated the correlations of the random forces in the frame of a general theory of stationary Gaussian-Markov processes~4). When these ran- dom forces are identified as produced by fluctuating heat flux and stresses, Fox-Uhlenbeck's results lead to the Landau-Lifshitz correlation formulae.

Also, a more fundamental point of view has been undertaken by some authors, as for instance Bixon and Zwanzig~5), Fox and UhlenbeckJ6), Logan and Kac~7), Tokuyama and Mori~8), Malek-Mansour, Brenik and Hor- sthemke~9). Indeed, by introducing a stochastic term in the linearized Boltz- mann equation, they can derive in a first-order Chapman-Enskog ap- proximation the heat flux and viscous stresses fluctuations predicted by Landau-Lifshitz. Peraita-Fabi and Garc/a-Colin 2°) have obtained recently the short-time behaviour of the random force autocorrelation functions for clas- sical many-body systems by using Mori's generalized Langevin scheme as a starting point.

A different approach to this problem has been proposed by Keizer2~-23), from the viewpoint of elementary molecular processes. In his approach, the dissipative contributions of heat flux and shear and bulk viscosity are written in terms of elementary processes. The fluctuations are described by a generalized fluctuation-dissipation theorem which relates random deviations from the average to the rates of the elementary dissipative processes. This allows to construct a non-linear fluctuation theory which, in the specific case of linearized regression equations near equilibrium, reduces to the Fox- Uhlenbeck's theory.

Recently, we proposed another treatment 24) for the description of the heat flux fluctuations in a rigid heat conductor. Here, it is developed in detail and extended to the fluctuations of the viscous pressure tensor in a one com- ponent isotropic fluid system. In section 2, we review the fundamentals of extended irreversible thermodynamics and we obtain the iinearized evolution equations describing the relaxation of the heat flux and of the viscous pressure. This leads to the identification of the new parameters appearing in the generalized Gibbs equation. In section 3, the Einstein relation giving the

590 D. JOU et al.

probability of a fluctuation is assumed to be valid with the generalized non-equilibrium entropy playing the role of the classical entropy. This is used to obtain the second moments of the fluctuations of the dissipative fluxes. In the limit case when the relaxation time vanishes, we recover the well-known results of Landau-Lifshi tz . In section 4, we add a random noise to the evolution equations of the dissipative fluxes, which become in this way Langevin-type equations. By assuming the validity of a generalized fluctua- t ion-dissipation theorem, we obtain the covariance matrix of the stochastic noise, which describes the effects of the rapid normal modes on the heat flux and the viscous pressure.

2. Review of extended irreversible thermodynamics

We assume that the state of a simple, one component , isotropic fluid is determined by the following set of independent variables: the specific internal energy u, the specific volume v, the velocity v, the heat flux ¢ and the viscous pressure tensor pv. In order to have a complete description of the system, we need evolution equations for these variables, and suitable initial and boundary conditions. The evolution of the classical variables v, v and u is given by the well-known balance equations of mass, momentum and energy, which are respectively

Ob = V • v, (2.1)

oi~ = - V , P + p F , (2.2)

pti = - V. q - P : V + or, (2.3)

where P = p U + I aV is the total pressure tensor, which is the sum of the scalar thermodynamic pressure p plus the viscous pressure tensor pv; F and r are respectively the body force per unit mass and the energy supply per unit mass, which describe the long range interaction between the system and the external world. In this paper we assume for the sake of simplicity that both F and r vanish identically, and also that pv is symmetric, because the fluid has no internal structure. As usual, V is the symmetric part of the velocity gradient and p is the mass density.

While these evolution equations are valid for every system, the evolution equations for the dissipative fluxes depend in a direct way on the particular system under consideration. The corresponding evolution equations for q and pv can be obtained in a thermodynamic f ramework similar to that of classical irreversible the rmodynamics~) . In this case, the specific nonequilibrium entropy s is assumed to be a function of u, v, q and pv, whose differential is given by the generalized Gibbs equation

H Y D R O D Y N A M I C A L F L U C T U A T I O N S IN E X T E N D E D T H E R M O D Y N A M I C S 591

ds = T -j du + T f p d v + T -I va l q " dq + T -j vao p v dp v + T - l vc~2 /~v : d p v,

(2.4)

where we have taken into account the usual splitting of pv in a scalar viscous pressure pV and a traceless tensor ~v. The absolute temperature T, the pressure p and the functions ai, which will be identified below, are given as equations of state by the corresponding partial derivatives of the entropy. In a first approximation, they take the respective forms

T -~ =- ( a s / a u ) v . , . r . ~ v = T ; " (u, v),

T Zp _(Os/av), ,q.pV, t~v= Telqpeq(u, v),

T -~ OOLlq ~ (cgs/Oq) . . . . pV, ~v, (2.5)

T -I vaop v = (Os/Op v) .... q. t~v,

where Teq and p,q stand for the local equilibrium values of T and p, respectively.

From here on, the development is standard: the balance equations (2.1) and (2.3) are introduced into the time derivative of s obtained from (2.4); the entropy flux is assumed to be given by

J, = T-I q + fl0P~ q +/3, P~" q (2.6)

with the/3i given as phenomenological coefficients, and the time derivative of the entropy is written in the standard form of a balance equation

p~ + V. J~ = cr (2.7)

which leads to a definite expression for the entropy production or. The second law of thermodynamics imposes that ~r has to be definite positive. This states on the evolution equations of the dissipative fluxes some restrictions that lead finally to the expressions s)

(1 = T a ~l{txloq + tzl lP ~ q + tx12 I ~ " q -- V T-~ - fl 1V" ~ v

-/30 Vp ~ - ~ p V V/30 - ~ I $~- V/3,}, (2.8)

1~ ~ = T a o' {/Xo, p~ - f~ll q: - ~ q " V/3o -/30 V. q + T-i V. v}, (2.9)

(~v ) " = Tot2,{/z20U + / x 2 , ~ v _ tz,2(qq)S + T - ' (/

- ~(V/3~qy - /3 , (Vqy}, (2.10)

where the /~i are positive phenomenological coefficients. For the sake of simplicity we limit ourselves in this paper to a linear theory, and we assume therefore that the coefficients/30,/3~,/z~ and/xz2 are vanishingly small. In this approximation, the evolution equations of the dissipative fluxes reduce to

592 D. JOU et al.

4 = (oq T A ) - I ( q + AVT),

,o ~ = (ao~r) -~ (pV + ~'V. v),

(#~)" = (2c~2/x)-'(0 v + 2/xV).

(2.11)

(2.12)

(2.13)

q = - A V T ,

p V = - s r V . v ,

# = -

Therefore , we can identify the coefficients ai of (2.4) as

ctl = - "c J TA ; ao=-7o /~ ' ; a2=- 'r2/2/a, ,

In writing these equations, the coefficients /z~0, /x0~ and /x2~ have been identified as/zl0 = (T2A) -j, P~01 = (T~r) -I and/x21 = (2Tpt) -t, where A, ~" and/z are the usual heat conductivity, bulk viscosity and shear viscosity, respectively. Eqs. (2.11), (2.12) and (2.13) are of the kind of Maxweli-Cat taneo equation and describe the relaxation of the dissipative fluxes to their classical linear values

(2.14)

(2.15)

(2.16)

(2.17)

where the r~s are the corresponding relaxation times, which can be measured in principle f rom the velocity of propagation of thermal and viscous waves.

In view of these identifications, the Gibbs equation (2.4) can be written in the form

d s = T -I d u + T -I p d v - ( ' r l v / A T 2 ) q • d q - ( r o v / ~ T ) p v d p ~

- ( r 2 v / 2 l x T ) # v : d # v. (2.18)

This equation may be interpreted as a definition or construction of a non- equilibrium entropy which, as the classical entropy, is not directly mensurable but which can be assigned a value for each state of the system by integration of (2.18). In particular, in the case when AT 2, ~'T,/zT and the relaxation times are constant, this integration leads to

s = Seq(U, v ) - ( r t v / 2 A T 2 ) q 2 - ( r o V / 2 ~ T ) p v 2 - (~-2 v / 4 1 z T ) #v : #v, (2.19)

where Seq is the local equilibrium entropy. The general form of the entropy production or obtained from (2.7) is

o, = XI • q + X0 p v + X2 : # L (2.20)

where

X1 =/£10 q "l- /U, II pv q + /~12 #v .q, (2.21)

Xo = ]-to0 +/£01 pv +/./,02 pv2 +//,03 q2 +/Zoa #v : #v, (2.22)

~2 = Ix20 U + P-21 #v +/z22 #v2 + tz23 (qq)% (2.23)

HYDRODYNAMICAL FLUCTUATIONS IN EXTENDED THERMODYNAMICS 593

Since we limit ourselves to linear constitutive relations, Xj, X0 and )G2 reduce

to

Xj --/xl0 q; X0 =/x01 pV; )(2 =/x21/~v. (2.24)

Taking into account the above identifications /~10 = (T2A) -~, /x0~ = (Ts r) i and ~z2~ = (2T~)-1~ (2.20) takes the form

tr = (T2A) -~ q2 + (T~,):-i pV2 + (2T~)-1/~v :/~v = 0-t + ~0 + try, (2.25)

where we have separed the independent terms or1, tr0 and tr2 respectively associated to the heat flux, the scalar viscous pressure and the traceless viscous pressure tensor.

When we compare (2.19) and (2.25) we can write for the generalized entropy the expression

s = seq - ('rl v /2 )o '1 - (r0v/2)~r0 - (r2v/2)tr2. (2.26)

In the hypotheses of the present paper, i.e., with constant relaxation times and constant phenomenoiogicai coetticients A T z, ~'T and/xT, the theorem of the minimum entropy production of Prigogine 2s) states that, with linear phenomenological laws and stationary boundary conditions, the entropy production has its minimum value at the steady state compatible with the prescribed conditions. Therefore , the difference s - s , q of (2.26) will be a maximum in the steady state, and it will take the value

(s - - S e q ) s t ~--- - - ( r l v / 2 ) ( A V T ) 2 - ( z o V / 2 ) ( ~ V " v ) 2 - (r2v/2)4/x z I~' : If. (2.27)

In the next section, we take advantage of this property of the difference s - Seq to describe the fluctuations of the dissipative fluxes q, p and ~6v around their steady state values (2.14)-(2.16).

3. Fluctuations of dissipative fluxes

Einstein calculated the second moments of the fluctuations of extensive parameters in thermodynamic equilibrium systems starting from two ap- proximations. The first one, based on the Boltzmann hypothesis for the entropy of a given macrostate, is the identification of the probability of a fluctuation as

Pr -- e x p ( A S / k ) , (3.1)

where k is the Boltzmann constant and A S is the change of the entropy, starting from the equilibrium, associated with the fluctuation.

The second hypothesis of Einstein is to expand the entropy around its value

594 D. JOU et al.

at a state of reference

A S = S - Se = (~S)e + (1/2)(82S)~ (3.2)

and neglecting higher-order terms. Since for an isolated system at equilibrium (8S)~ = 0, the probability density reduces to

Pr - exp{½(82S)Jk} . (3.3)

As it is well known, this Gaussian form of the probability density leads in a direct way to the correct expressions of the second-order moments of the fluctuations, but is inadequate for higher moments26). With this restriction in mind, the validity of the Einstein formula for small fluctuations from equili- brium states is well founded.

Recently, a different approach to thermodynamics far from equilibrium has been developed27-3°), based on the hypothesis of the validity of a relation as (3.3). The knowledge of fluctuations allows to define a generalized ther- modynamic potential (I) such that the probability distribution W of fluctua- tions away from the steady state is expressed as W - e x p ( - c b / k ) . In this way, the fluctuations determine the state functions rather than the state functions determining the fluctuations.

Here, on the contrary, it is our aim to show that the function p ( s - s J , defined by (2.26) and which has a maximum in the steady state of the system, provides a suitable description of the fluctuations of the dissipative fluxes, when it is used in a formula of the kind of (3.3). Indeed, when the second derivatives of this function as obtained from (2.26) are introduced into

Pr(Sq, 8p v, 8~ v) - exp{ ( l12k )82(ps - pSeq)} (3.4)

we obtain in the Gaussian approximation

Pr(Sq, 8I) v, 81 ~ ) ~ exp{[ - (zl/A T2)( Sq ) 2 - ( 7o/ ~T) ( Sp V) 2

- (rJ2tzT)(8~ ~ : 8~V)l12k}. (3.5)

This expression leads for the second moments of the fluctuations of the dissipative fluxes to the formulae

(Sqi(r , t )Bqj(r , t )) = kXT2"rll Sij,

(Sp~(r, t),~pV(r, t ) ) = k~TT~ I,

(SP~k(r, t)SPTm(r, t )) = kl~T~l(SUSkm + 8i~Skt), (3.6)

(SpVSqi) = ( ~p ~ ~P ~) = ( ,Sqi,~IS [t) = O.

We do not deal here with the problem of the spatial behaviour of the correlation functions of the dissipative fluxes, but we assume for simplicity that we are far from any critical point and that the fluctuations are delta

H Y D R O D Y N A M I C A L F L U C T U A T I O N S IN E X T E N D E D T H E R M O D Y N A M I C S 595

correlated in space. This too simple behaviour has been disscussed by some authors 2°) which show that, at least for very short times, the spatial depen- dence shows a definite non-local behaviour of the autocorrelat ion functions. However , we take this simple dependence in order to compare with the classical results of Landau and Lifshitz. Also, the problem of fluctuations near a critical point, which has a great interest 3~) is beyond the scope of this

paper. We specialize here to a consideration of the time dependence of the

correlation functions of the fluctuations of the dissipative fluxes. In order to make precise some ideas, we take first as an example the problem of heat fluctuations. The fluctuations of q around its steady state value q = - A V T s, where T s is the steady distribution of temperature, are of the form

( t~q) lo t = - A V S T + 6q, (3.7)

where 8 T is the temperature fluctuation associated with the fluctuation 6u of the internal energy, i.e. 6 T = c -~ Bu where c is the specific heat at constant volume. We assume that the time decay of 8u (or of S T ) is much longer than the characteristic time of decay of random independent heat fluctuations 8q.

Obviously, since ( 6 u 6 q ) = 0, the heat fluctuations accounted for in (3.4) are the latter random independent fluctuations of q. Therefore , when we speak of heat fluctuations we refer to random heat fluctuations around the in- stantaneous value q = - )tV(TS + 6 T ) , which can be considered as a constant during the decay of a random heat fluctuation.

With this remark in mind, the regression of the heat fluctuations, and of the viscous fluctuations, is described by

( ~ q i ) " = -- T I t ~q i ,

(SPY)" = - Zo ~ 8p ~, (3.8)

(~,~) = _ ,~, ~ .

As a consequence, the time correlation of the fluctuations of the dissipative fluxes in a stationary system is given by 3z)

{6qi(r, t )6qj(r , t + t')} = k A T 2 "rl' 6ii exp( - I t ' l ¢; ') ,

{6p~(r, t )6pV(r, t + t ')) = k (TTo ' exp( - I t ' l ¢o'), (3.9)

(6P~k(r, t)B/5~m(r, t + t')} = k~Tz~'(8i~Sk,. + 6i,.Skt)exp( - I t ' l ¢2-').

In the limit when the relaxation times ~'0, r~ and r2 tend to zero, expressions (3.9) reduce to the simple form

{Sq~(r, t )Sqi (r , t + t ')) = 2 k A T z 8~ i 8( t ' ) ,

{6pV(r, t )Sp~(r, t + t')} = 2 k ( T 6 ( t ' ) , (3.10)

(SPTk(r, t)815~,.(r, t + t ')) = 2k l zT(6u 6k,. + 8~m 6kDS(t').

596 D. JOU et al.

These results are in accordance with those of Landau-Lifshi tz ~2) and Fox-Uhlenbeck~3), who considered a priori vanishing relaxation time ~0, I-~ and ~'2. However, it is known that this hypothesis leads to instantaneous propagation of thermal and viscous signals Hi) and therefore it can only be considered as an approximation.

4. Langevin equations for dissipative fluxes

The evolution of the fluctuations of density, specific internal energy and velocity from their equilibrium values can be described in the form of a Langevin equation ~3)

d a i / d t = - Gi ja j + f i ( t ) , (4.1)

where ai is the deviation of the corresponding extensive variable ni away from its equilibrium value, G o is the phenomenological rate matrix and fi is a white stochastic noise. The covariance matrix of the random forces fl con- sistent with the Boltzmann-Planck equilibrium distribution of n,- is ~3)

( f~ ( t ) f i ( t + t ' ) ) = (Gik E k / + E ~ ~ Gik)tS(t ' ) , (4.2)

where

Ejk =- -- k - ~ ( 02 S / Oni Onk )eq, (4.3)

where eq represents evaluation at equilibrium. Once the covariance matrix (4.2) is found, the fluctuation theory is completely defined when the stochastic noise is stationary, Gaussian and Markoffian.

Our aim in this section is to find the covariance matrix of the stochastic noise consistent with the fluctuations (3.9) of the dissipative fluxes. We add stochastic noises to the evolution equations (3.8), which become

(Sq~) " = - "r~ I 8qi + f l i ( t ) ,

(6PV)" = - %' 8P v + f0 (t), (4.4)

(6/5~) = _ ~'2' 616~ + f2,~ (t).

Comparing (4.4) with the general form (4.1) and assuming that the fluctua- tion-dissipation theorem (4.2) also applies in the present situation, we obtain for the covariance matrix of the f ' s

( f t i ( t ) f l i ( t + t ' ) ) = 2kAT 2 1"] -2 6, 8(t'),

( f o ( t ) f o ( t + t ' ) ) = 2 k ~ T T o 2 6 ( t ' ) , (4.5)

( f 2ik ( t ) f 2, ,( t + t')) = 2kl~ T~ ~2( ,Sil,Skm + 6im'SU ),5( t ' ) ,

< / J o ) = (f,Y~k~) = (lo/~k~) = O,

HYDRODYNAMICAL FLUCTUATIONS IN EXTENDED THERMODYNAMICS 597

where we have assumed that the relaxation times of the random noises are

negligible compared to the r~s.

5. Conclusions

When a fluid sys tem is per turbed away f rom its equilibrium state, the per turbat ion can be decomposed in a large number of normal modes. Most of these modes decay in a very short time, so that the per turbed sys tem can be usually treated with reference to only five normal modes , corresponding to the usual five hydrodynamical variables v, u and v. Here , we go beyond the classical descript ion and take into account nine additional normal modes , corresponding to the independent variables q, p V and PV. The deve lopment presented here can therefore be considered as a second step in the stablish- ment of a hierarchy of hydrodynamical variables, based on the different orders of magnitude of their corresponding relaxation times.

In the classical theory, q, pv and Ov are not considered as independent variables, but they are given in terms of spatial derivat ives of v, u and v by means of the classical consti tutive eqs. (2.14)-(2.16), and their relaxation times are assumed to vanish. However , in a more detailed analysis, the dissipative fluxes are allowed to fluctuate around their values (2.14)-(2.16), and these fluctuations act as a stochastic noise in the equations describing the regression of the fluctuations of the classical variables u, v and v. Their second moments are obtained via the f luctuation-dissipation theorem (4.2).

Here , both u, v, v and q, p V and ~v are considered independent variables, and their respect ive fluctuations are described by the non-equilibrium ent ropy (2.4) via the Einstein relation (3.3). To complete the description of the fluctuations, we must take into account the remaining rapid normal modes of the system, which act as a stochastic noise in the evolution equat ion of the intermediate normal modes q, p V and Or. In analogy with the classical

description, the covar iance matrix of this random noise is obtained via the f luctuation-dissipat ion theorem (4.2)

Acknowledgements

We want to thank Prof. Dr. G. Lebon, of Liege Universi ty, for interesting commen t s and valuable suggestions.

References

1) 1. Muller, Z. fiir Phys. 198 (1967) 329. 2) J. Lambermont and G. Lebon, Phys. Lett. 42A (1973) 499. 3) G. Lebon and J. Lambermont, J. M6c. 15 (1976) 579.

598 D. JOU et al.

4) G. Lebon, Bull. Acad. Roy. Belgique, Clas. Sci. LXIV (1978) 456. 5) G. Lebon and J. Casas-Vfizquez, Phys. Lett. 55A (1976) 393. 6) G. Lebon, D. Jou and J. Casas-Vfizquez, J. of Phys. A 13 (1980) 275. 7) D. Jou, J.M. Rubi and J. Casas-Vfizquez, J. of Phys A 12 (1979) 2515. 8) D. Jou, J. Casas-Vfizquez and G. Lebon, J. Non-Equilib. Thermodyn. 4 (1979) 349. 9) !. Gyarmati, J. Non-Equilib. Thermodyn. 2 (1977) 233.

10) W. Israel, Ann. Physics 100 (1976) 310. I1) M. Kranys, Nuov. Cim. 8B (1972) 417. 12) L. Landau et E. Lifshitz, M6canique des Fluides (MIR, Moscou, 1971) p. 656. 13) R.F. Fox and G.E. Uhlenbeck, Phys. Fluids 13 (1970) 1893. 14) L. Onsager and S. Machlup, Phys. Rev. 91 (1953) 1505, 1512. 15) M. Bixon and R. Zwanzig, Phys. Rev. 187 (1969) 267. 16) R.F. Fox and G.E. Uhlenbeck, Phys. Fluids 13 (1970) 2881. 17) J. Logan and M. Kac, Phys. Rev. 13A (1976) 458. 18) M. Tokuyama and H. Mori, Prog. Theoret. Phys. 56 (1976) 1073. 19) A. Malek-Mansour, L. Brenig and W. Horsthemke, Physica 88A (1977) 407. 20) R. Peralta-Fabi and L.S. Garc/a-Col/n, Phys. Lett. 62A (1977) 7. 21) J. Keizer, J. Chem. Phys. 63 (1975) 398. 22) J. Keizer, J. Chem. Phys. 64 (1976) 1679. 23) J. Keizer, Phys. Fluids 21 (1978) 198. 24) D. Jou and J.M. Rub/, Phys. Lett. 72A (1979) 78. 25) P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluc-

tuations (Wiley, New York, 1971) p. 34. 26) H.B. Callen in Non-Equilibrium Thermodynamics, Variational Techniques and Stability,

Donnelly, Hermann, Prigogine, eds. (Univ. of Chicago Press, Chicago, 1966). 27) R. Graham, Phys. Rev. Lett. 31 (1978) 1479. 28) R. Graham, Phys. Rev. 10A (1974) 1762. 29) J. Keizer, J. Chem. Phys. 65 (1976) 4431. 30) F. Jahnig and P.M. Richter, J. Chem. Phys. 64 (1976) 4645. 31) V. Zaitsev and M. Shliomis, Sov. Phys. JETP 32 (1971) 866. 32) L. Landau et E. Lifshitz, Physique Statistique (MIR, Moscou, 1967) p. 451.