fluctuation theory and extended irreversible thermodynamics

Physica A 155 (1989) 221-231

North-Holland, Amsterdam

FLUCTUATION THEORY AND EXTENDED IRREVERSIBLE

THERMODYNAMICS

David JOU Departament de Fisica, Fisica Estadistica, Universitat Autbnoma de Barcelona, 08193 Bellaterra Catalonia, Spain

Received 4 February 1988

Revised manuscript received 13 September 1988

The moments of the fluctuations of dissipative fluxes are used to obtain explicit expressions

for the coefficients of a generalized entropy which depends on heat flux and viscous pressure.

The results are applied to quantum ideal gases and to classical dilute nonideal gases.

1. Introduction

‘The usual hydrodynamic theory is known to fail for high-frequency phenomena, as can be easily understood, because it is based on phenomenological transport laws (Fourier’s law for heat conduction, Newton- Stokes law for viscous pressure) which are only an approximate form useful for slow phenomena. Relaxation terms must be included in the transport equations in order to account for relatively fast phenomena, of a frequency of the order of the inverse of the collision time of the system. The search for a generalized version of irreversible thermodynamics able to deal with such relaxational transport equations has led some authors [l-5] to introduce the dissipative fluxes (heat flux, viscous pressure tensor, electric current) as independent variables of the macroscopic theory.

In such a theory, usually known as extended irreversible thermodynamics, the dissipative fluxes appear in the entropy and in the entropy flux, which take therefore a form different than that used in the local-equilibrium hypothesis [6--71. For a simple, unicomponent fluid, the dissipative fluxes to be taken into account are the heat flux q, the symmetric traceless viscous pressure tensor @” and the bulk viscous pressure p”. In this case, the generalized entropy density s per unit mass and the nonconvective part of the entropy flux J, have the form

[l-51

0378-4371/ 89/ $03.50 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)

222 Il. Jou I Fluctuation theory and extended thermodynamics

s = s&l, P) - (a,/2)q * q - (a,,/2)pV2 - (a2/2)BW (1.1)

and

J,= T~‘q+&p”q+/32~V.q. (1.2)

Here, s,~(u, p) is the local-equilibrium entropy per unit mass, which is a

function of the specific internal energy u per unit mass and of the mass density

P. The quantities (pi, cr(, and a2 may be given an explicit physical meaning in

terms of transport coefficients and of the relaxation times of the fluxes. It can

be easily shown [3,8,9] that

(Y1 = T,(pAT2)- ’ ) % = dPlT)F > a?_ = 72(2p77T)-’ 1 (1.3)

where h, 6 and rl are the heat conductivity, bulk viscosity and shear viscosity,

respectively, and 7,) T(, and 72 are the respective relaxation times of the fluxes

as defined through the Maxwell-Cattaneo equations [l-5,8,9]

r,i+q=-AVT, (1.4)

(1.5)

T2(FV). + 9 = -2779. (1.6)

The upper dot means the material time derivative, u is the baricentric local

velocity of the fluid and I? is the symmetric traceless part of the velocity

gradient.

Since the entropy plays a fundamental role in the thermodynamic theory, it

is very important to have procedures to obtain explicit expressions for (Y,, a,,

and (Ye, p, and & as functions of u and p. In fact, the equations for a,, a0 and

a2 play in the generalized theory a role analogous to the equations of state

T(u, p) and p(u, p) of the classical theory, which they complement in the

extended theory, since they are also obtained as partial derivatives of the

entropy.

One may proceed at three levels. A purely macroscopic method has been

devised by Liu and Muller [lo, 111, which is able to give expressions for these

coefficients for whose explicit calculation only the equilibrium equations of

state T(u, p) and p(u, p) are required. A second level of complexity is

provided by fluctuation theory [3,8,9,12-141, which relates these coefficients

to equilibrium fluctuations of the fluxes, so that the equilibrium distribution

D. Jou I Fluctuation theory and extended thermodynamics 223

function of the system is enough for an evaluation of the coefficients. A third

level of complexity requires the non-equilibrium distribution function of the

system, as obtained either from information theory [15] or by kinetic theory

[16].

‘The purpose of this paper is to clarify several fundamental points of the

application of fluctuation theory to extended thermodynamics and to enlarge

its field of applications. In section 2, we obtain the macroscopic expressions for

the coefficients of the entropy in terms of fluctuations of the fluxes and we

calculate them on a microscopic basis for quantum ideal gases and for classical

nonideal dilute gases. In the concluding remarks of section 3 we give the

corresponding results for the coefficients of the entropy flux.

2. Entropy coefficients and fluctuations of the fluxes

In this section we relate the coefficients appearing in the nonclassical part of

the entropy as expressed by (1.1) with the second moments of the fluctuations

of the fluxes. Here we neglect the bulk viscous pressure in our expressions

because it vanishes for monatomic ideal gases. The procedure is rather

straightforward: the second moments of the fluctuations SX, of extensive

parameters X, in a system under constant values of the intensive parameters

F,, ., . . . , F,, with Fk defined as Fk = (dSldX,> are given by [17]

(2.1)

with k the Boltzmann constant.

In the particular case we are dealing with we have from (1.1)

F,, = (dS/dq,) = -a,@q, > (2.2)

Fpt, = (dS/df’;) = -24@; (i+j),

(2.3) Fp,I = (cWdP;,> = -(2/3)a,Vp(2~;, - Pi2 - Pi,).

Here, V is the volume of the system, p the mass density, and S = Vps is the

total entropy, s being the entropy per unit mass. We have written explicitly the

derivatives of S with respect to p; (i # j) and Py, to remind the symmetric and

traceless character of the viscous pressure tensor. From (2.2,3) one obtains for

the second moments

(Sqi sqj) = (k/a,%)‘, 7 (2.4)

224 D. Jou I Fluctuation theory and extended thermodynamics

(@;8&) = (k/2a,Vp)Aij,, , (25)

where Aijkl = &$ + 6,,6,, - (2 /3)iSlj6,,. These expressions may also be written

in terms of (1.3) as

(Sq, Sq,) = kh T2(7,V)~ 16,j ) (2.6)

(Sk; Si’;,) = /c~T(T~V)~‘A~~~, . (2.7)

Relations (2.4)-(2.5) or (2.6)-(2.7) could be useful if one was able to

measure heat and viscous pressure fluctuations. This may be achieved indirect-

ly in light scattering in gases and in neutron scattering in liquids. But these

relations may also be useful from a theoretical point of view if we are able to

compute the equilibrium fluctuations of P’ and q, from equilibrium statistical

mechanics.

We will write explicitly the procedure for a gas at rest-the results are

immediately applicable to a gas in uniform motion. The expressions for the

pressure tensor P and for the heat flux q in microscopic terms for a monatomic

gas are

(2.8)

and

(2.9)

Here, X stands for the set of positions and velocities of the particles r,,

V 1,...f-N, v, and FN(X) is the N-particle distribution function.

The Hamiltonian of the system is

(2.10)

The momentum p, of the particle a is p, = mu,, with m the mass of the

molecule, and c#J(T,~) is the potential which describes the interactions between

molecules, which is assumed to depend only on the relative distance between

them, rah being the vector form molecule a to molecule b. The upper primma

in the summation signs of (2.10) means that the terms with a = b are excluded

from the summation.

The microscopic operators i),, for the pressure tensor and 4; for the heat flux

are given by [l&20]


(2.11)

and

- h $ uai . lZ=l

The symbol 4’ stands for the derivative of 4(r) with respect to its argument

and h is the enthalpy per particle,

(2.13)

U being the internal energy of the system. Eq. (2.12) takes into account that

the heat flux is the total energy flux minus the enthalpy transport (ref. [18], p.

305).

The second moments of the fluctuations are then given by

(‘qi’qj)=(l’V)j(g,(x)-iu,))(~,(x)-(q,))F,,,(X)dX. (2.15)

The angular brackets mean equilibrium average, so that (Pi.) and ( 4,) are the

macroscopic values Pt and qi defined in (2.8) and (2.9). Note that the

microscopic operator for the viscous pressure tensor, pi., is defined as

P; = P, - pa, (2.16)

with p the equilibrium pressure, given from the trace of the equilibrium

pressure tensor as

p = 4 trP,, = $ i mu: - 1 T $: rab4.(l,b)]~Ncq(X) dX . (2.17) a=1

We will proceed at first to evaluate the coefficients (or and CQ for quantum

ideal gases (4 = 0). In a second stage, we will obtain the first terms in the virial


development of the coefficients (Y, and LX* and we will calculate (r,, for classical

nonideal dilute gases. Nonideal gases have been dealt with in extended

irreversible thermodynamics by several authors [21-241, but expressions for the

coefficients studied here have not yet been provided explicitly.

2.1. Ideal gases

In the case of ideal gases, one may work in terms of the

distribution function, f,(r, u, r), which in equilibrium is given by

one-particle

(2.18)

with x = (mi2kT)“‘u. The upper and the lower signs refer respectively to

Bose-Einstein and to Fermi-Dirac statistics. The parameter y is y = (2s +

1)(21rh)) for fermions and y = 2s(2&)” for bosons with spin S, 27rh being

Planck’s constant. The coefficient CI is related to the chemical potential p as

(Y = -plkT. From the well-known relation for the second moments of fluctuations of the

occupation number nk of the state k [19,25],

(2.19)

and taking into account that a = -p/kT, we may write (2.19) as

mwN2) = -wl,<,ww.,v . (2.20)

From here, and in view of the independence of the particles, (2.14) and (2.15)

become

V(SPi, SPY,) = -y(J/aa) m2uTu?;(ea+x’ T l)-’ du (2.21)

and

V(6q, 6q,) = -Y(c~/~cY) J ( ~mu2u, - hu1)2(eatx2 3 1))’ du (2.22)

These integrals may be expressed in terms of the functions I,’ defined as

(2.23)

D. Jou / Fluctuation theory and extended thermodynamics

which satisfy the relation

227

aZ,(a)lacu = -(1/2)(n - 1)zn_2((Y).

In terms of these functions we obtain finally

(2.24)

a2 = (2PTP) l 9

ai = (5/2)PTP]Y - 5CPlP)I >

where p, the pressure, is given by

(2.25)

(2.26)

p( p, T) = (2/3)(RIA)T5’*Z&) , (2.27)

with R = klm and A = (4rrfiR3’2~y))1, p is the mass density and y is defined

as y = (14/5)RT(Z,/Z,). Note that CY may be obtained as a function of p and T through the relation T3”Zz(a) = Ap. The new equations of state (2.25) and

(2.26) of the generalized entropy coincide with those obtained by Liu and

Miiller [lo], though the method of obtaining them is completely different.

2.2. Nonideal dilute gases

We consider now classical idea1 gases. Expressions (2.24) and (2.15) could

be developed in series of powers of the density. We will restrict ourselves to

the case of not very high densities, where the first virial coefficient is sufficient

for an accurate description of the system. In this case, the integrals

and (2.15) may be approximately described in terms of the one-

particle distribution functions, defined according to

F,(r,, u,, . . . r4, uq) = N![(N - q)!]-’ I

FN(X) drq+, . . .du, .

It is more usual to refer F, and F, as f, = F,IV and in terms of

correlation function g(r) defined as

F2(r,, vi, r2, u2) = vn(r,)n(r,)g(r,,) exp[-(mui + mvi)/2kTl ,

where n(r) is the particle number density at the position r.

in (2.14)

and two-

(2.28)

the pair-

(2.29)

In terms of g(r), (2.14) and (2.15) may be expressed, up to the second order

in the density, as C.=

V(SPy, SPy2) = nk2T2 + (2n/15)n2 1 r4$‘(r)2g(r) dr (2.30)


and CC

V(6q, Sq,) = Gnk3T3m-’ + 2mn2kTmp’ r’g(r)+(r)” dr

7. r

+ j r44’(r)2g(r) dr - 2 _f r34(r)+‘(Mr) dr] 0 0

J r

- 2nn3kTmp’ [I r24(r)g(r) dr - f 1 r”+‘(r)&) dr] (2.31) 0 0

These expressions may be written in a more compact form if we note that,

up to second order in the density, the pair correlation function is given by

g(r) = exp(- 4(r) /kT), so that +‘(r)g(r) = -kTg’(r), and integrating by parts

the corresponding terms we define

cc z

I, = (2vnikT) I

g(r)+‘(r)r3 dr , I, = (2wtikT) I

+(r)g(r)r’ dr , 0 0 7. CC (2.32)

Z, = (2TnikT) 1 g(r)+“(r)r” dr , I, = (2nn/k2T2) 1 +(r)‘g(r)r’ dr

0 0

In terms of these expressions, the final results are

V(SP;, SP,y2) = nk2T2(1 + &Z, + &Z,) (2.33)

and

5 nk3T’ v@q, 6%) = 5 -y [1+ $(Zz + Z4) + ;z, - FZ,] . (2.34)

We have neglected higher-order terms in the density in (2.34). When (2.32)

and (2.33) are introduced into (2.4) and (2.5) we get

a2 nonld =a 2,d(l - A4 - AZ,> (2.35)

and

ol nonid = a1 id[l - $(2Z, + I, - 61, + Z,)] , (2.36)

where a2 id and CY, ,d are the values corresponding to the classical ideal gas,

namely a2 id = (2pTp)-’ and (Ye id = fs ( p2T)-‘. Note that the results for (Y, and

cy2 are not those which one could have expected by merely introducing into the

D. Jou I Fluctuation theory and extended thermodynamics

ideal-gas expressions cyi jd, (Y* id the virial development for p,

obtained from (2.17) and which reads

p = nkT(l- fr,> .

229

which may be

(2.37)

It remains to obtain the coefficient CQ related to the bulk viscous pressure.

This task is subtler than in the case of the previous coefficients. One must take

into account that the purely kinetic part of the viscous pressure tensor does not

contribute to the bulk viscous pressure [16,20], so that the contribution comes

only from the interactive part of the tensor. We have then

~ch’(~,d) I %JX) dX . (2.38)

The fluctuation expression for (Ye is [8]

a” = k(V(Gp’ 6pv))-’ .

From (2.38) and (2.39) we obtain, after tedious calculation,

a0 = g(ppT)-‘(I, + $r,>-’ .

(2.39)

Note that the value for CY(, diverges for ideal gases. This is not any problem,

because for such a gas the bulk viscous pressure is identically zero, so that the

term in cq,p”* does not contribute to the entropy.

3. Concluding remarks

In the present paper we have used the theory of fluctuations of dissipative

fluxes to obtain explicit numerical values for the coefficients appearing in the

nonclassical terms of the entropy of extended irreversible thermodynamics.

The present paper enlarges the previous works on this subject because we

apply this technique to quantum ideal gases and to classical non-ideal gases.

The results for non-ideal gases are completely new. In the case of quantum

ideal gases, fluctuation theory was applied by considering as the basis of the

fluctuations the fluctuations in the distribution function itself [26].

It would be desirable, for the sake of completeness, to compute also the

coefficients PO and p2 appearing in the nonclassical terms of the entropy flux

(I .2). This is not at all such a direct task as the obtainment of the coefficients

appearing in the entropy itself. An expression for the & coefficient was

obtained in ref. [13]. The result may be written as


k(W, WZ’3) LA = (Sq, Sq,)(Szy, szq,) (3.1)

with 6( q,c2) standing for

%q,c,) = (1 w J- k(wgX) - ( 4i~,wlvw) dX. (3.2)

In the case of ideal gases, classical or degenerate, (3.1) leads to

PIid = -$(PT)-’ 3 (3.3)

a result which agrees with the macroscopic one by Liu and Miiller [lo] and with

Grad’s result in kinetic theory [16]. In the case of dilute nonideal gases, (3.1)

leads, up to second order in the density, to

P ?non,d = Pz,<,[l - ;(!I, + :I, -61, + &)I. (3.4)

The expression for /3,, analogous to (3.1) would be

k(M’ 67X)) &= (Sp’6p’)(Sq, Sq,) ’ (3.5)

which leads to

The limiting value of PO for ideal gases is not immediate in this case, because

both the numerator and the denominator vanish, but it is not of special interest

because, as we have commented before, the bulk viscous pressure for

monatomic ideal gases is zero so that the corresponding term does not

contribute to the entropy flux.

In summary, fluctuation theory appears as a useful complement of the

macroscopic theory in the domain of extended irreversible thermodynamics.

This is not very surprising, because fluctuation theories have played an

important role in previous formulations of nonequilibrium thermodynamics

(the reciprocity relations of Onsager were obtained from fluctuation theory)

and of nonequilibrium statistical mechanics (Green-Kubo expressions for

transport coefficients are based on the time correlation function of the fluctua-

tions of the fluxes). It is then satisfactory to uncover the important role of

fluctuation theory in such a nonclassical nonequilibrium thermodynamic theory

as extended irreversible thermodynamics.


Acknowledgements

The author acknowledges the important role of discussions with Prof. Ingo

n/Killer (Technische Universitat Berlin), which stimulated and inspired the

developments of the present paper. He also acknowledges Prof. Muller’s

invitation to Berlin, and fruitful discussions with Profs. Wilmansky, Dreyer and

Weiss. Such a visit was made possible by a grant of the German DAAD

(Deutsche Akademische Austauschdienst) during June and July 1986. Partial

financial support of the Spanish Direction General de Investigation Cientifica

under grant PB86-0287 is also acknowledged.

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fluctuation theory and extended irreversible thermodynamics

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