Contribution to Statistical Mechanics far from Equilibrium. II

84

Progress of Theoretical Physics, Vol. 52, No. 1, July 1974

Contribution to Statistical Mechanics far from Equilibrium. II

--Extension to Non-Steady States and to General Stochastic Models--

Kyozi KAWASAKI

Research Institute for Fundamental Physics, Kyoto University, Kyoto

(Received February 18, 1974)

The new approach to non-equilibrium statistical mechanics initiated in Part I of this series is extended to cover non-steady states as well as more general stochastic models. The logarithm of the solution to the stochastic equation for the probability distribution function of gross variables is obtained as a systematic perturbation series starting from the logarithm of the suitably defined Gaussian part of the local equilibrium distribution function, which is best expressed in terms of the characteristic function of the probability distribution function. Again the effects of heat baths which surround the system enter only on the level of fully renormalized macroscopic equations of motion. A self-consistent scheme was derived which determines the macroscopic law of evolution and the probability distribution function of fluctuations.

§ I. Introduction

In the first of this series1l (referred to as I hereafter) a new apprbach · to statistical mechanics far from equilibrium was proposed · by treating a simple stochastic model. One of the merits of this new approach· is the fact that influences of heat reservoirs enter only on the level of fully renormalized (and hence experimentally observable) macroscopic equations of motion, thereby allowing us to suppress irrelevant details about nature of the contacts with heat reservoirs. Of course if the nature of such contacts happens to be important, perhaps one ought to regard a portion of heat reservoirs as a part of the system under consideration. On the other hand, we restricted ourselves to steady states independent of time and also to rather special model systems. However, from the viewpoint of applying such a method we must extend the method to nonsteady states (chemical oscillations,2l nonlinear phonon interactions,3l electrical oscillations, etc.4l) and to more general types of stochastic models5l (e.g., time-dependent Ginzburg-Landau type models or more general master equation type models). Here we intend to accomplish both extensions.

In I we made use of the projection operator techniques in order to obtain steady state distribution function out of the local equilibrium one. We have tried to do the same for more general situations under consideration in this work without success.*l However, the fact that in I th.e projection operator could be

*) We shall see later that the projection operator can be included with no harm to the theory. See § 5.

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Contribution to Statistical Mechanics far from Equilibrium II 85

eventually dropped suggests an approach which does not use a projection operator from the outset as we shall describe in this paper.

§ 2. Kinetic equations

Our starting point IS the following master equation for the distribution function g ({a}, t) of the gross variables {a} that includes the variation due to reversible drift vi ({a}):

-g( {a}, t) =- :E -vi({a})- Wt( {a},{x} )g( {a}, t)dx 8 8 s 8t i 8ai

+ S Wt( {a-x}, {x} )g'( {a-x}, t)dx, (2·1)

where wt ({a}' {x}) denotes the (in general time-dependent) transition probability . per unit time from a state {a} to another state {a+ x} which accounts for

irreversible change. *l This may be written in accordance with Kubo et al. as

_J_g ({a}, t) =Ht[ {_]_}, {ai}Jg ({a}, t), 8t , 8ai

(2·2)

where

Ht[{_J_},{ai}J=- ~_]_vi( {a})- Jax{l-exp[- ~xi_]_]} Wt( {a}, {x}) 8at • 8ai • 8ai .

(2·3)

with

m1t(i;{a}) =vi( {a})+ S dx XtWt( {a}, {x} ), (2·4)

mnt(id2"'in;{a}) =J dx Xi 1Xi0 "Xin Wt( {a},{x} ), n>l. (2 ·4')

Here we have used the Kramers-Moyal expansion in the second step. The stochastic equation (2 • 3) is more general than the one considered in I not only because (2 · 3) has a more general form but also because a part of the effects of heat reservoirs can be included in mnt, for instance one can include in this way the effects of heat reservoirs which spatially overlap with the system under consideration, e.g., chemical reservoirs in chemical reactions or lattices in the case. of spin lattice relaxation. We may call these types of heat reservoirs the

spatially ov~rlapping heat reservoirs. On the other hand, there are also heat reservoirs which contact with the system under consideration only at the boundanes of the system and specify the boundary conditions, which we may call

*> Here we generally follow the notations of I unless otherwise stated.

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86 K. Kawasaki

surrounding heat reservoirs. This latter type of heat reservoirs cannot be included in the stochastic equation in a simple way (that is, without including irrelevant, ambiguous details). In I we have avoided this difficulty by formulating the problem in such, a way that the influences of the surrounding heat reservoirs enter only on the macroscopic level.

Our strategy proposed in I to handle (2·2)' is not to attack it directly but to construct first the local equilibrium distribution function g1 ({a}, t) having the same average values {c} of {a} as for g ({a}, t).

In most situations encountered in physics there are well-defined prescriptior{s to construct a local equilibrium state knowing {c}. For instance, one can imagine applying a field conjugate to every gross variable at time t, and simultaneously cuts ·off contacts with heat reservoirs entirely and leaves the system until an equilibrium is reached in the presence of the conjugate fields. One then repeats this process with different values of the fields until final average values of the gross variables obtained coincide with their initial values {c (t) }. This state is identified with the local equilibrium state at time t. The probability distribution function of the gross variables in this state is denoted as g1 ( {p;}; {c (t)}) or simply as g1 (t). Since local equilibrium states can be dealt with by equilibrium statistical ~echanics, in this work we shall assume knowledge ~f g, (t). Our problem could then be formulated as a problem of obtaining the non-equilibrium distribution function g (t) knowing g1 (s), (s<t).

The first step taken in I to obtain g (t) was to consider a "wave function" ¢ = g£ 112g so that in a local equilibrium state ¢1 =gif2 takes a Gaussian form. Here we have some complications since we no longer assume g1 to be Gaussian. Let us first consider the "wave function" ¢ (t) defined by

¢ (t) =g£1 (t) g}~2 (t) g (t)' (2·5')

where g10 (t) is a Gaussian part of g1 (t) and will be specified more precisely later on. Equation (2 · 2) then takes the form

with

_J___¢ (t) = ._g{t¢ (t) at

(2·6)

.j{t=g£1 (t) g}~2 (t) Htg 1 (t) gi(//2 (t) - g£1 (t) (/z (t) +! gi(/ (t) flzo (t), (2 · 7') where dots denote time derivatives. Now, g1 (t) depends upon time through {c£ (t)} or the corresponding conjugate fields {b£ (t)} which appear through the factor exp[L:;t b£(t)at*] and also through the ,normalization factor in g1(t). Hence we obtain

_2_ In gz (t) =I; tt (t) ob, (t) 0 In gz (t) =I; tt (t) [X-1 (t) ]1,ata1*, at 'i Bet (t) ab, (t) ij

(2·8)

where atai=ai- Ci (t) and X (t) is the susceptibility tensor whose ij-element is

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Contribution to Statistical Mechanics jar from Equilibrium II 87

(t) - OCt (t) <~ ~ *) XtJ - abJ (t) utawtai u . (2·9)

Here < · · · )u is the local equilibrium average at time t. Hence x (t) in fact is

- some known function of {c(t)}. Now, there is certain arbitrariness in the choice

of the Gaussian part g10 (t), and one is free to choose g10 (t) which suits him best. We find it convenient to choose g10 (t) to be the normalized Gaussian

probability distribution function having the same averages of at and the same

susceptibility tensor as those for g1 (t). Then (2 · 7) takes the following form:

!f{~-=H~[.{l+ 8ln[gz(t)g!Olf2(t)] ·}, {at}] 8at Oat

(2 ·10)

Here the complication due to non-Gaussian g1 (t) surfaces up again by- the

fact that the simple operator 8/8at in H 1 is replaced by a complicated operator. If one wants to avoid this complication, one must replace the original problem

by another problem of obtaining g (t) knowing g10 (s), O<s<t, where an additional

assumption is introduced. See ( 4 · 23) below. Then in the foregoing formulation

we may replace g1 (t) everywhere by g10 (t). In particular (2 · 5'), (2 · 7') and

(2 ·10) become

cfJ (t). = gw112 (t) g (t),

!JCt=g!Ot;2(t)Htg~~2(t) _! 8lnto(t) t '

' (2·5)

= H 1 [ { 8~t + ! 8 1~~:0 (t)} {at} J - ! ~ t. (t) [X-1 (t) ]1.tJ1a1*. (2 · 7)

§ 3. Second quantization representation

One complication which we encounter in generalizing the second quantization

representation of 'I here is the fact- that the susceptibility tensor x (t) can change

in time. Thus we cannot choose a suitably normalized set of {a} unless we

a1low explicit time dependence of {a}. Thus we introduce a normalized set of

gross variables {a1} such that (a/)u = 0 and (at11Z~*)u = (JtJ> in terms of which

{a} are expressed as

a,=ct(t) + L: Ut1 (t)a/, . j

where the matrix (uu (t)) IS related to X (t) through

x(t)=u(t) ·ut(t).

(3·1)

(3·2)

The second quantization representation is then obtained by introducing the time

dependent Boson creation and annihilation operators a,1 and a/, respectively,

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88 K. Kawasaki

which are related to a/ through a/= a/+ a~. and 8 /8a/ =i- (a~.-a/). Thus, in this representation, we have

8 "' [ -I ( ') J 1 ( t' - t') --;---=..::...... u t Ji-2 af*-a1 uai J

with the following commutation relations:

where

[a/, a/']= -tUw(tt') +tU1i•(t't),

[a/, a/] =tUi,(tt') +tU,.i.(t't),

[ - t - t'] 1 u ( 1) 1 u ( I ) ai ,a1 = 2 i*J tt -2 J*i t t,

U(tt') =u-1 (t) · u (t').

(3 · 3a)

(3·3b)

(3 ·4a)

(3·4b)

(3·4c)

(3·5)

Note that at and av behave like Bose operators only for particular times t=t'. Thus it is convenient to obtain relationships between a and a referring to different times. By means of the matrix notation, (3 ·1) is written as

iJta = u (t) ·at or at= u-1 (t) · ifta .

Therefore we find

at= U(tt') ·av +u-1 (t) · [c(t') -c(t)J. (3·6) Similarly we obtain ,

(3·7)

Using these results together with the definitions of at and at, we finally obtain

a/= t~ [ uij (tt') + uj*i* (t't) J a/'+ t ~ [ uij (tt') - uj*i* (t't) J a~. j j

(3 ·Sa)

(3·8b)

In this representation !J(I, (2 · 7), becomes

!J{t=flt( {a}, {a}) -t ~ ti(t) cx-1 (t)],iiJtaj* (3·9) ij

with

(3 ·10)

where ai is given by (3 · 3a). The average of any function X of the gross variables in the non-equilibrium

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state ¢ (t) can be expressed in the second quantization representation as

(X)t= S X( {a}) g ({a}, t)da = (1/Jo(t) IX( {a}) I¢ (t)), (3 ·11)

where we have used the Dirac bra and ket notation to represent states. The

normalization condition is then

(1/Jo (t) I 1/J (t)) = 1 . (3 ·12)

The fact that the average value of (Jtai vanishes m I¢ (t)) is expressed as

0=(1/Jo(t)I(Jta;II/J(t)) or 0=(1/Jo(t)la/lf/J(t))=O. (3·13)

This implies that ¢ (t) contains no single particle states.

§ 4. Non-equilibrium distribution function

Our strategy to obtain a non-equilibrium distribution function g ({a}, t) at

time t is as follows. First we assume the knowledge of time variation of

averages {c(s)} of the gross variables prior to the time t. We also assume

that at some distant past, say at t 0 which may be taken to the infinite past, the

system is in a local equilibrium state compatible with {c (t0)}. As time goes on

the system evolves from its initial local equilibrium state at t 0 towards the true

non-equilibrium state by the stochastic equation (2 · 2) or (2 · 6) which are valid

at least well inside the system where surrounding heat reservoirs have no direct

effects. We construct the solution g (t) of the stochastic equation in such a

way that it depends only on the values of {c} at times very close to t. The

macroscopic equation of motion becomes

ti (t) = J a;Htg (t) da . (4·1)

Then we solve ( 4 ·1) with the boundary conditions specified by the surrounding

heat reservoirs. If the initially assumed {c (s)} were correct, they must c;:oincide

with the {c(s)} obtained by solving (4·1). In this way {c(s)} are determined

self-consistently. The distribution function g (t) obtained in this manner is iden

tified as the true non-equilibrium distribution function at the time t.

Now, one may doubt the procedure outlined above because effects of the

surrounding heat reservoirs do not enter the stochastic equation. Ho~ever, if

one notes that the major role of the surrounding heat reservoirs is to specify

the boundary. conditions on {c}, any other residual effects of the surrounding

heat reservoirs will be confined to the immediate vicinity of the boundary and

the non-equilibrium distribution function obtained here should correctly describe

· fluctuations of the gross variables occurriqg well inside the system.

Let us now carry out the program- outlined above in the second quantization

representation. First note that for a local equilibrium state whose distribution

function is approximated to be Gaussian, ¢ (t) reduces to the Gaussian form

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90 K. Kawasaki

g}~2 (t) which is denoted as ¢0 (t) implying a free vacuum state of Bose. particles described by {a'} and {a'}. We then expand H', (3 ·10), in a' and a' as follows:

(4·2)

where {i} denotes the set of indices ii, is, · · ·, i,. and the same for {j}. Note that the form of the general kinetic equation (2 ·3) requires that

Ji,. ( {i} {j}) = 0 . (4·3) ,g.{' contains a part denoted as !IC;/ which describes a slow time variation of ¢0 (t):

(4·4)

Since ¢0 (t) changes with time only through {c(t)}, we obtain by (2·8) which is valid for g10 (t) as well, and also by (3 · 3a) and (3 · 2),

_]__1/Jo (t) = _!_ a In gzo (t) 1/Jo (t) =_!_ ~ t, (t) [x-1 (t) J,,~,a,•I/Jo (t) at 2 at 2 ,,

= ~ ~ t,.(t) [u-1 (t)T]H (a~*+ a/) 1/Jo (t), (4·5)

where the superfix T denotes transpose of a matrix .. , Thus .!J(0' takes the general form

(4·6) where

(4·7) and

(4·8)

We choose !IC01 so that the remainder

!JC'' = !JC'- !fCo' (4·9) describes rapidly varying processes. As we shall see later [see the discussion following Eq. (5 · 8)] it turns out that we can choose

!ICo'= ~ JA(i,j)a/a/+t ~ t, (t) [u-1 (t)]1,(a/-a~.). iJ iJ

We then find, noting ( 4 · 3), that

where

!fC '' = !JC/ + !fCs' '

nr '-"-Mota- t L/1,1=£..-J i ,, i

. (4·10)

(4·11)

(4·12)

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Contribution to Statistical· Mechanics far from Equilibrium II 91

(4·13)

and

Mt0t=Jfo(i) -:E[u(t)-1]tA(t). (4·14) j

Here :E~,. is the sum in which the terms with m = 1, n = 0 and m = n = 1 are excluded. Note that !/{'t contains the processes in which two modes are created m vacuum at the time t and also those in which more than three modes interact at the -time t.

We can find Jfo(i) by comparing (2·3) with (4·2) where there is a problem with the {a}-dependence of m,t. The best way to handle this is to expand m,t npt in powers of Ota' s but to express it in terms of properly normalized Hermite polynomials ¢,.' of Ota. Then ¢n is equal to: ITt (ai + ai*) "i: apart from some numerical coefficients where :: denotes a normal product of operators a and a.*> Hence only the lowest order term of this expansion contributes to the J{0 term of · ( 4 · 2) . In this manner we find

J1~ (i) = E [ u-1 (t) ]itm/ (j, {c(t)}), ( 4 ·15) j

where m11 (j, {c (t)}) is the average of m11 (j, {a}) in the state g10 (t). Therefore ( 4 · 4) becomes

(4·16)

Since m1t (j, {c (t)}) is the rate of change of ci in a Gaussian local equilibrium state, the equation MZ1 = 0 1s nothing but the so-called "bare" macroscopic equations of motion,**>

t~. (t) = m11 (i, {c (t)}), (4·17)

provided, of course, the ~atrix u (t) is non-singular, which we assume to be the case. However; the true macroscopic equation of motion that one observes in macroscopic measurements is not (4 ·17) but the fully renormalized one***> which r

can be obtained from ( 4 ·1) or

(4·18)

*l -Further details of this will be found in K. Kawasaki, in Phase Transitions and Critical Phenomena Vol. V, eds, C. Domb and M. S. Green (Academic Press, London, 1974).

**l There are exceptional cases when the starting master equation (2·1) contains effects of partial renormalization of fluctuations. Such ,an example is given by Kawasaki in S:ynergetics, ed. H. Haken (B. G. Teubner, Stuttgart, 1973).

***l For instance, for the model adopted in I, (4·17) contains only the unrenormalized Onsager kinetic coefficients. For the TDGL model, (4·17) coincides with the result of the local equilibrium approximation which gives the wrong critical exponent, and the memory effects contained in the second term of (4·19) is crucial to obtain the correct critical exponent (K. Kawasaki, to be published). In our opinion (4·18) and (4·19) should contain the so-called turbulent viscosity provided that ¢(t) correctly describes the probability distribution of turbulent flow where (4·17) can be taken as the Navier-Stokes equation, although this is yet to be seen.

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92 K. Kawasaki

By means of (3 ·12), (3 ·13), ( 4 · 2), ( 4 ·15) and (3 ·1), this becomes

ti(t) ='ffirt(i, {c(t)}) + E UiJ(t)~ - 1- 'EJfn(j, {l}) j n=2 n! {!}

(4 ·19)

where the second term represents the renormalization contribution to m/. We now proceed to the construction of cjJ (t), which can be related to cjJ (t0)

·at some distant past t 0 by formally solving (2 · 6) as

where exp+ (or exp_) denotes the usual time ordered exponential in which !}{' with greater s always comes to the left (or right) of those with smaller s. Then one can easily obtain the following identity:

where

(4·22)

and

(4·22')

We now make one basic assumption here: We assume that at the distant past t 0 the distribution function g (t) was given by the Gaussian local equilibrium distribution function g zo (t0), hence

cjJ Cto) = c/Jo Cto) (assumption). (4 ·23)

Since g zo (t) has been constructed so that the first and second moments of at for gzo (t) coincide with those for g (t), the difference between gz (t0) and gzo (t0) will quickly disappear, and in this sense our assumption is practically equivalent to assuming the true local equilibrium state at the distant past t 0• Then, noting that from (4·4) we have

cjJ0 (t) =exp+{ i:ds 3fo'}¢0 (t0), (4·24)

( 4 · 20) reduces to

cjJ (t) = S (tto) cp0 (t), (4·25)

where S (tt0) plays the role of an S-matrix which transforms the Gaussian local equilibrium distribution function at time t to the true non-equilibrium distribution

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Contribution to Statistical Mechani~s far from Equilibrium II 93

function at the same t.ime. We are, however, not yet ready to start calculations using ( 4 · 25) because

by ( 4 · 22') S (tt 0) still contains Bose creation and annihilation operators a! (t, s), a! (t, s) referring to different times s which are defined by

a!(t;s)=exp:{ fds'3Ct}a; exp_{- fds'3Ct}, etc. (4·26)

We must express them in terms of the Bose operators a/ and a/ referring to the time t. First note the following equations of motion readily odtained from (4 · 26):

_l__a; (t, s) = :E JA(ij) a/ (t, s) +a/ (t, s) + _l L:; [u-1 (s) ];A(s), (4·27a) OS j 2 j

where

(4·28)

It is convenient to employ the matrix notation hereafter. First we introduce the composite column vector At defined by

At= [:1 (4·29)

where at is the column vector whose i-th element is a/ and at is another column vector whose i-th elements is a~.. ( 4. 27) is then simplified to

(4·30)

where A' (ts) and A• (ts) have similar meaning as a• (ts) and a• (ts) etc. and !N1

is the composite matrix defined by

!!'= [ Ji1 0 J o -JN .

(4· 31)

J{1 being the submatrix with its ij-element given by Ji1 (ij), and JN its Hermitian conjugate. - The last term of ( 4 · 30) is meant to be a double column vector cons-isting of two identical column vectors also denoted as tu-1 (s) · c (s).

We shall follow similar convention hereafter. In order to obtain A• (ts) we go back to (3·8) which becomes in matrix notation,

At= CU (tt') A~'+ tu-1 (t) · [ c (t') - c (t)],

where CU (tt') is a composite matrix defined by

[HU(tt') + Ut(t't)]

CU(tt') = t[U(tt')- Ut(t't)]

t[U(tt') -Ut(t't)]J

t[U(tt') + Ut(t't)]

(4·32)

(4· 33)

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94 K. Kawasaki

with U(tt') defined by (3 · 5). Differentiation with respect to t of the last term of ( 4 · 32) yields - tu-1 (t) : t (t) after setting t' = t. Similarly we have

Je=_J_CU (tt') I t'=t at = [-tu-1 (t)u(t) +tut(t) [ut(t)]-1

-tu-1 (t)u(t) -tut(t) [ut(t)]-1

-tu-1 (t)u (t) -tut(t) [ut(t)]-1]

-tu-1 (t)u(t) +tut(t) [ut(t)J-1 •

In this way we obtain from (4·32)

At=JCtAt-tu-I(t) ·t(t) ..

Substituting this into ( 4 · 30) we finally obtain

_J_A'(ts) = (S'+JC') ·A'(ts) OS

(4·34)

(4·35)

(4·36)

(4·37)

In this way a/ (ts) and a/ (ts) contained in S (tt0) can be expressed as linear combinations. of {a/} and {a/}, and we are almost ready to obtain ¢ (t). However, we need certain properties of the propagator G (ts) entering ( 4 · 37) defined by

(~·38)

T being the transpose. For this purpose we consider the following commutation relation expressed in matrix notation:

' [A'•(ts1), A'•(tsg)] =GT(ts1) ·E·G(ts2), (4·39)

where E is given by

E= [At, At]= [~I ~]. (4·40)

I being the sub matrix whose ij element equals (Jid*· The differentiation of ( 4 · 39) with respect to t yields

with

fDt=_[t.E +E ._[tT = [- .£f2·~ + ~ . .£;[ .£f1·l + l.·.£;[] - . .G2 ·1 -1 . ..Cf[ ..c;1 ·l-l· ..Cf[ '

(4·41)

(4·42)

where ..C~fJ is a sub matrix of ..Ct. By ( 4 · 31) and ( 4 · 34) we notice the following properties:

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. .n2 =Lit =.LtJ,

.Yti = - .Li2 .

(4·43a)

(4·43b)

Using these, we immediate,ly verify that gy vanishes identically ensuring the independence of (4 · 39) upon t. Thus, choosing t = s1> s2 in ( 4 · 39) and noting that G (tt) is an identity matrix, we obtain

GT(ts1) ·E-G(ts2) =E·G(s1s2). (t>s1>s2)

Decomposing this into submatrices, we finally find

- Gf1 (tst) ·l· Gu (ts2) + Git (tst) ·l· G21 (ts2) = l· G21 Cs1s2),

- G~ (tst) ·l· G12Cts2) + Gi1 (ts1) ·l· G22 (ts2) = l· G22 (s1s2),

Gf2 (tst) ·l· Gu (ts2) - Gi2 (tst) ·l· G21 (ts2) = l· Gu Cs1s2),

Gf2 (ts1) ·l· G12 (ts2) - Gi2 (tst) ·l· G22 (ts2) = l· G12 (s1s2),

where G~p denotes the transpose of Gap, not~ the a#-submatrix of GT.

(4·44)

(4·45a)

(4·45b)

(4·45c)

(4·45d)

For the special cases in which x(t) and hence u(t) do not change in time, _[t equals !;t and thus ..£12, ..£21 and hence G 12 and G21 vanish. ( 4 · 45) then become for s1>s2

Gi1 (ts1) ·l· G22 (ts2) =I· G22 (s1s2),

Gf2 (tst) ·l· Gu (ts2) = l· G11 (s1s2).

(4·46a)

(4·46b)

Here G's have very simple interpretations which cail be best illustrated by assuming that !!' is independent of time and is diagonal: (Jtt)t1 = (iwt -Lt0) 1Ji1

where Wt an.d L/ are real. Then [Gi1(ts)]t1 =1Jt1 exp {(t-s) ( -iWt+L/)} and [Gf2 (ts)]t•f*=IJt1 exp{(t-s) (iWt-Lt0)}. Therefore, Gf2(ts) can be interpreted as the forward propagation of a mode from the past to the future whereas Gi1 (ts) is the backward propagation of a mode from the future to the past as shown in Fig. 1 where time is ordered from the right to the left. The relations ( 4 · 46) can then be easily understood: for instance, the left-hand side of ( 4 · 46a) describes the propagation of a mode from s2 to t and then from t back to s1, which is clearly equivalent to the propagation from s2 to s1 as shown on the right-hand side (see Fig. 2). When x (t) changes in time, G 11 and G22 are no longer independent but are connected through G12 and G 21• Here we use doubly directed lines to represent Gap as shown in Fig. 3. Then, similar interpretations as those given for (4 · 46) can be given for each of ( 4 · 45). Figure 4 illustrates such a diagrammatic a! interpretation for ( 4 · 45a).

-5~ 52 51 )•

$2 51 52 c( 52

1 j = [G22 (51:52l]ij [G11 (51 52l]ij St

Fig. 2. The diagram representing the both Fig. 1. Diagrams for Gu and G.,. sides of (4·46a).

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96 K. Kawasaki

)t (

Fig. 3. Diagrams for GaP·

t : 52

-~~--~--s-2 = <:::--~---. s

~2 s, s,

Fig. 4. The diagram representing the both sides of (4·45a).

§ 5. Diagrammatical method

The construction of a non-equilibrium distribution function from now on can ·be most easily achieved through diagrammatical method. Here we would like to explain this method by expressing S (tt0) as a sum of normal products of the Bose operators. First, we represent each term in !}{'1 of ( 4 ·11), ( 4 ·12) and ( 4 ·13) by a vertex from which a dashed line directed from the vertex to be called a-line hereafter emerges to the left for each operator a£ and a dashed line directed toward the vertex. to be called a-line hereafter emerges to the left for each operator a£,*> where the lines are ordered from the top to 'the bottom in accordance with the order in which the operators a's and a's appear from the left to the right in !}{'1• A typical such diagram is shown in Fig. 5. On the either hand, !}{'' (t, s)' instead of !}{" enters s (tto)' where a• =(a/) and a• =(a:.) are to be replaced by a'(ts) and a'(ts) defined by (4·26) or by the following which are derived from ( 4 · 37) and ( 4 · 38),

a'(ts) =Gi1Cts) ·a+Gf1Cts) ·a,

a• (ts) = Gi2(ts) ·a+ Gf2 (ts) ·a.

(5 ·1a)

(5 ·1b)

Accordingly, a term in !}{' 1 corresponding to a diagram such as that shown in Fig. 5 is changed in !}{'' (t;) to a sum of terms represented by diagrams which are obtained from the original one by replacing each dashed line by either of the lines representing each term of (5 ·1) as shown in Fig. 6 where time is ordered from the rrght to the left, and solid lines run from s to t. A typical

L --. --~ J ---"'(' ....... :\

:; ,'' k ___ or./ , .

.( ----Fig. 5. The diagram of a vertex representing

a term in .5{".

---->----- __ ,... ___ ~---~

____ ..,____ -->--~ -"'(--~

Fig. 6. The diagrams in the right column are derived from those in the left column.

*> From now on we simply write a;, etc. instead of al, etc. unless otherwise specified since in the following we express all the operators in a''s and a''s.

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~~:~~

----;:1 --~'r

Fig. 7. One of the diagrams derived from the vertex diagram of Fig. 5.

-~~ ~

--<.cA +' ~s ->--~. ',,.,i; ;-----, --,.,, + ~

s, Sz s, Sz

Fig. 8; Two examples showing elimination of an inverted pair of lines.

~=->-~~> +Q_:>+ » +_~ - /~- _ _._.

Fig. 9. Typical second order diagrams related to each other through contractions.

such diagram obtained from that of Fig. 5 is depicted in Fig. 7. Then a diagram representing a term in the perturbation expansion of S (tt0) expressed as a normal product of operators is constructed by the following steps: (1) Draw n vertices of the type shown in Fig. 5 representing terms of ( 4 ·12) and ( 4 ·13) where n is the order. of the perturbation and each dashed line is labelled by indices specifying the mode. These vertices are time ordered from the right to the left of the diagram as the time s of !]{" (ts) increases. (2) Replace every directed line in the diagram by one of the two diagrams shown in the right column of Fig. 6 where, how eyer, .the labels are omitted for simplicity. We now define the ordering of all the dashed lines contained in a diagram as follows; within a vertex the ordering goes from the bottom to the top, and among the dashed lines belonging to different vertices the ordering is from the right to the left. The third step is (3) to pick up any pair of lines in which an a-line comes after an a-line which may be. called an inverted pair, and to add a diagram in which this pair of dashed lines is replaced by a solid line with ® in the middle (contracted diagram). See Fig. 8 for two such examples. This process of contraction is repeated until all the inverted pairs of lines are eliminated. Typical second order diagrams which are related to each other through contractions are shown in Fig. 9.

Given a diagram, the corresponding term in the perturbation series of S (tt0)

is obtained by the following rule: (1) Associate appropriate factors a and a for each a-line and a-line, respectively, and form a normal product. (2) Associate a factor [Gill (ts1) ·l·.G2r (ts2) ]iJ with each line containing ® of the. type shown in Fig. 10, where dots indicate arrows whose directions are specified by {3 and r ( = 1 or 2) and when s1 = s2, the part of the line emerging at s2 has been assumed to precede the one emerging ·at s1• (3) Associate an appropriate

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98 K. Kawasaki

{3~~ L 51 52

Fig. 10. The diagram representing [Gft,(ts,) ·l·G,r(ts,)],,.

propagator Gap(ts) shown in Fig. 3 with every remaining solid line. (4) Associate appropriate vertex factors like · MZt and J!n ( {i} {j}) appearing in ( 4 ·12) and ( 4 ·13) with every vertex. (5) Integrate over all s, over the domains which extend from t 0 to t but are restricted by the particular time ordering of the vertices.

The entire contribution to the n-th order term in S (tt0) is now obtained by summing the contributions obtained above over all the ind'ices i, j, etc. specifying modes which are divided by the· symmetry number of the diagram (see I), and then by collecting the contributions of all the topologically distinct unlabelled (i.e., without indices specifying modes and times) diagrams containing n vertic,es of the type described earlier. ·

As an example, we display below the contributions from the diagrams of the form shown in Fig. 9 where the two diagrams on the right are topologically identical,

X [G22 (ts2) ]!nJio' (ij) Jill (mn) ak.a!*

+ :E :E :E ft ds1 f"ds2[G22 (ts1) ],k[G~l (ts1) ·l· G22 (ts2Hn ij kL mn Jt 0 Jto

X [G22 (ts2) ]imJio' (kl)Ji6 (mn) a,.aJ*. (5·2)

The part of S (tt0) -which contributes to ¢ (t), ( 4 · 25), is immediately obtained by discarding those terms in S (tt 0) which contain at least one a. For example, in (5·2) only the last two terms remain, which is equivalent to discarding the first diagram on the left in Fig. 9.

As in I, it is also possible to extend the Hubbard linked cluster theorem to the present more general situation. *l The result is that the operator S (tt0)

can be written as

S(tt0) = exp SL (tto), (5·3)

*> The generalization of the Hubbard linked cluster theorem for our time-ordered diagrams · proceeds almost parallel to the proof of Hubbard [Proc: Roy. Soc. (London) A240 (1957), 539] . except that the number of diagrams belonging to the sa/Tie class is a bit more complicated. Note that time ordering among disconnected diagrams is in fact immaterial.

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where SL (tt0) is the part of S (tt0) which arises from connected diagrams only.

For example, in Fig. 9 only the two diagrams on the right reJ?ain in SL (tt0).

We now .turn to the renormalized macroscopic equation of motion ( 4 ·19) which we may rewrite using (4 ·16) as well as ( 4 ·12) and ( 4 ·13) as

0 =MZt + f; __!___ :E Jfn (i, {j}) (ifio (t) I a~,a~, .. ·a~nl if! (t)) n=2 n! {J}

(5·4)

The diagram for the parts of 3-{'tl ¢ (t)) which contribute to the right-hand side of (5 ·4) has the form shown in Fig. 11 where the dot stands for, 3-{'t. The

hatched circle is a collection of connected diagrams of the kind described earlier

which represent the renormalized macroscopic equations of motion. As in I the

effects of contact with surrounding heat reservoirs only enter here as the boundary

conditions in solving this macroscopic equations of motion. This generalizes the

result of I that processes represented in Fig. 11 vanish due to the macroscopic

equations of motion. There is one difference, however, when x (t) changes in time. (5 · 4) tells us the absence of processes shown in Fig. 11 which end at

the time t in the second quantization representatibn in terms of at and ce. Thus

the story can be different for processes ending at different times and in the

second quantization representation using a• and a• with s=/=t. Nevertheless, of

course, (5 ·4) itself holds for any .. time t equal to s greater than t 0 in the representation that uses a• and a•. Then, for each time s one can define the in

general, time-dependent projection operators P. and P. of the type considered in

I which projects g(s) onto g10 (s) and cp(s) onto cp0 (s), respectively. See I (3·7) or I ( 4 ·15). Let us consider the effects of replacing 3-['• by (1- P.) 3-{,. in

( 4 · 25) and ( 4 · 22). For this purpose we suppose that in S (tt0) we change 3-['•

in a very small time interval around some particular time s to (1- P.) 3-['•. The

change induced in S(tt0) I ¢0 (t)) is then proportional to

exp+ { fds' 3C'"' (ts')} R (ts) F,3-{,.R-1 (ts)

X exp+ { fds' 3C'"' (ts')} I ¢0 (t) ), (5·5)

where

R(ts) =exp+ { fds' 3Ct}. (5·6)

By means of ( 4 · 22') and (4 · 24) as well as (4 · 22) and ( 4 · 25), we find

Fig. 11. The diagram representing vanishing contributions due to the macroscopic equations of motion. ·

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100 K. Kawasaki

R-1 (ts) exp+{ fds' 3{''' (ts')} l¢0 (t)) =exp+{ fds' 3C''' (ss')} l¢o(s)) = l¢(s)).

(5·7)

Therefore (5 · 5) becomes

exp+ { fds' 3{''' (ts')} R (ts) P,3C''I ¢ (s)). (5·8)

Since P. is the projection operator onto the vacuum state and the single particle states in the representation using a• and a•, we see that (5 · 8) vanishes in view of the structure of 3{', (4·12) and (4·13), and the property (5·4). This process can be repeated· to show that the expression for ¢ (t), ( 4 · 22) and ( 4 · 25), is unaffected by replacing 3{'' by (1-P,) 3{''. Namely, the vacuum and single particle states are again excluded from the intermediate states in a suitable representation. This conclusion generalizes the similar results obtained in I. If we make a plausible assumption*l that the difference between the Gaussian local equilibrium distribution function g10 (t) ~nd the true local equilibrium one g1 (t) varies rapidly in time compared to the time variation of the average (a.), 3C''(ts) in (4·22), which we may replace by R(ts) (1-P,)3C''R-1 (ts), changes rapidly in time s and will average out if s is separated from t by some microscopic time if we ignore the problems associated with the long time' tails.7l Hence ¢ (t) will not depend on {c (s)} and x (s) with s far distant from t, which we already asserted in § 4 when we chose the particular form, (4·10), for 3C01•

On the other hand, P. in front of 3{" in S(tt0) becomes P,(ts)=R(ts)P,R-1 (ts). In other words, for an arbitrary state lx) we have

P,(ts) lx) =R (ts) {(¢o (s) IR-1 (ts) lx)l¢o (s))

+ ~ (¢o(s) la(R-1 (ts) lx)a/l¢o(s))}. i

By making use of the relations (5 ·1) and the following:

(¢o(t)l = (¢o(s) IR-1 (ts), I ¢o(t)) =R(ts) I ¢o(s) ),

(5·9)

(5·10)

where we have used the fact that the choice of 3C.fo is immaterial m (5 ·10) as long as ( 4 · 7) is satisfied, we find

P,(ts) lx) = (¢o(t) lx)l¢o(t))

+ ~ (¢o(t) I a/lx)all¢o(t)) [Gu (ts) · l · G~(ts) ]1z•. (5·11) j!

When x(s) is independent of time, namely, J{'=O, we have, using (4·43b),

G~2 (ts) = [exp+ {- fds' _[z';T} r =exp_ {- fds' _[z';*} =exp_{ fds'.£KT}.

(5·12) Therefore, since G11 (ts) =exp+{-J:ds'..L'~.l} by (4·38), we obtain

*> The spirit of this assumption als~· underlies our basic assumption (4· 23).

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= [Gn(ts) ·G22t(ts)]1t=a1~. (5·13)

Here the second term of (5 ·11) becomes

E (<Po(t) la/lx)a/I<Po(t)). (5 ·14) i

Namely, in this simple case we have

P.(ts) =Pt. (5 ·15)

Therefore, there is no vacuum or single particle intermediate states also in the

representation using at and at, which allows us to ignore diagrams of the type

Fig. 11 all the time. Note, however, that this simple situation no longer exists

when X changes in time. In this manner we are again led to the following expression for ¢ (t):

I¢ (t)) = exp SL (tto) I <Po (t) ),

where SL (tt0) takes the general form

(5·16)

(5·17)

and snt Cit, i 2• • ·, in, t 0) represents the sums of contributions of connected diagrams which start from the vacuum state and end up with the state in which n particles

it, i 2, ···,in are excited, which· can be represented schematically by Fig. 10 of I. (5 ·16) and (5 ·17) are the generalizations of I ( 4 ·18), I (5 · 4) and I (5 · 5). Thus,

defining the characteristic function for the distribution function g (t) by

Et ( {~}) = J da eiEJ<iaJg ({a}, t), (5 ·18)

and, similarly defining E1o( {~}) for g10 (t), we obtain, by making use of (3·3b) and ¢0 (t) a ;'<Pot (t) = -a /87itt, the final expression for Et ( {~}) as follows:

Et ( {~}) = [exp SL ( {i"L; u (t)tJ'~t} )] ·E10 ( {~}), (5 ·19) !

where we have expressed SL as a function of a1 ; SL=SL( {a1} ). This result

generalizes I (6 · 22).

§ 6. Concluding remarks

Here we have obtained an expression for the non-equilibrium distribution

function of the gross variables where chara'cteristic function is expressed in the

general form (5 ·19) where SL is a sort of "extensive" quantity. This suggests

that the distribution function itself g ({a}, t) will take a general form exp [- fb ({a}, t)] where W ({a}, t) will again be an "extensive" quantity6l and 1s m some

sense a non-equilibrium analogue ·of the thermodynamic potential for describing

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102 K. Kawasaki

the probability distribution of fluctuations although we do not necessarily suppose that such close parallel should hold in every case.*' Thus in a subsequent publication we shall discuss the explicit form of (/) ({a}, t) and its properties.

Acknowledgements

This work was undertaken as part of the annual research project "Statisti" cal Mechanics of· Non-Linear and Non-Equilibrium Phenomena" sponsored by the Research Institute for Fundamental Physics. The financial support of the Scientific Research Fund of the Ministry of Education is also gratefully acknowledged.

References

1) K. Kawasaki, Prog. Theor. Phys: 51 (1974), 1064. 2) K. Tomita and H. Tomita, Prog. Theor. Phys. (to be published) and the references

quoted therein. 3) K. Nakamura, J. Phys. Soc. Japan 32 (1972), 365; 33 (1972), 1273 and the references

quoted therein. 4) T. Kawakubo, S. Kabashima and M. Ogishima, J. Phys. Soc. Japan 34 (1973)•, 1149.

T. Kawakubo, S. Kabashima and K. Nishimura, ibid. 34 (1973), 1460. 5) e. g., H. Mori, H. Fujisaka and H. Shigematsu, Prog. Theor. Phys. 51 (1974), 109. 6) R. Kubo, K. Matsuo and K. Kitahara, J. Stat. Phys. 9 (1973), 51. 7) T. Yamada and K. Kawasaki, Prog. Theor. Phys. 38 (1967), 1031.

Y. Pomeau, Phys. Rev. A5 (1972), 2569 and the references- quoted therein. 8) D. H. Gage et a!. in Non-Equilibrium Thermodynamics, Variational Techniques and

Stability, edited by R. J. Donnelly, R. Herman and I. Prigogine (University of Chicago Press, Chicago, 1966) .

*l For nonuniform states, the word "extensive" in general requires careful definition. Here we suppose that this property applies to the spatially uniform directions. For instance, for the case of a layer, .(f) would be proportional to the area of the layer. An argument against the existence of non-equilibrium thermodynamic potential was given in Ref. 8).

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Contribution to Statistical Mechanics far from Equilibrium. II

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