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Nonlinear response theory for inhomogeneous systems in various ensembles
Physica 86A (1977) 475-489 (~ North-Holland Publishing Co.
NONLINEAR RESPONSE THEORY FOR INHOMOGENEOUS SYSTEMS IN VARIOUS ENSEMBLES* D. RONISt and I. O P P E N H E I M
Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
Received 7 September 1976
Nonlinear equations governing the relaxation of some macroscopic quantities, a(r, t) are derived using Kubo's response theory, in isolated inhomogeneous systems. The resulting equations contain microcanonical time correlation functions. It is shown how to express these in terms of arbitrary ensemble correlation functions and corrections arising from their infinite time behavior are discussed. In addition it is shown that the concept of local equilibrium must be modified somewhat.
1. Introduction
In a previous paper t) one of us showed how additional care must be taken in using Kubo's 3) response theory for homogeneous systems. The problem there concerned the long-time behavior of certain canonical correlation functions. Earlier treatments 2) were shown to be lacking in that the forms of the response equations were not correct and that their local equilibrium conjecture was not valid. Both these problems were overcome by showing that a modified local equilibrium assumption holds, providing the initial average displacement from equilibrium of certain dynamical variables are constrained. In this case the transport equations are identical to those obtained previously2). These additional requirements also justified the replacement of canonical by grand canonical averages. Unfortunately many of the simplifications used in ref. 1 are only valid for homogeneous systems. In addition, all the previous works l°z) have taken the system to be canonically distributed in the infinite past. Since the systems they consider are mechanically isolated this presupposes the equivalence of the average response of a canonical ensemble of systems to a microcanonical one. As we shall show, this need not be the case. * Supported in part by the National Science Foundation, t Supported in part by the National Research Council of Canada. 475
476
D. R O N I S A N D I. O P P E N H E I M
In this paper the average relaxation of an isolated, inhomogeneous system not far from equilibrium shall be described to quadratic order in the average displacement of the dynamical variables from equilibrium. We shall show that the results of ref. 1 can be extended to inhomogeneous systems and that generalized local equilibrium and transport equations hold. These are needed before a molecular theory describing the nonequilibrium behavior of many important systems (for example surfaces4), interracial phenomena, gravitational systemsS)) can be obtained. Section 2 contains the nonlinear response theory for microcanonical ensembles. This relates the average behavior of certain dynamical variables to microcanonical time correlation functions. In section 3 we show how these equations may be reexpressed in terms of arbitrary ensemble averages and give a condition sufficient to insure that the various ensembles give equivalent results. An outline of the derivation of the transport equations is given in section 4 and a discussion of the results of this work is contained in section 5. Throughout this paper multilinear variables and long time tails are not considered, since they have been shown 6) to be unimportant in three dimensions.
2. Nonlinear response in isolated systems In response theory a system at equilibrium in the infinite past is adiabatically removed from equilibrium by time-dependent forces (either real or fictitious) which are coupled to pertinent microscopic dynamical variables. The only purpose of these forces is to create an initial non-equilibrium distribution function. At t = 0 the forces are suddenly switched off and macroscopic relaxation begins. We consider a classical system of N particles confined by walls or other external static fields to some region of space V. The state of the system at any time t is given by the phase point
X N ( t ) ~- {qU(t), pU (t)}, whose time dependence is governed by the hamiltonian H T ( X u, t). This total hamiltonian may be written as H T ( X N, t) =-- H ( X N ) + H 1 ( X u, t),
H , ( X N, t) ~- - A ( X u, r) * F(r, t),
(l)
where H ( X u) is the hamiltonian of the unperturbed system, A ( X N, r) the set of dynamical variables whose macroscopic response is desired,
A ( X N, r) * F(r, t) - f d3r A ( X N, r) • F(r, t) V
(2)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
477
and where the time d e p e n d e n c e of the forces is
F(r,t)=-
f °
F ( r ) e ~'
tO
÷
/
'
(3)
as required by the switching-on p r o c e d u r e described above. The set of variables A(X N, r) should include all p e r t i n e n t slowly varying variables. The distribution f u n c t i o n f ( X N, t) satisfies Liouville's e q u a t i o n
,~f(X ~, t) - - tgt
iLT(t)f(X', t),
(4)
where the Liouville o p e r a t o r LT(t) is defined as LT(t)
-- L + Ll(t),
L--= i[H, - ],
Lt(t)-i[Hl(t),
(5) ],
the symbol [,] indicating a P o i s s o n bracket. Eq. (4) has the f o r m a l solution t
f(XN, t ) = T + e x p [ - f d t , i L ( t l ) J f ( X N , - o o ) ,
(6)
-oe
where T+ is a time ordering operator. W e a s s u m e that the t = -oo system was at equilibrium which is equivalent to
iLf(X N, -oo) = 0.
(7)
E x p a n d i n g (6) to second order in the forces and using (7) we find that t
f ( X N, t) = f ( X N, -oo) + j dt, e -iu'-t0
(8)
-ot~ t
t1
×[f(xN,--oo),A(XN, rl)] * F(rt, tl)+ f dtl f dt2e iL"-lP[(e-iLttl-t'~) --oo
-oo
x [f(X N, -oo), A(X N, r:)]}, A(X N, r0]~ F(rl, tOF(r2, t2) + ~7((NF)3). Eq. (8) is not a suitable e x p r e s s i o n for the full N b o d y distribution function, as the e x p a n s i o n p a r a m e t e r is NF n o t F, and this is c o n s i s t e n t with o b j e c t i o n s
478
D. RONIS AND I. OPPENHEIM
raised by othersT). However when (8) is used to compute the average macroscopic response of the dynamical variables, an expansion in powers of F is obtained. The usual procedure L2) is to make f ( X , - o o ) a canonical distribution function and to assume that certain canonical time correlation functions factorise at long times. Since L, the Liouville operator governing the relaxation, is that of an isolated system, this method yields the average relaxation of a canonical ensemble of systems. All experiments are performed on either isolated systems or on those in contact with heat or particle reservoirs. The former requires that f ( X , -oo) be a microcanonical distribution function, while the latter necessitates the inclusion of reservoir degrees of freedom. In this paper only isolated systems are considered. Thus we take
fo(X N) = - f ( X N, t = - ~ ) -- 3(E
-
H(XU))tJ2(E),
(9)
where 6 ( X ) is the Dirac 6 function and
f
d X N 3 ( E - H ( X N)).
(lO)
In the next section the question concerning the equivalence of the relaxation in any isolated system to the average relaxation for some canonical ensemble of systems shall be examined. Noting that in force-free systems d A(r, t)=--7-A(XN(t), r ) = iLA(r, t), dt
(11)
(8) and (9) imply
f ( X N, t) =fo(X N) + g] '(E) - ~ (~(E)fo(XN)) t
x f dh A(r~, t, - t) * F(r,, t~) -- 3C t
t I
[a2(O(E)/o(X N))
x A(r,, tt - t)'~F(r~, t~)F(r2, t9 + O(O(E)f°(XN)) e -ic"-q~ OE ×[A(r,, t 2 - t,), A(r2, O)]*F(r2, t2)F(r,, t l ) ] } + ~((NF)3).
(12)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
479
We want to calculate the average deviation from equilibrium of the dynamical variables A(X N, r). That is of a(r, t) -- f d X N f ( X N, t ) A ( X N, r) - (A(r)),
(13)
where ( ) denotes an equilibrium microcanonical average. Using (12) for f(XN, t), (13) becomes !
a(r, t) ~ O-~(E)
dt, - ~ O ( E ) ( A ( r , O),~(r,, t, - t)) t
F(r,, t,) +
*
t1
ff{o
at2 ~-~ I"~(E)(A(r, O)A(rl, t, - t).'~(r2, t2 - t))
dt,
_r~
- m
0 * F(r2, t2)F(r,, t,) + --~ I~(E)(A(r, t - t,)[,tL(rl, t2 - t,), A(r2, 0)]) ~F(r2, t2)F(rl, tl)}].
(14)
We now assume that the system is mixing. That is, for variables A(r, t): lim (A~,(r, t )A~,(r')} = (A~(r))(A~,(r')).
(15)
As was discussed in refs. 1 and 11, we can expect eq. (15) to hold only if the ensemble is constructed such that all normal constants of the motion do not fluctuate. Thus, should the system have other conserved quantities (for example total linear or angular momentum) the correct infinite past distribution function must include additional delta functions pertaining to them. We shall return to this point in section 5. By making use of the definition (3) for the forces in relaxation processes, eq. (15), and integrating (14) by parts it is easily shown that for t -> 0
a(r, t) ~ a - ' ( E )
~/2(E)(A(r,/)A(rl)
) *
F(r,)
1 a2 + ~ aE z ~O(E)(,~(r, t),~(r,).4(rz))*F(r:)F(r,) 0
dE
dtl lim 12(E)(A(r, t - tO[A(r2, t2), A(rO])*F(r,)F(r2)] + G(e), (16)
480
D. RONIS AND I. OPPENHE1M
w h e r e we have written A ( r ) for A(r, t = O) and (17)
gi(r, t) =-- A(r, t) - (A(r)).
The m i c r o c a n o n i c a l f o r m u l a t i o n has the a d v a n t a g e that the mixing a s s u m p t i o n (eq. 15) is well studied within the c o n t e x t of ergodic theoryS). In fact Sinai 9) has s h o w n that a system consisting of a finite n u m b e r of hard spheres in a box is mixing. Thus we can e x p e c t eq. (15) to hold before the t h e r m o d y n a m i c limit is taken, a crucial r e q u i r e m e n t if finite systems are to be studied. P e r f o r m i n g an integration by parts in the phase averages in the last f o r m in eq. (16) and using eqs. (10) and (15), we find that 0
dt, lim (A(r, t - t,)[A(r2, t2), A ( r , ) l ) * F ( r , ) F ( r O
J~-t(E) - ~ n ( E )
12--.-
zc
,.~
= -~-'(E)
0
o
~-~ ~ ( E )
I [dt,
([A(r, t - tO, A(r,)|)(A(r2))
ee
o ] t - tOA(r,))(A(r2)) * F(r2)F(r~) o[ /-~(E)(,~(r, t)A(r,)) * F ( r , ) - ~o( A ( r 2 ) ) * F(r2) ]• =-~-'(E)~--~ + / 2 '(E) ~ / / ( E ) ( A ( r .
(18)
C o m b i n i n g (16) and (18) gives
a(r, t ) ~ J 2 - 1 ( E ) - ~ x
g~(E)(A(r, t),4(rO) * F ( r , )
[1o- ~ ( A ( r 2 ) ) ,o
* F(r2)
] ]
+-l~(E)(.4(r, t).,~(rO.4(r2)}~F(r2)F(r~) • 20E
(19)
The d e r i v a t i o n of eq. (19) depends on the mixing a s s u m p t i o n (eq. 15), and on the neglect of the ~7(F 3) terms (which should be valid, providing each of the variables A may be e x p r e s s e d as a sum w h o s e terms d e p e n d only on a small n u m b e r of single particle variables). N e i t h e r of these restrictions need d e p e n d on the size of N or E and so eq. (19) should hold for small systems. For w h a t will follow we assume that the system is large, that is, E - (7(N) with N - > 1. N o t i n g that the t e m p e r a t u r e , T ( N , E, V), of a m i c r o c a n o n i c a l
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
481
ensemble is defined as
~(N,E, V ) - ( K B T ( N , E , V)) - ' = (c~ In ( O ( E ) ) ) aE
(20) N.v'
where KB is Boltzmann's constant, eq. (19) becomes, to lowest order in N -~, a(r,
[
t) ~ 13(,ii(r, t)Ji(rO) * F(rO 1
0(A(r2)) OE
* F(r:)
]
+ ½fl2(,ii(r, t),~(rO,iA(r2))*F(r2)F(rO + (7(N-~).
(21)
In analysing the order of magnitude of the various terms in eq. (19), we make use of the fact that the correlation functions therein decay to ~ ( N -t) whenever any of their arguments are separated beyond some microscopic correlation length. Should the variable As(r, t) be conserved, eq. (21) implies that
f dr a~(r, t) = 0.
(22)
V
In the next section we show how eq. (21) may be rewritten in terms of canonical or grand canonical correlation functions and averages.
3. Transformation to other ensembles
For many applications and for the sake of comparison we would like to express eq. (21) in terms of ensemble averages other than the microcanonical. To do this we extend the procedure for finding the asymptotic forms of reduced canonical distribution functions of Lebowitz and Percus ~°) to arbitrary correlation functions. We consider two ensembles F and F' which differ in the fact that the set of variables C(X N) fluctuate in the latter but not in the former. In addition it is required that the variables C are ¢?(N) in the F ensemble, and that (C)r = (C)r., where the symbols ( )r and ( )r' represent averages in the F and F ' ensembles, respectively. Averages in the two ensembles are related by f
(B)r,-- J dC (B)rP (C, r), P ( C , ~/) --
[g(C)/A(,y)] exp [ - 3 ' • I2].
(23) (24)
The ~,,~are intensive variables conjugate to the C,, g(C) is a degeneracy factor, and A(-r) is a normalization constant.
482
D. RONIS AND I. OPPENHEIM
F o r large systems at finite t e m p e r a t u r e and away from critical points, the f u n c t i o n P(C, y) should be very sharply p e a k e d near C = (C)r,. This fact allows us to e x p a n d (B)r in eq. (23) in a T a y l o r series about C = (C>r,. Thus the two e n s e m b l e averages may be related from eq. (23) by
(B>r, = (B)r + ~(dd)r,: acacl c=., I
~..
a3(B>r
]
+ 3--'.(COO>r, i aCaCa~ c=~,+
•(N-3),
(25)
where we have retained terms to 0'(N-2). R e a r r a n g i n g eq. (25) and iterating, we obtain
r = (B>r,- ~(d~>.: 1 <~(.(.>r' 3! +-~(CC)r':
aZ(B)r,
a(C>,..a(C>r, O3(B>~, a( C )r.a( C)r,O( C)r,
O(C>r,a(O-,>r,
O((',)r,O(g',)r/
~?(N-3)"
(26)
F r o m eqs. (23) and (24) it is easily s h o w n that
a(C,)r, (C,~'j>r,,
(27)
ayj
and (27')
_
cgyk We first consider some binary correlation f u n c t i o n of the form
(BD)r = (BD)r-(B)r(D)r,
(28)
where B and D are any d y n a m i c a l variables. E m p l o y i n g eq. (26) to ~ ( N - ' ) it follows that o2r,
o(C>c,O(C)r, O(B>r,O(D>r,
- (Co)r,:
a~r,+ ~(N-2)'
(29)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
483
which by eq. (27) may be written as = (/~(1 - P ) D ) r
- -~(d¢)r:
+ ~(N-2).
(29')
a(C)r,a(C)r
In this last expression the operator P is defined as
PB
-
(BC)r,
"
(30)
(C(~)r)" ¢
and is simply the projection operator onto the fluctuating parts of the variables C in the F' ensemble. We also need to know to 6 ( N -2) how triple correlation functions of the form {/~/5/~)r may be expressed in terms of F' quantities. This is accomplished by using eq. (26) in a rather tedious calculation, and the result is given in the appendix. Talking the F' ensemble to be microcanonical and using eqs. (26) and (29') and the appendix, eq. (21) becomes a(r, t) =/3(A(r, t)(1 - P),i~(r,))r, * F(rO
x [l
O(A(r2))r.a(E)r, F(r2)] +½13z[([(1-P)A(r,t)],i~(r,)A(rz))r ,
- (,i,r, t)[P.ii(rl)]A(rz))r- (A(r, t)A(r])[PA(r2)])r . . ^ a(A(r))r, O(A(r]))r a(A(rO)r + 2(CFiCDr a
a2(A(r))r
+ ½((%G)~,(G G )~, Ua( C,)~a(G)~
x 0(A(r,))r, O(A(r2))r + 4cg2(A(r))r, O(A(rL))r, a(A(r2))r, a(Ck), o(G),, a(C,>r,a(Ck)r, a(G>r, a(C,>r, 2 other permutations of the ]] + position of the double derivatives_lJ ~ F(rz)F(r0 + (9(N-I),
(31) where repeated indices are to be summed. If the set of variables C includes the energy and we impose the condition
O(A(r))r a(G>r,
• F(r) = 0
(32)
for all i, eq. (31) simplifies to a(r, t) = fl(A(r, t)A(r,)>r * F(rO + ½/3z([(l -P)A(r, t)]A(r]),ii(rz))r * F(rz)F(rl).
(33)
484
D. RONIS AND I. OPPENHEIM
It should be noted that eq. (32) is equivalent to
(34)
( ~ , A ( r ) ) r , * F ( r ) -- 0,
which is a c o n s e q u e n c e of eq. (27). When F' is a canonical ensemble (C = E) in the homogeneous limit, the response equations (33) and constraint equation (32) are equivalent to those obtained by one of us 1) through correcting for projections onto conserved variables. Let e(r, t) be the average displacement from equilibrium of the densities of the variables O. According to eqs. (33) and (34)
(35)
d3r c(r, t) = ¢3(~,.~(r))r, * F ( r ) = O.
Since the actual relaxation occurs in an isolated system the integrals of the densities of the nonfluctuating quantities must never change from their equilibrium values, independent of the choice of ensemble used to express the correlation functions. This is shown by eq. (35). In the next section an outline of the derivation of the h y d r o d y n a m i c equations shall be presented. Owing to the similarity between eq. (33) and an equation derived by one of us earlier, most details will be omitted as they are contained in ref. 1.
4. Hydrodynamic equations The first step in obtaining the transport equations is to express the fictitious forces in terms of the variables a (r, t). This is most easily accomplished when F' is a grand canonical ensemble. Here the correlation functions can be expected to decay exactly to zero beyond some microscopic correlation length and the linear transformation a~)(r, t ) - - / 3 ( A ( r ,
(36)
t)A(r,))o.c~ * F ( r , )
can be inverted. In this section the symbol ( )G.c. denotes a grand canonical average. Defining the inverse of the kernel in eq. (36) through K-l(rlr~; t) * (A(r,, t)A(r'))G~c. = 6 ( r - r')l
(37)
and using eq. (33), it is easily shown that [3F(r) = K-l(rlr~; t) * [a(r~, t) -½([(1 - P ) A ( r ,
t)JA(r2)A(r3))~.c.
• (K-t(r3[r4; t) * a(r4, t))(K ~(r2lrs; t) * a(rs, t))] + ~7(a3).
(38)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
485
Since for a grand canonical e n s e m b l e the new fluctuating variables t2 are the total n u m b e r and energy, both constants of the motion, eq. (33) implies ~i(r, t) =/3(A(r, t),4(r0)G.c. * F(rO (39)
1 + ~(A(r, t),4(r,),'~(r2))Gc_ * F(r2)F(rd,
which by eq. (38) b e c o m e s to 6~(a 2)
a(r, t) = M(rlrl; t) * a(rl, t) (40)
+ N(rlrtlr2; t) * a(rz, t)a(rl, t).
In the last equation (41)
M(rlr,; t) ~ (A(r, t),/~(r2))G.c. * K-i(rzlr~; t) and
N(rlr,lr2; t ) - ( A ( r , t)A(r3)A(r,))c.c. ** K ](r41rl; t)K-l(rslr2, t) - ½(A(r, t)A(rs))~.c. * K-'(rslr4; t) • ([(1 - P),4(r4, t)],i~(rs)A(r6))c.c.** K-~(rsIr,; t)K-t(r61r2; t). (41') As was shown in ref. 1, the projection operator appearing in eq. (41') m a y be omitted since both n u m b e r and energy are c o n s e r v e d quantities whose densities must be included in the set of variables A. H e n c e the relaxation equation (40) is identical to that obtained by Weare and Oppenheim2), starting with a canonical distribution, providing eq. (35) holds. The set of variables A(r, t) was a s s u m e d to be slowly varying. Therefore eq. (40) is rewritten, exactly as in ref. 1, to make as m a n y time derivatives as possible explicit. The result is to second order in the smallness parameter characterizing the change of the slow variables
a(r, t) = M(rlrl; 0) * a(rt, t) + N(rlrllr2; O ) ,* a ( r 2, t ) a ( r l , t) l
- f dtt(l(r, tOl(rO)a.c * K-t(rllr2; 0) * a(r2, t) 0 t
+ f dt, [½(l(r, t01(r0)a.c. *
K-'(rdr2; 0)
0
• (,4(r2),4(r3),4(r4)}6.c.- ½(l(r, tOl(r3),4(r4))a.c. - ½(l(r, tl)1~(r3)i(r4)}Gc.]* (K-'(r41rs; 0) * a(rs, t))(K-t(r31r6; O) * a(r6, t)), (42)
486
D. RONIS AND I. OPPENHEIM
where the generalized dissipative current, /, is defined as I ( r , t ) ---- A ( r ,
t) -
M(rlr~; O) * A(X N, rt).
(43)
These dissipative currents have the property of being orthogonal to the slowly varying quantities. That is,
(l(r, t)A(r'))~.c. = 0
(44)
to the first order in the smallness parameter characterizing the magnitude of the time derivative of A. Correlation functions involving the dissipative currents should decay on a molecular time scale. Therefore the limits of the integral in eq. (42) may be extended to some time, r0, much longer than the duration of any experiment. As in ref. 1, we define force-like quantities, ~ ( r , t), by a(r, t) =-- fl(,4(r).4(r,))~.c. * ill(r1, t)
+ ½flz(,i~(r)J~(r,),ii(r2))c.c. * ¢11(r2~ t ) ~ ( r , , t),
(45)
or, using eq. (37),
fiR(r, t) = K l(r[rl; 0) * [a(rl, t) -~(A(rOA(r2)A(rs))G.C"
~ (K-l(r3]r4; 0) * a(r4, t))
× (K-l(r2]rs; 0) * a(rs, t))] + ~(a3).
(46)
Eq. (42) in terms of the ~ ' s becomes
a(r, t) = fl(A(r)ll(rO)~.c * ~(rt, t) + ½fl2(A(r)A(rl)Ji(r2))G.c. ~, ~(r2, t ) ~ ( r l , t) "r0 i-
-
J dt.[13(l(r, t,)l(rj))c,c * q~(r~, t) 0
+ ½/3Z[(l(r, t)l(rO,4(r2))c.c. + (l(r,, t)A(rl)l(r2))~.c.] * q~(r2, t)~(r~, t)]. (47) The force-like quantities may be used to construct a local equilibrium distribution function, fL(X N, t). AS was shown in ref. 1, / ~ c ( X ~) exp [J3A(X ~, r) * ,i~(r, t)]
/L(X N, t)----
/-
,
(48)
| d X N f~.c.(X N) exp [/3A(X N, r ) * q)(r, t)] N=0
d
where/G.c.(X N) is the grand canonical distribution function. From eq. (45) it
NONLINEAR
RESPONSE
FOR INHOMOGENEOUS
SYSTEMS
487
follows that
a(r, t) = (A(r))L(t) + ~7(q~3)
(49)
and from eq. (47) rO
,el(r, t) = (A(r))L(t)-- fl f dtl(l(r, tOi(rl))L(t) * eP(r,, t) + ~(q~3),
(50)
o
where ( ) L ( t ) represents an average using the distribution function given by eq. (48). The quantities ~ ( r , t) are now seen to be the thermodynamic forces conjugate to the macroscopic variables n(r, t), as was shown in ref. 2. The actual forms of the hydrodynamic equations for given sets of variables A(r, t) have been discussed in ref. 2 and will not be presented here. In the next section we shall summarize the results of this paper and discuss some of the finer points of this theory.
5. Discussion A careful study of response theory and generalized hydrodynamics has shown that the expressions obtained in homogeneous systems are applicable to inhomogeneous ones. The transport equations are in general local in time but nonlocaI in space. The choice of ensemble used to describe the relaxation is irrelevant, providing variables such as the total number and energy are never displaced from equilibrium. Strictly speaking, this work is only applicable to inhomogeneous systems. In homogeneous ones, as shown in ref. 1, the total momentum is conserved and the microcanonical ensemble will not be the correct ensemble for which the mixing assumption, eq. (15) will hold (as discussed in section 2). The transformation theory of section 3 would include projections onto the total m o m e n t u m [as in eqs. (33)] but would not affect the final transport equations, form of the local equilibrium distribution function, or definition of the conjugate thermodynamic forces [eq. (46)]. In a later paper 4) the specific forms of the transport equations for some inhomogeneous systems shall be discussed. There the conditions needed to obtain forms local in space shall be presented.
Appendix In this appendix we shall show how triple correlation functions of the form (/3/)/~)r transform to the F ' ensemble. For our purposes only corrections to (?(N -2) need be retained.
488
D. RONIS AND I. O P P E N H E I M
Eq. (26) may be rewritten as F + ~ ' , ? - -
,, =
a2(B)r
a
a(C)r,a(C)r,a(C~)r,
a~(B)r ,
r(4)
O~r,
~- I , . ' 2 -
+ -ijkt a (C~)r,a< Q)r,a (Ck)to (C,)r
+ C( N -~3),
(A..~)
where repeated indices are summed, (A.2)
aro/ ijk
a(G)r,
(A.3)
3!
and (A.4)
By using eqs. (27) and (A.2) it is easily shown that
l~i~' = ~( C, Cidk>r,.
(A.5)
Note that
r-(B)rr . . . .
(A.6)
+ 2(B)rrr,
where the symbol . . . represents the other two permutations of the variables B, D, E. By using eq. (A.1) in each of the averages appearing in (A.6) and retaining terms to ~(N 2), we obtain after some lengthy manipulations
(BI~E>r =r'+ ~ ) L a i r , a 3r,
~-2 - - - - 8 ( Ca(Cj)r, i)r, 4- . . . a2(B)r,
a(/5/~)r +...
+ ~i3~ a(C,)r,ar, ~ 3 ar, + ' ' " a(B)r
a4
+ 4 a
+ ~i~ a(C)r,a(C)ra
a3(B>r , a
aF]
a(B)r, ar, + 6 a(C,)r-~--~,a(C~)r, a-~-Ck>~,l
ar, a(G)r,
a3r,
+ •. .
a.aF -a,~, -
N O N L I N E A R R E S P O N S E FOR I N H O M O G E N E O U S SYSTEMS
4
02(B)r,
o(C)r,a(Cj)r, a(cor, a(C~)r,
+4
+''
which
02(DE)r,
+...
O(C~)r,O(Ck)r, a(Cj)r,O(Ct)r, "+8
O2(B)r ,
489
+...+2
cg(D)r, 3(E)r,
a(C~)r,a(CDr, a(Cj)r, c9(Ct)r.
~2(B)r,
O(D)r, O(E)r,
o(c,)F,a(cj)r, a(COr, a(C,)r, ~-" ' " / + _1
G(N
3),
(A.7)
is t h e d e s i r e d r e s u l t .
References 1) I. Oppenheim, in Topics in Statistical Mechanics and Biophysics: A Memorial to J.L. Jackson, R.A. Piccireili, ed. (A.I.P., New York, 1976) p. 111. 2) J.H. Weare and I. Oppenheim, Physica 72 (1974) 1, 20. 3) R. Kubo, J. Phys. Soc. Japan 12 (1957) 570. R. Kubo, M. Yokota and S. Nakajima, J. Phys. Soc. Japan 12 (1957) 1203. 4) R. Pasmanter and I. Oppenheim, Physica 84A (1976) 507. 5) D. Ronis, J. Kovac and I. Oppenheim, to appear in Physica. 6) T. Keyes and i. Oppenheim, Phys. Rev. A7 (1973) 1384. I.A. Michaels and I. Oppenheim, Physica glA (1975) 522. 7) N.G. Van Kampen, Physica Norwegica 5 (1971) 10. 8) J.L. Lebowitz, Hamiltonian Flows and Rigorous Results in Non-equilibrium Statistical Mechanics, in the Proc. I.U.P.A.P. Conference on Statistical Mechanics, S.A. Rice, K.F. Freed and J.C. Light, eds. (Univ. of Chicago Press, 1972). 9) Ja.G. Sinai, Ergodicity of Boltzmann's Gas Model, in Proc. I.U.P.A.P. Meeting, Copenhagen, 1966, T.A. Bak, ed. (Benjamin, New York, 1967) p. 559. 10) J.L. Lebowitz and J.K. Percus, Phys. Rev. 122 (1961) 1675. ll) For a discussion of ergodic theory and the choice of ensemble-, see: A.I. Khinchin, Mathematical Foundations of Statistical Mechanics, chapter III (Dover Publ., New York, 1949).
NONLINEAR RESPONSE THEORY FOR INHOMOGENEOUS SYSTEMS IN VARIOUS ENSEMBLES* D. RONISt and I. O P P E N H E I M
Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
Received 7 September 1976
Nonlinear equations governing the relaxation of some macroscopic quantities, a(r, t) are derived using Kubo's response theory, in isolated inhomogeneous systems. The resulting equations contain microcanonical time correlation functions. It is shown how to express these in terms of arbitrary ensemble correlation functions and corrections arising from their infinite time behavior are discussed. In addition it is shown that the concept of local equilibrium must be modified somewhat.
1. Introduction
In a previous paper t) one of us showed how additional care must be taken in using Kubo's 3) response theory for homogeneous systems. The problem there concerned the long-time behavior of certain canonical correlation functions. Earlier treatments 2) were shown to be lacking in that the forms of the response equations were not correct and that their local equilibrium conjecture was not valid. Both these problems were overcome by showing that a modified local equilibrium assumption holds, providing the initial average displacement from equilibrium of certain dynamical variables are constrained. In this case the transport equations are identical to those obtained previously2). These additional requirements also justified the replacement of canonical by grand canonical averages. Unfortunately many of the simplifications used in ref. 1 are only valid for homogeneous systems. In addition, all the previous works l°z) have taken the system to be canonically distributed in the infinite past. Since the systems they consider are mechanically isolated this presupposes the equivalence of the average response of a canonical ensemble of systems to a microcanonical one. As we shall show, this need not be the case. * Supported in part by the National Science Foundation, t Supported in part by the National Research Council of Canada. 475
476
D. R O N I S A N D I. O P P E N H E I M
In this paper the average relaxation of an isolated, inhomogeneous system not far from equilibrium shall be described to quadratic order in the average displacement of the dynamical variables from equilibrium. We shall show that the results of ref. 1 can be extended to inhomogeneous systems and that generalized local equilibrium and transport equations hold. These are needed before a molecular theory describing the nonequilibrium behavior of many important systems (for example surfaces4), interracial phenomena, gravitational systemsS)) can be obtained. Section 2 contains the nonlinear response theory for microcanonical ensembles. This relates the average behavior of certain dynamical variables to microcanonical time correlation functions. In section 3 we show how these equations may be reexpressed in terms of arbitrary ensemble averages and give a condition sufficient to insure that the various ensembles give equivalent results. An outline of the derivation of the transport equations is given in section 4 and a discussion of the results of this work is contained in section 5. Throughout this paper multilinear variables and long time tails are not considered, since they have been shown 6) to be unimportant in three dimensions.
2. Nonlinear response in isolated systems In response theory a system at equilibrium in the infinite past is adiabatically removed from equilibrium by time-dependent forces (either real or fictitious) which are coupled to pertinent microscopic dynamical variables. The only purpose of these forces is to create an initial non-equilibrium distribution function. At t = 0 the forces are suddenly switched off and macroscopic relaxation begins. We consider a classical system of N particles confined by walls or other external static fields to some region of space V. The state of the system at any time t is given by the phase point
X N ( t ) ~- {qU(t), pU (t)}, whose time dependence is governed by the hamiltonian H T ( X u, t). This total hamiltonian may be written as H T ( X N, t) =-- H ( X N ) + H 1 ( X u, t),
H , ( X N, t) ~- - A ( X u, r) * F(r, t),
(l)
where H ( X u) is the hamiltonian of the unperturbed system, A ( X N, r) the set of dynamical variables whose macroscopic response is desired,
A ( X N, r) * F(r, t) - f d3r A ( X N, r) • F(r, t) V
(2)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
477
and where the time d e p e n d e n c e of the forces is
F(r,t)=-
f °
F ( r ) e ~'
t
÷
/
'
(3)
as required by the switching-on p r o c e d u r e described above. The set of variables A(X N, r) should include all p e r t i n e n t slowly varying variables. The distribution f u n c t i o n f ( X N, t) satisfies Liouville's e q u a t i o n
,~f(X ~, t) - - tgt
iLT(t)f(X', t),
(4)
where the Liouville o p e r a t o r LT(t) is defined as LT(t)
-- L + Ll(t),
L--= i[H, - ],
Lt(t)-i[Hl(t),
(5) ],
the symbol [,] indicating a P o i s s o n bracket. Eq. (4) has the f o r m a l solution t
f(XN, t ) = T + e x p [ - f d t , i L ( t l ) J f ( X N , - o o ) ,
(6)
-oe
where T+ is a time ordering operator. W e a s s u m e that the t = -oo system was at equilibrium which is equivalent to
iLf(X N, -oo) = 0.
(7)
E x p a n d i n g (6) to second order in the forces and using (7) we find that t
f ( X N, t) = f ( X N, -oo) + j dt, e -iu'-t0
(8)
-ot~ t
t1
×[f(xN,--oo),A(XN, rl)] * F(rt, tl)+ f dtl f dt2e iL"-lP[(e-iLttl-t'~) --oo
-oo
x [f(X N, -oo), A(X N, r:)]}, A(X N, r0]~ F(rl, tOF(r2, t2) + ~7((NF)3). Eq. (8) is not a suitable e x p r e s s i o n for the full N b o d y distribution function, as the e x p a n s i o n p a r a m e t e r is NF n o t F, and this is c o n s i s t e n t with o b j e c t i o n s
478
D. RONIS AND I. OPPENHEIM
raised by othersT). However when (8) is used to compute the average macroscopic response of the dynamical variables, an expansion in powers of F is obtained. The usual procedure L2) is to make f ( X , - o o ) a canonical distribution function and to assume that certain canonical time correlation functions factorise at long times. Since L, the Liouville operator governing the relaxation, is that of an isolated system, this method yields the average relaxation of a canonical ensemble of systems. All experiments are performed on either isolated systems or on those in contact with heat or particle reservoirs. The former requires that f ( X , -oo) be a microcanonical distribution function, while the latter necessitates the inclusion of reservoir degrees of freedom. In this paper only isolated systems are considered. Thus we take
fo(X N) = - f ( X N, t = - ~ ) -- 3(E
-
H(XU))tJ2(E),
(9)
where 6 ( X ) is the Dirac 6 function and
f
d X N 3 ( E - H ( X N)).
(lO)
In the next section the question concerning the equivalence of the relaxation in any isolated system to the average relaxation for some canonical ensemble of systems shall be examined. Noting that in force-free systems d A(r, t)=--7-A(XN(t), r ) = iLA(r, t), dt
(11)
(8) and (9) imply
f ( X N, t) =fo(X N) + g] '(E) - ~ (~(E)fo(XN)) t
x f dh A(r~, t, - t) * F(r,, t~) -- 3C t
t I
[a2(O(E)/o(X N))
x A(r,, tt - t)'~F(r~, t~)F(r2, t9 + O(O(E)f°(XN)) e -ic"-q~ OE ×[A(r,, t 2 - t,), A(r2, O)]*F(r2, t2)F(r,, t l ) ] } + ~((NF)3).
(12)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
479
We want to calculate the average deviation from equilibrium of the dynamical variables A(X N, r). That is of a(r, t) -- f d X N f ( X N, t ) A ( X N, r) - (A(r)),
(13)
where ( ) denotes an equilibrium microcanonical average. Using (12) for f(XN, t), (13) becomes !
a(r, t) ~ O-~(E)
dt, - ~ O ( E ) ( A ( r , O),~(r,, t, - t)) t
F(r,, t,) +
*
t1
ff{o
at2 ~-~ I"~(E)(A(r, O)A(rl, t, - t).'~(r2, t2 - t))
dt,
_r~
- m
0 * F(r2, t2)F(r,, t,) + --~ I~(E)(A(r, t - t,)[,tL(rl, t2 - t,), A(r2, 0)]) ~F(r2, t2)F(rl, tl)}].
(14)
We now assume that the system is mixing. That is, for variables A(r, t): lim (A~,(r, t )A~,(r')} = (A~(r))(A~,(r')).
(15)
As was discussed in refs. 1 and 11, we can expect eq. (15) to hold only if the ensemble is constructed such that all normal constants of the motion do not fluctuate. Thus, should the system have other conserved quantities (for example total linear or angular momentum) the correct infinite past distribution function must include additional delta functions pertaining to them. We shall return to this point in section 5. By making use of the definition (3) for the forces in relaxation processes, eq. (15), and integrating (14) by parts it is easily shown that for t -> 0
a(r, t) ~ a - ' ( E )
~/2(E)(A(r,/)A(rl)
) *
F(r,)
1 a2 + ~ aE z ~O(E)(,~(r, t),~(r,).4(rz))*F(r:)F(r,) 0
dE
dtl lim 12(E)(A(r, t - tO[A(r2, t2), A(rO])*F(r,)F(r2)] + G(e), (16)
480
D. RONIS AND I. OPPENHE1M
w h e r e we have written A ( r ) for A(r, t = O) and (17)
gi(r, t) =-- A(r, t) - (A(r)).
The m i c r o c a n o n i c a l f o r m u l a t i o n has the a d v a n t a g e that the mixing a s s u m p t i o n (eq. 15) is well studied within the c o n t e x t of ergodic theoryS). In fact Sinai 9) has s h o w n that a system consisting of a finite n u m b e r of hard spheres in a box is mixing. Thus we can e x p e c t eq. (15) to hold before the t h e r m o d y n a m i c limit is taken, a crucial r e q u i r e m e n t if finite systems are to be studied. P e r f o r m i n g an integration by parts in the phase averages in the last f o r m in eq. (16) and using eqs. (10) and (15), we find that 0
dt, lim (A(r, t - t,)[A(r2, t2), A ( r , ) l ) * F ( r , ) F ( r O
J~-t(E) - ~ n ( E )
12--.-
zc
,.~
= -~-'(E)
0
o
~-~ ~ ( E )
I [dt,
([A(r, t - tO, A(r,)|)(A(r2))
ee
o ] t - tOA(r,))(A(r2)) * F(r2)F(r~) o[ /-~(E)(,~(r, t)A(r,)) * F ( r , ) - ~o( A ( r 2 ) ) * F(r2) ]• =-~-'(E)~--~ + / 2 '(E) ~ / / ( E ) ( A ( r .
(18)
C o m b i n i n g (16) and (18) gives
a(r, t ) ~ J 2 - 1 ( E ) - ~ x
g~(E)(A(r, t),4(rO) * F ( r , )
[1o- ~ ( A ( r 2 ) ) ,o
* F(r2)
] ]
+-l~(E)(.4(r, t).,~(rO.4(r2)}~F(r2)F(r~) • 20E
(19)
The d e r i v a t i o n of eq. (19) depends on the mixing a s s u m p t i o n (eq. 15), and on the neglect of the ~7(F 3) terms (which should be valid, providing each of the variables A may be e x p r e s s e d as a sum w h o s e terms d e p e n d only on a small n u m b e r of single particle variables). N e i t h e r of these restrictions need d e p e n d on the size of N or E and so eq. (19) should hold for small systems. For w h a t will follow we assume that the system is large, that is, E - (7(N) with N - > 1. N o t i n g that the t e m p e r a t u r e , T ( N , E, V), of a m i c r o c a n o n i c a l
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
481
ensemble is defined as
~(N,E, V ) - ( K B T ( N , E , V)) - ' = (c~ In ( O ( E ) ) ) aE
(20) N.v'
where KB is Boltzmann's constant, eq. (19) becomes, to lowest order in N -~, a(r,
[
t) ~ 13(,ii(r, t)Ji(rO) * F(rO 1
0(A(r2)) OE
* F(r:)
]
+ ½fl2(,ii(r, t),~(rO,iA(r2))*F(r2)F(rO + (7(N-~).
(21)
In analysing the order of magnitude of the various terms in eq. (19), we make use of the fact that the correlation functions therein decay to ~ ( N -t) whenever any of their arguments are separated beyond some microscopic correlation length. Should the variable As(r, t) be conserved, eq. (21) implies that
f dr a~(r, t) = 0.
(22)
V
In the next section we show how eq. (21) may be rewritten in terms of canonical or grand canonical correlation functions and averages.
3. Transformation to other ensembles
For many applications and for the sake of comparison we would like to express eq. (21) in terms of ensemble averages other than the microcanonical. To do this we extend the procedure for finding the asymptotic forms of reduced canonical distribution functions of Lebowitz and Percus ~°) to arbitrary correlation functions. We consider two ensembles F and F' which differ in the fact that the set of variables C(X N) fluctuate in the latter but not in the former. In addition it is required that the variables C are ¢?(N) in the F ensemble, and that (C)r = (C)r., where the symbols ( )r and ( )r' represent averages in the F and F ' ensembles, respectively. Averages in the two ensembles are related by f
(B)r,-- J dC (B)rP (C, r), P ( C , ~/) --
[g(C)/A(,y)] exp [ - 3 ' • I2].
(23) (24)
The ~,,~are intensive variables conjugate to the C,, g(C) is a degeneracy factor, and A(-r) is a normalization constant.
482
D. RONIS AND I. OPPENHEIM
F o r large systems at finite t e m p e r a t u r e and away from critical points, the f u n c t i o n P(C, y) should be very sharply p e a k e d near C = (C)r,. This fact allows us to e x p a n d (B)r in eq. (23) in a T a y l o r series about C = (C>r,. Thus the two e n s e m b l e averages may be related from eq. (23) by
(B>r, = (B)r + ~(dd)r,: acacl c=
~..
a3(B>r
]
+ 3--'.(COO>r, i aCaCa~ c=
•(N-3),
(25)
where we have retained terms to 0'(N-2). R e a r r a n g i n g eq. (25) and iterating, we obtain
r = (B>r,- ~(d~>.: 1 <~(.(.>r' 3! +-~(CC)r':
aZ(B)r,
a(C>,..a(C>r, O3(B>~, a( C )r.a( C)r,O( C)r,
O(C>r,a(O-,>r,
O((',)r,O(g',)r/
~?(N-3)"
(26)
F r o m eqs. (23) and (24) it is easily s h o w n that
a(C,)r, (C,~'j>r,,
(27)
ayj
and (27')
_
cgyk We first consider some binary correlation f u n c t i o n of the form
(BD)r = (BD)r-(B)r(D)r,
(28)
where B and D are any d y n a m i c a l variables. E m p l o y i n g eq. (26) to ~ ( N - ' ) it follows that o2
o(C>c,O(C)r, O(B>r,O(D>r,
- (Co)r,:
a
(29)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
483
which by eq. (27) may be written as = (/~(1 - P ) D ) r
- -~(d¢)r:
+ ~(N-2).
(29')
a(C)r,a(C)r
In this last expression the operator P is defined as
PB
-
(BC)r,
"
(30)
(C(~)r)" ¢
and is simply the projection operator onto the fluctuating parts of the variables C in the F' ensemble. We also need to know to 6 ( N -2) how triple correlation functions of the form {/~/5/~)r may be expressed in terms of F' quantities. This is accomplished by using eq. (26) in a rather tedious calculation, and the result is given in the appendix. Talking the F' ensemble to be microcanonical and using eqs. (26) and (29') and the appendix, eq. (21) becomes a(r, t) =/3(A(r, t)(1 - P),i~(r,))r, * F(rO
x [l
O(A(r2))r.a(E)r, F(r2)] +½13z[([(1-P)A(r,t)],i~(r,)A(rz))r ,
- (,i,r, t)[P.ii(rl)]A(rz))r- (A(r, t)A(r])[PA(r2)])r . . ^ a(A(r))r, O(A(r]))r a(A(rO)r + 2(CFiCDr a
a2(A(r))r
+ ½((%G)~,(G G )~, Ua( C,)~a(G)~
x 0(A(r,))r, O(A(r2))r + 4cg2(A(r))r, O(A(rL))r, a(A(r2))r, a(Ck), o(G),, a(C,>r,a(Ck)r, a(G>r, a(C,>r, 2 other permutations of the ]] + position of the double derivatives_lJ ~ F(rz)F(r0 + (9(N-I),
(31) where repeated indices are to be summed. If the set of variables C includes the energy and we impose the condition
O(A(r))r a(G>r,
• F(r) = 0
(32)
for all i, eq. (31) simplifies to a(r, t) = fl(A(r, t)A(r,)>r * F(rO + ½/3z([(l -P)A(r, t)]A(r]),ii(rz))r * F(rz)F(rl).
(33)
484
D. RONIS AND I. OPPENHEIM
It should be noted that eq. (32) is equivalent to
(34)
( ~ , A ( r ) ) r , * F ( r ) -- 0,
which is a c o n s e q u e n c e of eq. (27). When F' is a canonical ensemble (C = E) in the homogeneous limit, the response equations (33) and constraint equation (32) are equivalent to those obtained by one of us 1) through correcting for projections onto conserved variables. Let e(r, t) be the average displacement from equilibrium of the densities of the variables O. According to eqs. (33) and (34)
(35)
d3r c(r, t) = ¢3(~,.~(r))r, * F ( r ) = O.
Since the actual relaxation occurs in an isolated system the integrals of the densities of the nonfluctuating quantities must never change from their equilibrium values, independent of the choice of ensemble used to express the correlation functions. This is shown by eq. (35). In the next section an outline of the derivation of the h y d r o d y n a m i c equations shall be presented. Owing to the similarity between eq. (33) and an equation derived by one of us earlier, most details will be omitted as they are contained in ref. 1.
4. Hydrodynamic equations The first step in obtaining the transport equations is to express the fictitious forces in terms of the variables a (r, t). This is most easily accomplished when F' is a grand canonical ensemble. Here the correlation functions can be expected to decay exactly to zero beyond some microscopic correlation length and the linear transformation a~)(r, t ) - - / 3 ( A ( r ,
(36)
t)A(r,))o.c~ * F ( r , )
can be inverted. In this section the symbol ( )G.c. denotes a grand canonical average. Defining the inverse of the kernel in eq. (36) through K-l(rlr~; t) * (A(r,, t)A(r'))G~c. = 6 ( r - r')l
(37)
and using eq. (33), it is easily shown that [3F(r) = K-l(rlr~; t) * [a(r~, t) -½([(1 - P ) A ( r ,
t)JA(r2)A(r3))~.c.
• (K-t(r3[r4; t) * a(r4, t))(K ~(r2lrs; t) * a(rs, t))] + ~7(a3).
(38)
NONLINEAR RESPONSE FOR INHOMOGENEOUS SYSTEMS
485
Since for a grand canonical e n s e m b l e the new fluctuating variables t2 are the total n u m b e r and energy, both constants of the motion, eq. (33) implies ~i(r, t) =/3(A(r, t),4(r0)G.c. * F(rO (39)
1 + ~(A(r, t),4(r,),'~(r2))Gc_ * F(r2)F(rd,
which by eq. (38) b e c o m e s to 6~(a 2)
a(r, t) = M(rlrl; t) * a(rl, t) (40)
+ N(rlrtlr2; t) * a(rz, t)a(rl, t).
In the last equation (41)
M(rlr,; t) ~ (A(r, t),/~(r2))G.c. * K-i(rzlr~; t) and
N(rlr,lr2; t ) - ( A ( r , t)A(r3)A(r,))c.c. ** K ](r41rl; t)K-l(rslr2, t) - ½(A(r, t)A(rs))~.c. * K-'(rslr4; t) • ([(1 - P),4(r4, t)],i~(rs)A(r6))c.c.** K-~(rsIr,; t)K-t(r61r2; t). (41') As was shown in ref. 1, the projection operator appearing in eq. (41') m a y be omitted since both n u m b e r and energy are c o n s e r v e d quantities whose densities must be included in the set of variables A. H e n c e the relaxation equation (40) is identical to that obtained by Weare and Oppenheim2), starting with a canonical distribution, providing eq. (35) holds. The set of variables A(r, t) was a s s u m e d to be slowly varying. Therefore eq. (40) is rewritten, exactly as in ref. 1, to make as m a n y time derivatives as possible explicit. The result is to second order in the smallness parameter characterizing the change of the slow variables
a(r, t) = M(rlrl; 0) * a(rt, t) + N(rlrllr2; O ) ,* a ( r 2, t ) a ( r l , t) l
- f dtt(l(r, tOl(rO)a.c * K-t(rllr2; 0) * a(r2, t) 0 t
+ f dt, [½(l(r, t01(r0)a.c. *
K-'(rdr2; 0)
0
• (,4(r2),4(r3),4(r4)}6.c.- ½(l(r, tOl(r3),4(r4))a.c. - ½(l(r, tl)1~(r3)i(r4)}Gc.]* (K-'(r41rs; 0) * a(rs, t))(K-t(r31r6; O) * a(r6, t)), (42)
486
D. RONIS AND I. OPPENHEIM
where the generalized dissipative current, /, is defined as I ( r , t ) ---- A ( r ,
t) -
M(rlr~; O) * A(X N, rt).
(43)
These dissipative currents have the property of being orthogonal to the slowly varying quantities. That is,
(l(r, t)A(r'))~.c. = 0
(44)
to the first order in the smallness parameter characterizing the magnitude of the time derivative of A. Correlation functions involving the dissipative currents should decay on a molecular time scale. Therefore the limits of the integral in eq. (42) may be extended to some time, r0, much longer than the duration of any experiment. As in ref. 1, we define force-like quantities, ~ ( r , t), by a(r, t) =-- fl(,4(r).4(r,))~.c. * ill(r1, t)
+ ½flz(,i~(r)J~(r,),ii(r2))c.c. * ¢11(r2~ t ) ~ ( r , , t),
(45)
or, using eq. (37),
fiR(r, t) = K l(r[rl; 0) * [a(rl, t) -~(A(rOA(r2)A(rs))G.C"
~ (K-l(r3]r4; 0) * a(r4, t))
× (K-l(r2]rs; 0) * a(rs, t))] + ~(a3).
(46)
Eq. (42) in terms of the ~ ' s becomes
a(r, t) = fl(A(r)ll(rO)~.c * ~(rt, t) + ½fl2(A(r)A(rl)Ji(r2))G.c. ~, ~(r2, t ) ~ ( r l , t) "r0 i-
-
J dt.[13(l(r, t,)l(rj))c,c * q~(r~, t) 0
+ ½/3Z[(l(r, t)l(rO,4(r2))c.c. + (l(r,, t)A(rl)l(r2))~.c.] * q~(r2, t)~(r~, t)]. (47) The force-like quantities may be used to construct a local equilibrium distribution function, fL(X N, t). AS was shown in ref. 1, / ~ c ( X ~) exp [J3A(X ~, r) * ,i~(r, t)]
/L(X N, t)----
/-
,
(48)
| d X N f~.c.(X N) exp [/3A(X N, r ) * q)(r, t)] N=0
d
where/G.c.(X N) is the grand canonical distribution function. From eq. (45) it
NONLINEAR
RESPONSE
FOR INHOMOGENEOUS
SYSTEMS
487
follows that
a(r, t) = (A(r))L(t) + ~7(q~3)
(49)
and from eq. (47) rO
,el(r, t) = (A(r))L(t)-- fl f dtl(l(r, tOi(rl))L(t) * eP(r,, t) + ~(q~3),
(50)
o
where ( ) L ( t ) represents an average using the distribution function given by eq. (48). The quantities ~ ( r , t) are now seen to be the thermodynamic forces conjugate to the macroscopic variables n(r, t), as was shown in ref. 2. The actual forms of the hydrodynamic equations for given sets of variables A(r, t) have been discussed in ref. 2 and will not be presented here. In the next section we shall summarize the results of this paper and discuss some of the finer points of this theory.
5. Discussion A careful study of response theory and generalized hydrodynamics has shown that the expressions obtained in homogeneous systems are applicable to inhomogeneous ones. The transport equations are in general local in time but nonlocaI in space. The choice of ensemble used to describe the relaxation is irrelevant, providing variables such as the total number and energy are never displaced from equilibrium. Strictly speaking, this work is only applicable to inhomogeneous systems. In homogeneous ones, as shown in ref. 1, the total momentum is conserved and the microcanonical ensemble will not be the correct ensemble for which the mixing assumption, eq. (15) will hold (as discussed in section 2). The transformation theory of section 3 would include projections onto the total m o m e n t u m [as in eqs. (33)] but would not affect the final transport equations, form of the local equilibrium distribution function, or definition of the conjugate thermodynamic forces [eq. (46)]. In a later paper 4) the specific forms of the transport equations for some inhomogeneous systems shall be discussed. There the conditions needed to obtain forms local in space shall be presented.
Appendix In this appendix we shall show how triple correlation functions of the form (/3/)/~)r transform to the F ' ensemble. For our purposes only corrections to (?(N -2) need be retained.
488
D. RONIS AND I. O P P E N H E I M
Eq. (26) may be rewritten as F + ~ ' , ? - -
,, =
a2(B)r
a
a(C)r,a(C)r,a(C~)r,
a~(B)r ,
r(4)
O~r,
~- I , . ' 2 -
+ -ijkt a (C~)r,a< Q)r,a (Ck)to (C,)r
+ C( N -~3),
(A..~)
where repeated indices are summed, (A.2)
a
a(G)r,
(A.3)
3!
and (A.4)
By using eqs. (27) and (A.2) it is easily shown that
l~i~' = ~( C, Cidk>r,.
(A.5)
Note that
(A.6)
+ 2(B)r
where the symbol . . . represents the other two permutations of the variables B, D, E. By using eq. (A.1) in each of the averages appearing in (A.6) and retaining terms to ~(N 2), we obtain after some lengthy manipulations
(BI~E>r =
~-2 - - - - 8 ( Ca(Cj)r, i)r, 4- . . . a2(B)r,
a(/5/~)r +...
+ ~i3~ a(C,)r,a
a4
+ 4 a
+ ~i~ a(C)r,a(C)ra
a3(B>r , a
a
a(B)r, a
a
a3
+ •. .
a
N O N L I N E A R R E S P O N S E FOR I N H O M O G E N E O U S SYSTEMS
4
02(B)r,
o(C)r,a(Cj)r, a(cor, a(C~)r,
+4
+''
which
02(DE)r,
+...
O(C~)r,O(Ck)r, a(Cj)r,O(Ct)r, "+8
O2(B)r ,
489
+...+2
cg(D)r, 3(E)r,
a(C~)r,a(CDr, a(Cj)r, c9(Ct)r.
~2(B)r,
O(D)r, O(E)r,
o(c,)F,a(cj)r, a(COr, a(C,)r, ~-" ' " / + _1
G(N
3),
(A.7)
is t h e d e s i r e d r e s u l t .
References 1) I. Oppenheim, in Topics in Statistical Mechanics and Biophysics: A Memorial to J.L. Jackson, R.A. Piccireili, ed. (A.I.P., New York, 1976) p. 111. 2) J.H. Weare and I. Oppenheim, Physica 72 (1974) 1, 20. 3) R. Kubo, J. Phys. Soc. Japan 12 (1957) 570. R. Kubo, M. Yokota and S. Nakajima, J. Phys. Soc. Japan 12 (1957) 1203. 4) R. Pasmanter and I. Oppenheim, Physica 84A (1976) 507. 5) D. Ronis, J. Kovac and I. Oppenheim, to appear in Physica. 6) T. Keyes and i. Oppenheim, Phys. Rev. A7 (1973) 1384. I.A. Michaels and I. Oppenheim, Physica glA (1975) 522. 7) N.G. Van Kampen, Physica Norwegica 5 (1971) 10. 8) J.L. Lebowitz, Hamiltonian Flows and Rigorous Results in Non-equilibrium Statistical Mechanics, in the Proc. I.U.P.A.P. Conference on Statistical Mechanics, S.A. Rice, K.F. Freed and J.C. Light, eds. (Univ. of Chicago Press, 1972). 9) Ja.G. Sinai, Ergodicity of Boltzmann's Gas Model, in Proc. I.U.P.A.P. Meeting, Copenhagen, 1966, T.A. Bak, ed. (Benjamin, New York, 1967) p. 559. 10) J.L. Lebowitz and J.K. Percus, Phys. Rev. 122 (1961) 1675. ll) For a discussion of ergodic theory and the choice of ensemble-, see: A.I. Khinchin, Mathematical Foundations of Statistical Mechanics, chapter III (Dover Publ., New York, 1949).