-

1. NONLINEAR NONEQUILIBRIUM
STATISTICAL THERMODYNAMICS
F. M. Kuni and B. A. Storonkin
A study is made of strongly nonequilibrium systems for which the hypothesis of local equilib-
rium, on which ordinary linear thermodynamics is based, does not apply. A method is pro-
posed that is valid far from the critical point and allows one in a unified manner to allow for
not only the static correlations described by the quasiequilibrium distribution function but
also the dynamic correlations associated with the dissipative correction to this function.
Zubarev's nonequilibrium distribution function is the mathematical basis of the method. A
formula is derived for the mean nonequilibrium fluxes in the second order, i.e., inclusive
of the second derivatives and the squares of the first derivatives of the thermodynamic para-
meters with respect to the coordinates and the time at the point of observation. A method of
successive approximations similar to the Chapman-Enskog procedure in the theory of a
Boltzmann gas is constructed; it enables one to eliminate the time derivatives of the para-
meters in the thermodynamic forces in the expressions obtained for the mean nonequilibrium
fluxes.
In strongly nonequflibrium systems, the mean densities cease to depend on only the local values of the
thermodynamic parameters. Physically, this means the breakdown of local equilibrium, the assumption of
which is the basis of all of ordinary nonequilibrium thermodynamics [i]. The consistent construction of a
nonlinear statistical theory of strongly nonequilibrium systems is the aim of this paper.
Zubarev's nonequilibrium statistical distribution function [2, 3] is the mathematical basis of our meth-
od. Important in this connection is the assumption of sufficiently rapid correlation weakening needed to in-
troduce some of the lower moments of the correlation functions; it is evidently true far from the critical
point. For the single-time correlation functions of the particle number densities this assumption has been
rigorously proved in [4] for systems far from the critical point with van der Waals interactions of particles
at large distances.
To be specific, we give the entire exposition for classical systems.
The breakdown of local equilibrium leads to an interesting effect: the appearance of static correlations
described by the quasiequilibrium distribution function. This effect and the associated component in the
mean nonequilibrium fluxes were first considered by a method that differs from ours in [5] and were entirely
ignored in the earlier papers [6, 7] devoted to nonequilibrium processes of second order.
The method proposed in this paper enables one to treat from a unified point of view not only the static
correlations but also the dynamic correlations described by the two- and three-time retarded correlation
functions.
/
i. Zubarev's Nonequilibrium Distribution Function
We consider a nonequilibrium r-component system whose state may be characterized by the field of
the reciprocal temperature, ~(x, t) - I/kBT(x, t), the fields of the chemical potentials ~n(X, t) (n = 1 .....
r) (we use chemical potentials that refer to unit mass) and the field of the mass velocity v{x, t). Let ~(n)
(x, t) be the densities of the basic mechanical quantities: the energy (n = 0),._.,:_the masses of the components
(n = 1 ..... r), and the three momentum components (n = x, y, z), and let J~1)(x, t) be the (~-components
A. A. Zhdanov Leningrad State University. Translated from Teoreticbeskaya i Matematicheskaya
Fizika, Vol. 14, No. 2, pp. 220-234, February, 1973. Original article submitted November i, 1971.
9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g'est 17th Street, Neu, York, .'. Y'. 10011.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
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copy of this article is available from the publisher for $15.00.
164

2. (a = x, y, z) of the fluxes corresponding to these densities. We shall use the symbol + to denote quantities
that depend on the canonical variables: the momenta and coordinates of the particles in the system. For
what follows, it is convenient to label the first r + 1 values taken by a Latin superscript in brackets by a
Latin subscript without brackets and the last three values by a Greek subscript. Summation over repeated
indices is understood: for a Latin subscript without brackets, over all values from 0 to r; for a Greek sub-
script, over all three Cartesian components; and, finally, for a Latin superscript in brackets, over all
values from 0 to r and all three Cartesian components.
An important role in what follows is played by the transformation formulas for the dynamical variables
(densities and fluxes) on the transition from a fixed coordinate system to one moving with constant velocity
(to be specific, we shall assume that the two systems coincide at the time t). These formulas can be ex-
pressed eampaetly in the form
P<~)(x'+ (t'--t)v, t') = B ("'~>(v)ff'(m)(x', t'), (1)
]~") (x' + (t' -- t)v, t') = B ~ (v) [v~fi'~>(x', t') + f~(~)(x', t') ] (2)
by means of the matrix
/ U~ U2
i ~-"'T v+ v,, v,
0 I ... 0 0 0 0
R(v) : | o o . . :( oo o ' , (3)
0 v.,.: ... v~ I 0 0 j
0 % ... vv 0 l 0
0 v.... v+ 0 0 i
which satisfies the obvious group properties R(v t +v 2) =R(vl) R(v2) and R(0) = 1. The prime for dynamical
variables means that the canonical variables on which they depend are taken in the moving system "(in which
they depend on the canonical variables in the same way as in the fixed system).
Suppose we are interested in the mean values of the densities and the fluxes at a point x, t, which we
shall call the point of observation. We introduce a coordinate system moving with constant velocity equal
r
to the mass velocity v&(x, t) = (t)a(x))t/Z <Pn (x)> t of the system at this point (a vanishing time argu-
ment of dynamical variables will generally be omitted), where ()t denotes averaging with respect to the
nonequilibrium distribution function at the time t. Then
<PJ(x) >'= o. (4)
In the variables of the moving system, Zubarev's nonequilibrium distribution function [2, 3] has the
fo rm
p~(t) = •-' exp {-2' - #3,
A" = f dx'F'"~ (x', t)P 't'~) (x'),
, 0 ,(~
t~'=-- i dt'e~CS dx'[/3q"'(x"t )-~F '(x',t + t')
-o
(5)
(6)
+ 1^'(') (x' t') VjF'" (x', t + t') ]. (7)
Here e ~ +0 (after the thermodynamic passage to the limit), and E is a normalization factor; the thermo-
dynamic parameters F '(m)(x', r') 0" =t +t'), which are associated with the reciprocal temperature, the
chemical potentials, and the mass velocity by the equations (the dependence on x and t, the coordinates and
time of the point of observation, is generally not indicated)
, t
- ,)Av (x,x),Y,."(x','r')= F,. (x', ,: )+-~-(i &.o)Fo(x', . . . . (sa)
165

3. F.'r "d) =-Fo(x', "~')Av.(x', 1:'),
Fo(x', a:')-~ 13(x', "r F,~ (x', "r ~ -13 (x', "r t~,~(x', "r
F~(x', -d) -~ O,
Av (x', "r --- v (x', "r - v,
m ~ l,...~r,
(Sb)
(9)
(io)
are determined from the additional condition that the averaging <>qt of the densities P'(m)(x) by means of
the quasiequilibrium distribution function
~(t) = ~-' ~p {-~'} (11)
(E~ is a normalization factor) leads to the mean nonequilibrium values:
<~o,<~)(x,) >,. = <p,<m>(x,) >q,,; (12)
the partial derivative with respect to the time, 8/ST', is related to the total derivative d/dT, by
d /dT' = a / off + hv~(x', T') v~'. (13)
Using (3), we can rewrite Eq. (8) compactly in the form
F'~'~)(x', "d) = F<"~(x', "d)R('~) (--Av (x', "d) ). (14)
We shall not attach a prime to the distribution functions, since they are by their very nature invariant with
respect to the choice of the coordinate system. [Thus, substituting into (6)-(7) the transformation formulas
that are the inverses of (1) and (2), we can express the functions fi(t) and tiq(t) in the canonical variables of
the fixed system. They will then have their old form but with the parameters F '(m) and partial derivative
0/8~' obtained from (8), (13), and (14) by replacing AV(X', T') by v(x, ~").]
2. Locally Comoving Equilibrium System at the
Point of Observation
We introduce an equilibrium system that is locally eomoving at the point of observation x, t and moves
as a whole with constant velocityv(x, t) (it is at rest in the moving coordinate system); this system is
described by the distribution function
P~o(x, t)= Eo-iexp{- f,,,(x, t) j" dx'/3m' (x')} (15)
(E0 is a normalization factor); we have here determined
AF~ ~-F.,(x, t) -]~(x, t) (16)
from the condition that the averaging ( )xt of the densities P '(x) with respect to this distribution function
111 t
be equal to their averaging with respect to the quasiequilibrium function fiq( ):
<>W(x) >q'= <bU(x) >0". (17)
Since <t'g'(x)>0xt, obviously, vanishes [it was for this reason that the velocity of the locally comoving equi-
librium system was taken equal to v(x, t)] and therefore is automatically equal to the mean value <t' '(x)>ut/3. -~
[which vanishes because of (4) and (12)], from (12) and (17) we obtain for the mean densities in the moving
coordinate system at the point of observation
p(m) ~ <~)/(m)(X) >< : <p,'(rn)(X) >q[ = <~),(m)(X)>0xt = <~),(m)(O) >0it, (18)
where we have used the translational invariance of the equilibrium state (at the same time p/~ = 0).
Now we can obviously replace the primed dynamical variables in the averaged expressions and the
distribution function (15) by the unprimed variables:
<L~,M~,...>o:,= <D~M ...>o,,, (19)
where ()c xt denotes averaging with the equilibrium distribution function tic(X, t) obtained from (15) by re-
placing Pm'(X') by Pm(X'); we therefore also have
p<..) = <p(m)(x) >c~' = </~> (0) >o~' (20)
In contrast to ti0(x, t), the function tic(X, t) describes an equilibrium system at rest and is an ordinary grand
canonical function whose parameters fm however are different for each point of observation.
166

4. Formulas (18) and (20) reduce the mean values of the densities P'(m)(x) with respect to the nonequilib-
rium state to mean values with respect to the equilibrium state with the parameters fm" At the same time,
the nonequilibrium mean values of the densities P'(n)(x) in the fixed system, which in accordance with (1)
are equal to
</3(~)(x) >t = B(~m)(v)p(,~), (21)
will, of course, also depend on the mass velocity v. Thus, it is more convenient to use the parameters
fm than the original parameters Fm. Therefore, our task is now to find expressions in terms of the para-
meters fm and v for the mean values of the nonequilibrium fluxes: (J *~(m)(x)>t. We shall solve this prob-
lem in the second order, i.e., we shall allow for not only the first derivatives, as in ordinary linear thermo-
dynamics [1], but also for the second derivatives and the squares of the first derivatives of the parameters
fin and v with respect to the coordinates and the time at the point of observation x, t. (Ultimately, this is
equivalent to allowing for the same derivatives of the old parameters F m and v in the expressions for the
mean equilibrium densities and fluxes.)
Turning to the determination of AFro, we use for f?q(t) in the second order the expansion
1
P~oexp{--/~}/<exp{--/~}>. = po[t-- AoL+ ~- Ao(A.L)2§ ] (22)
(AoI~ -= I~- (L}0), taking ~ in it in accordance with (11) and (15) to be the quantity
C' ~-A'-/~(x,t) ] dx'/5./(x'), (23)
Expanding in the neighborhood of the point of observation, we write
F,~)(x,,t)=F,(,~)+(x~ _x~)X(,.) t ,' +T(x~ - .~) (zr - x.)ZD + ....
X~(~)_~%F'(% Z~$)=- V~V~F'(~).
Taking into account (8) and also the fact that AFm is a quantity of second order (at least, AFm is not lower
than second order; this follows immediately from the fact that it vanishes in the ordinary linear nonequilib-
rium thermodynamics) and that in the terms of first order and higher Fm can be replaced by fm, we obtain
F'(~) = F(~'), (26)
X~.~ = V~],,, X~ = -]oV~v~, (27)
Z~ = V~V.]~ + (l + 5.o)/o(V~) V~v~, (28a)
Z~. = -]0V~V.v. - (V~]0)V.v~ - (V~5) V~v~. (28b)
Substituting (24) and (26)-(28) into (16) and then the result into (23), we find, allowing for (16),
&'=xT' [. (29)
~, 1 (=) . . . . . ;"(~)" '" ^' ~dxP~ (x)=-~Z.~,, I dx (xv --x~,)tx., --x.,)t" ~x ), C~ = AF~ "^ " ' .
We then find for ~q(t) in the second order
^ +-2-1 ho(AoC~')q_~(t) = ~0(~, t) [ 1 - ao~' (30)b
To simplify the following arguments, we make a transformation from the moving system to a locally
comoving coordinate system (in contrast to the previously considered moving coordinate system this system
for a chosen point of observation x, t is introduced separately for each current time t'):
p/' = p/- m~hv(x/, t + t'), x[' = x~', (31)
where Pi, xi and m i are the momentum, radius vector, and mass of the i-th particle of the system, and the
double prime denotes variables in the locally comoving coordinate system; the ordinary prime, as before,
those in the moving system. Since the densities and fluxes at the point x', t' (except the energy flux) * de-
pend only on the momenta of the particles at the same point, the transformation for them to the locally co-
moving system is made in accordance with formulas (1)-(2), in which it is merely necessary to replace v
by Av(x', t + t') and omit the displacement of the spatial coordinates:
*If we use for it not the smoothed but the exact expression with integration with respect to the parameter
(see, for example, [8]).
167

5. P'(")(x', t') = R("") (Av(x', t + ff))P"("~(x', t'), (32)
and accordingly for the fluxes (except the energy flux).
Hence, in particular, it can be seen that
~'(~ (x) = b "(~) (x), (33)
]'(~) (x) = ]~(~) (x), ra ~ 0. (34)
It is convenient to use this invariance property to calculate the mean values of the dynamical quantities with
respect to a quasiequilibrium state, since, as can be seen by substituting (14) and (32) into (6) and allowing
for the relation R(-Av)R(Av) = 1, the quasiequilibrium distribution function (11) in the canonical variables
of the locally comoving system has the particularly simple form
which corresponds to the vanishing of the velocity AV(x', t) in the thermodynamic parameters.
Since the dynamical quantities retain their old form in the new canonical variables, replacing the can-
onical variables and the dynamical quantities with two primes by the same quantities with a single prime,
we obtain the result
</fi'(~) (x) >q' = <fi'(~)(x) )qot, (36)
<]'('~)(x))q'= <f'(~)(x)>,J, ra v~ 0, (37)
where the subscript 0 of the subscript q refers to the quasiequilibrium state with &v(x', t) - 0. Using (36),
we can rewrite condition (17) in the form
<~6j(,,) >~o,= <hJ(x) >o~'. (3s)
We substitute (30) together with (27)-(29) into (38), and also assume that the gradients of the mass
velocity vanish. Then, with allowance for the identity
(L'ao~/')J' = (aoL'Ao~/'>o~' = (aoL'M'>o~', (3 9)
the relation (19), and the translational invariance of the equilibrium state, we have
^ ^ i . . . . i
- (V~/,) <e~]yP,>- -~-(V~V.f,) <P~]yvP,>- AF,(P,.IP,)+ ~-(VJ0 (V./,) </hm]y/~,]vihh)= 0, (40)
where we have introduced the following notation for the moments of the sIngle-time correlation function: *
..... fd,,'xo'x;...
... <A~(0) A~/(x') Ac_~(x') ... >J', (41)
acL =-L- <L)cxt
Differentiating (20) with respect to fl by means of
0L/0I, = - LAo t42)
we obtain
Op., / 0]~ = - (fi,. [D~>. (43)
It follows from the invariance of an equilibrium state under spatial reflection that the moment of the
correlation function in the first term on the left-hand side of (40) vanishes, and the moments of the correl-
ation functions in the second and the fourth terms are proportional to 6Tv; therefore, multiplying (40) by
*In fact, we shall require only some of the lower moments (41) and ha what follows (64)-(65). For such
moments to exist one requires the well-known rapidity of correlation weakening; this is evidently available
far from the critical point. Thus, it has been rigorously proven [4] that in this case the single-term corre-
lation functions of the particle number densities, for example, for systems with van der Waals interaction
at a large r, decrease at r -6.
168

6. Ofn/OPm, summing over m, and using (43) and the obvious identity(afn/aPm) (3pm/Ofl) = 5nl
t Of,,
AF,~= [ (W/,) G~.dWF,)- (V,/,) (Vv/k)(PmJvP,IvPk>].
60p,~
we obtain
(44)
3. Mean Nonequilibrium Fluxes. Static Correlations
Averaging (2) with respect to the nonequilibrinm state at the point of observation* and allowing for
(18), we obtain
<]~")(x) >'= R(,~, (v) [vop(~) +<)"~' (x) >q (45)
We find the mean nonequilibrium (j~m) (x))t by using for p(t) the expansion (22), in which we take ]~' + C'
as L in accordance with (5), (11), (15), and (23). Then, taking into account (29) and (39) and also (19), we
have
^?(m} t
<J~ (x) > =<):(=) (x) 20.....TI~(~), (46)
-(=) ,~(~) "(=) (47)
= <" z(m)(x)h~C>i~t+-:-<hr (x), ~ ,, o , (4S)
2
02~, ,,~ <A=~),=,=(x)A~>o-, +<Ao)2'~ (x)AohAr
l ^(,n)
+-~-<A0]~ (x) (&h)z)J t, (49)
where C, C1 and B are obtained from C', and Cl', and B' by replacing ~,(m)(x', t')and J'(m)(x', t')by
15(m)(x ' , t') and J(m)(x ' , t').
In (46), (J~m)(x)}0xt is the flux characteristic of an ideal fluid. Therefore, ja(m) is the nonideal
part of the flux. Thus, Ja0 is the flux of heat, Jam' m = 1 ..... r, is the diffusion flux of the m-th compo-
nent, and Jag is the viscous flux. Further, the quantity (J (m))s defined by (48), which in accordance with
(30), (19), a~'d (39) is none other than
(L'~)), = 0: `~, (x) >0,- <):+' (~)>~ (50)
can be called the static component of j(m) due to the static correlations described by the quasiequilibrim~
distribution function ~q(t) [and ultimately the single-time correlation functions; see (51) below]. We can
call (ja(m))D the dynamic component of j(m).
We defer the calculation of (] (m))D in the second order to the subsequent section. Here we calculate
the static component (j(m)) s.
Substituting the explicit expressions for C and C1 into (48), taking into account (44), and the transla-
tional invariance of the equilibrium state, we obtain in the second order
" '= =-5--" = -T ap-7
- (vJ=) (v~/~) <~1 v~,~fv?~>] <)(~=>Is ~-x~ ')x<," <)~'~ Iv? + I~<>, (51)
where we have also used the fact that (~(m) iy~)(/))= 0 as a consequence of the equilibrium state's being in-
variant under space and time reflection.
Equation (51) is much simpler in the case m ~ 0, when, as can be seen from (50) and (37), the velocity
gradients can be taken equal to zero in the thermodynamic forces Z(/), X!/) and x+k) and one can there-
Tp ' '
fore assume that the superscripts take only the first r + 1 values. If, in addition, m = 1 ..... r, then in (51)
the moments of the correlation functions that contain the flux J~m vanish, and we have
0~)s =0, m = t ..... r. (52)
*Since we use a representation in which the mean values depend on the time through the distribution func-
tions rather than through the averaged dynamical quantities, the mean value of ~(n)(x' + (t'-t) v, t') at the
current time t' is (J(n)(x' + (t'-t)v, 0))t' , which gives (~(n) (x))t at the point of observation.
169

7. The relation (52) for the mass fluxes can be obtained directly from (50) and (37) by using the invariance of
the quasiequilibrium state Pq0 under time reflection, i.e., by noting that ~qQ is an even function of the part-
icle momenta while the mass flux is an odd flmction.
4. Dynamic Correlations
We write down expansions in the neighborhood of the point of observation:
0 p,(~ (x' t + t') = Y(~)+ (x~' - x~)Q(O+ t'T(z~+ (53)
Ot7- , ,~ ...,
v~'r'(" (x', t + t') = x~" + (~' - x~)z~(,', + t'Q~ ') + .... (54)
X(/) and v(l) are given by (25) and (27)-(28) andwhere
~Tt,
Y") =--~---~F'(~), Q~")~ 7~-~F T (~)=~,0~,,,,F"(t 9 (55)
Ot Ot2'
Substituting (8) into (55) and replacing Fl by fl (which gives corrections of third and higher order) in
the terms of first order and higher, we obtain
y~_ Oil y~ = f =---Ov~ (56)
at ' --~
Ot
Ov~ Ov~ 01~ Vvvx, (57b)
O~= -/~ (vd~ ot
0~1~0~ 8v~ Ov~ (58a)
L = ~ + (t - 6,o)Io----
ot" 8t 8t '
rx = - 1oo~vx- 2~176ov~ (58b)
Ot2 8t 8t
Using the relations
(59a)O/Ot=d/dt, V~O/Ot= V~d/dt- (V~v~)V,,
02lOt~= dZ/dt2- (dv~/dt) V. (59b)
which follow from (13) for the point of observation, to go over to an invariant form of expression (which
does not depend on the coordinate system's moving or being at rest) in the total derivatives with respect
to the time and then regrouping certain terms, we obtain the thermodynamic forces
d/~ dv~ (60)
Y,=-~/-, Y~=-io d---i-'
Q~,= V~~ - (V~v,) V,/, + (1 - 5~0)(V~v~)/0
dv,__L, (610)
dt '
! dv~ ,, d[o
O,~ = -Vv (fo--~-) +/o(V~v~) v,v~---~- Vvvh, (61b)
T~ d2f~ dr, V~lz+(i- dv, dr, (62a)
dt~ d~- 6t0)f0-~ dt '
d ( dv~ dv~ d/o dvx (62b)
T~=--~-[ [o--~-]+fo'-ffi-V.v~ dt dt
Substituting (53)-(54) into the expression for B, w~ obtain
B=- ~ dt'e w "dx'{[Y(~ +(x~'-x~)Q~ z: + t'T(Z)lfi(')(x',t') +[X~"' +(x/- x~)Z(~,) + t'Q(~z' ])(z) (x',t')}.
We substitute (63) and the expression for C1 into (49) and take into account the translational iavarianee
of the equilibrium state. Introducing the following notation for the moments of the two- and three-time
(retarded) correlation functions:
0
((L]O..... 0, cr ..... ~If~,v ..... !~1...})= ~dt't'...t'eW~ dx'x~'x~'... ~ dx"x./'x./'...
^ ^ z t ^ t!
...<AcL(0) AcM(x,t)h~K(x ,0)...)Jr, (64)
170

8. (((L Io .... , o, c~,~..... il 0,..., 0, "r,,: .... , K I~,, ~..... N[... )))
0 o
= f dt't'...t'e~e I dt"t"...t"e*t"Idx'xa'x~'...fdx"xx"x('...
--co --o~
"" "I dx xxxa ... (ACL(0) A~.~?(x', t') A~K(x",t ) ACN(x , 0)... )oxt, (65)
we obtain for the dynamical component of the mean nonequilibrium fluxes in the second order
(1^~(~'),= Y")(()Y)tP(*'))+X(~', (()Y) 1),"' ))
(l) ^(m} ^(m) )(m)
+Q, [(<J~ Ivb")>>+(</= ]O)~')>>]+T.,<<= 10/3(',>>
+ Z (o ((~ (.o ";(0 ^
~ ..... w, >>- Y"'x(, ~>[<<)~' It'("l~,P(">>
_ <<<]2 ~, I~,,,,ij;~'>>>l_ x:"-(" [ el, ,~,x, ,, o I)? i,:x;(~,>>
L
'<<<)='l):" ^"' ]--- I], ~>> + Y")Y('<<<?2r~'l~'"'l~(%>>. (66)
2
In contrast to (51), Eq. (66) contains linear terms (the first two terms), which are characteristic of
ordinary nonequilibrinm thermodynamics. The remaining terms in (66) are, as in (51), of second order.
Equation (66) is also different in that it contains only two- and three-time correlation moments, whereas
(51) contains only single-time moments.
Equations (5l) and (66) in conjuction with (27)-(28) and (60)-(62) for the thermodynamic forces provide
the basis for our subsequent treatment. Since the coordinates of the point of observation and the total deri-
vative with respecttothe time in the expressions (27)-(28)and (60)-(62)are the same in the moving and the fixed
coordinate system; the expressions themselves are valid in not only the moving system, in which they were
derived, but also in the fixed system.
5. Chapman--Enskog Procedure
Our next task is to express the time derivatives of the parameters fm and the velocity v, which enter
j(m) through the thermodynamic forces (60)-(62), in terms of their spatial derivative. Then substitution
of the resulting expressions for j~ (m) in the second order in the spatial derivatives at the point of observa-
tion into the conservation laws for the energy, masses of the components, and momentum gives a closed
system of equations of motion for fm and v. Since the conservation laws are already required when one eli-
minates the time derivatives, we shall begin with them.
We write down the conservation laws
8t
(67)
which are obtained by averaging the corresponding microscopic laws for the dynamical quantities with res-
pect to the nonequilibrium distribution function.
j(n),
Rewritten in terms of the total derivatives with respect to the time and the nonideal parts of the fluxes
these conservation laws have the form [i-3]
dpn
(p~ + &0II) v~,,~ - v=j~ - ~0j=~v~v~, (68)
dt
dv,i
9 'dr=-- VJI- V~j~n, (69)
where II is the equilibrium pressure (for given parameters frn) defined by means of the relation
<]:~ (x) >:t = soXI. (70)
Equations (68) in conjunction with the identity
(71)
171

9. which express the vanishing of the sum of the diffusion fluxes, lead in their turn to the conservation law for
the total mass:
dp
dt pV~v~, (72)
where
is the mean mass density.
Using (68), we can find dfn/dt;
dt
o = p.--- <b.'_x = </~.(~)
),
n=t n~! n~t
(73)
for allowing for the identity dfn/dt = (afn/SPk) (dPk/dt). we obtain
~(p~+~0II) V~v~-~-~/"vd~ ~ Of...~- o~o~ ]~v~v.. (74)
ph pk
Using the thermodynamic relations [3]
OpdOh = OpJOl~, (75)
OH~O/.= -]o-' (p. + g.oII), (76)
we can rewrite (74) and (69) in the form
dt --]O~p~ Opk -- 8~~O-~-h/~V~ "' (77)
dr. = 9-' (78)10-T (p,+ 8~0ii)vd~- ]0~-' Yds..
A decisive factor in the elimination of the time derivatives is the circumstance that Eqs. (77)-(78)
enable us to determine the time derivatives of the parameters f and the velocity v in an approximation that
ja(n) n
is an order higher than the known nonideal fluxes that occur on the right-hand side of the equations.
In principle, this enables us to develop a method of successive approximation that is analogous to the
Chapman-Enskog procedure in the theory ofa Boltzmann gas. For our purposes it will be sufficient to re-
strict ourselves to the second step in this method, substituting the values of j(n) found in the first order
into the right-hand sides of (77)-(78).
Since the right-hand sides of (77)-(78) contain the quantities Vc~j~(m), we require the gradients of the
correlation moments ((f-I 1VI)). Taking into account (42) and the definition (64), we have for these gradients
V~((/]]M)) = -((/~].~[fi,)) VJ,. (79)
Using (47), (51), and (66), we write in the first order
9(~" Y" dF'l-,,6'">> +-(" (dF'IF' >>, (80)
and we then obtain, using (79),
V . (m)
d= = (V~Y (~))<<J2") [fi"'>> - Y(')<(]~(")1['~')[t)~>> V~].
+(vox;" )<<YF't??>> -x;" <<)F'I)? IE>>vz.. (8~)
To find VaY(/) in (81), we must, as can be seen from (60), calculate the gradient of the time deriva-
tires dfl/dt and f0dvx/dt. Determining these in the first order from (77)-(78), we find the desired gradients
in the second order
dl, 011 ~ v.v.+to-~pV~V.v~, (82)
V~ dt Op~ (VJ~ V"v~+ ]~ap, Op.
I-~ ]+P-'LOI. a,0/0-'(p.+6~ (VdJVd,+o-'(p,+6,o~)V~Vd.
where we have used (73) rewritten in the form p = (1-6n0)P n, and we have taken into account (76).
(83)
172

10. We substitute (82)-(83) into (81) and then our result together with (80) into (77)-(78). We find the
remaining time derivatives in the thermodynamic forces Y(/) [which occur in (80) and (81)] by means of (77)-
(78) in the first order, which is here sufficient. We then obtain the desired expressions for the thermody-
namic forces y(l) in terms of the spatial derivatives in second order:
8p,~
OH +{P-' [-0-'(P~+5~om (i-- <0) 8:,
8p~
+ 6/~ -<o]o-'(p.+<on) ] 8/, <<)~t?~>>
J 8pk
_ o-*(p, + ~,0~) o/, <<J~lbdP~>>+ 9I, <<Y~iL~iE>>}(vz~) v4,
' Op~ 8p~
+ :f ' [p -' (p, + 6,olI ) <<i~ i['~)>- <(].~ l]~,>)] v ~v ~?,
6~oO1,/o an ^ <<L.,I)~>>] (v.~)v~,.- e~,~ [6o.--~p~ <<J~IP,.>>- (84)
dr, {[ OH
r~ ~ --lo--~ = --p-' (p~+ 8~oF[)V~.]~+ lop-~ 5~o6o,,8p~
Opt,8p,, 81~ J 8pk
-<o <<J~[L.>>+ to <<L~Ii~1 E>>} (v.t~)vo,~
+/o2p-' [6o~ OH ((I~1/3~))- <<)~1)o~))] V~Vov..
t Opt,
(85)
In (84)-(85) we have allowed for the fact that, as a consequence of the state's being invariant under reflec-
tion of the coordinates,
<<L,I&>> = (<L,I f.>> = o, (s6)
<<L,IL.>> = <<L,IL~>>= o, (87)
and, similarly, for the correlation moments that contain in addition the even quantity Ps"
We now turn to the calculation of the thermodynamic forces QT.(/) . Substituting (82)-(83) into (61) and
then taking into account (77)-(78) in the first order, we find expressions for these forces in terms of the
spatial derivatives in the second order:
OH 8qI 8p.
91t
+ *~ v~v~v. - [6,, - (l - ~,o)V-' (p, + 6~o1I) 1(v~v~) v#. (88)
Op.~ 9-t Op~
OH
x (vd,) V,l~ - 0 -t (p, + 6,oH) V,V,/, + lo(V,vO V.v~ - ]o~-L-(V,v~) V,v,,. (89)
Finally, we calculate the thermodynamic forces T(/) . To find the second time derivatives of the
parameters fl and the velocity v in these forces, we can use (77)-(78) in the first order for the first deri-
vatives. Then, allowing for the relation
d d
V~= V,-=- - (V~#~)V...
d-7 (90)at
Eq. (72) and Eq. (76), we obtain
dZl, OH d]o
dt2 8pz dt
0zII dp~ V~v~+0H / dye
,v~,~+to Op~Op~ dt "aTv~[s~
Ope dt 8p~
(91)
173

11. d dv~ dp,
dt (Io )=o '(p,+6,on)(v~,.)v~t,+p-'--~--v~t,
8 -' -' <oH)d--~-'v4+p-'(p,+6,om [v~7-'#' 1- ,op ]0 (po+ -(v~v~) v ..
J
,(92)
Here we calculate the gradients of the time derivatives of the parameters fi and the velocity v by means of
(82)-(83). Then, substitution of (91)-(92) into (62) and calculation of the remaining first derivatives by
means of (68), (77), and (78) in the first order lead to the desired expressions for the thermodynamic forces
T(l) in terms of the spatial derivatives in second order:
T,=/0 [ OH OH 02H ]
Op, Opo Op,Op. (p,,+6,~oII) (V.v~,)2
+p-' --p-'(p~ + 6~oH)(1 -- 6~0) 0/, O],
+ (:t - 6,o)Io%-' (p, +-6,;[f) (p, + 6,2H) - 6,,1o--' (p, -i-:6;;U) ~.
)
, OH OH
X (V,,]0 V~].+ p- __'-~P(pi + &oil) V2]~-/o 0-~z(V~,v~)V.v., (93)
T~= -p-' [/o(p~+ 6,oH) O2H Op,L+ OH..
Op,Op,~ Of. -~p (p +6~olI)] (V~f~)V.v~
OH
- o-']o(p~ + 6MI)--~p-Tp*VxV~v.+ p-' (p, + 6~oH)(V4,) (V~v.+ V~vQ. (94)
This completes the elimination of the time derivatives in the second order. Equations (77)-(78) in
conjunction with the expressions (47), (51), and (66), in which the thermodynamic forces are given by
(27)-(28), (84)-(85), (88)-(89), and (93)-(94), form a closed system r + 4 equations of motion (in partial
derivatives) for the r + 4 unknown functions fn and v~? of the three spatial coordinates and the time. This
system of equations is suitable for describing strongly nonequilibrium systems far from the critical point.
1,
2.
3.
4.
5.
6.
7.
8.
LITERATURE CITED
S. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam (1962).
D. N. Zubarev, Dokl. Akad. Nauk SSSR, 140, 92 (1961); 162, 532, 794 (1965); 164, 65 (1965).
D. N. Zubarev, Nonequilibrium Statistical Thermodynamics [in Russian], Nauka (1971).
F. M. Kuni, Vestn. LGU, No. 4, 30 (1968); Dokl. Akad. Nauk SSSR, 179, 129 (1968); Phys. Lett.,
26A, 305 (1968).
F. M. Kuni and B. A. Storokin, Teor. Mat. Fiz., 9, 122 (1971).
S. Grossman, Z. Phys. 233, 74 (1970).
V. A. Savchenko and T. N. Khazanovich, Teor. Mat. Fiz., 4, 246 (1970).
R. A. Piccirelli, Phys. Rev., 175, 77 (1968).
174