PHYSICAL REVIEW B VO LUM E 11, NUM BER 3 1 FEBRUA RY 1975

First-order approximation for the time-dependent Ising model*

Huey W. HuangPhysics Department, Rice University, Houston, Texas 77001

{Received 19 June 1974; revised manuscript received 5 August 1974)

Bethe's method of equilibrium statistical mechanics is generalized to treat Glauber's time-dependent

Ising models in two and three dimensions. This constitutes the first-order approximation which

improves the mean-field theory {the zeroth-order approximation) by incorporating specific short-range

correlations into the distribution function. We first calculate the single-spin expectation value. Although

the dynamic equation may meed to be solved by using a numerical method, its approximate solution in

the pretransitional region can be studied in detail. We then calculate the two-spin correlation function.

This requires yet another extension of Bethe s method. Again approximate solutions can be obtained in

the pretransitional region. Our results for the static pair correlation functions are equivalent to that ofElliott and Marshall, but our derivation is simpler. We also calculate the time-dependent correlation

function for the magnetization, and reproduce the Auctuation-dissipation relation.

I. INTRODUCTION

In this paper we will discuss some approximatemethods for solving the time-dependent statisticalproblems of lattice systems. Since we have foundtha, t many well-known techniques originally invent-ed to deal with equilibrium problems can be gen-eralized to tackle the time-dependent problems aswell, we shall briefly review the methodology ofstatistical mechanics related to our work.

The difficulty of producing an exact solution instatistical mechanics is well known. In most casesuseful results are obtained by using some approxi-mate methods. One of the often used methods isclosed-form approximations. '~ The starting pointof the closed-form approximations is the mean-field theory. This method gives a qualitative de-scription of critical phenomena. However its quan-titative descriptions, depending on systems, varyfrom being very poor to very good. The systemswell described by the mean-field theory includesuperconductors, liquid crystals" and ferroelec-trics."

The major drawback of this theory arises fromits fa,ilure to include the effects of fluctuations. In-deed large fluctuations in the pretransitional region,are a typical critical phenomenon. These anoma-lous fluctuations a,re clearly due to strong localcorrelations. Therefore all the methods subse-quently developed to improve the mean-field theorywere designed to incorporate the short-ra, nge cor-relations into the distribution function. The approx-imations are then based on construction of a, smallba, sic cluster. The intera, ctions among the sites ofthe cluster are taken into a,ccount exactly, whilethe interaction between the cluster sites a.nd therest of the lattice is approximated by a mean field.The sequence of approximations is obta, ined bysteadily increasing the size of the basic cluster.

The zeroth-order approximation which uses thesmallest cluster, i. e. , single site, is obviouslythe mean-field theory. The first-order approxi-mation, which uses the second smallest (symmet-rical) cluster, i. e. , a site plus its nearest neigh-bors, is the Bethe method. The higher-order ap-proximations are, however, not unique; among theseveral versions, the Kikuchi method"" perhapshas been most successful. It is important to pointout that higher-order approximations have consis-tently produced better predictions.

We now turn our attention to the time-dependentstatistical mechanics. In particular, we a.re in-terested in the master-equation approach to spinrelaxation in the Ising model (Glauber's model). "'2The a,dditional dependence in time obviously makesexact solutions even more difficult to find. Felder-hoff' has shown that Glauber's (time-dependent)one-dimensional model is equivalent to a two-di-mensional model in equilibrium. Thus an exact andcomplete solution to Glauber's equation for a one-dimensional Ising model exists in the absence ofmagnetic field, '3 while the same model with a, mag-netic field cannot be (or has not been) solved exact-ly, similar to the situation of Onsager's two-dimen-sional model. Fortunately the approximate methodsmentioned above are applicable to any dimensional-ity and it seems likely that they are applicable totime-dependent problems as well. As a rnatter offact the zeroth-order approximation of Glauber'smodel has been studied by Suzuki and Kubo. " Thisauthor has applied the first-order approximationto the one-dimensional model with a magnetic field;in this pa, rticular case the first-order solution pos-sesses an exact equilibrium limit. "

In the sections that follow we will generalizeBethe's method (the first-order approximation) totreat the two- and three-dimensional time-depen-dent Ising models. '6 Since most of the interesting

1144

FIRST-ORDER APPROXIMATION FOR THE TIME-DE PENDENT. . .properties of a lattice system may be constructedin terms of the one- 2nd two-spin expectation val-ues, we shall concentrate our discussion on theirtime-dependent behavior. Matsudaira, '7 has dis-cussed these problems by using a decoupling pro-cedure where the rnultispin correlation functionsare approximated as sums of products of pair-cor-relation functions. This approximation seems togive a better static result than Bethe's method.However, this method as a rule gives rise to alarge set of simultaneous equations. To solve thema. further approximation is often used. In principlethis method can also be systematically improvedby effecting the decoupling at higher orders, butthe subsequent calculations for a huge set of simul-taneous equations look formidable. It is useful toestablish the dynamic Bethe approximation becauseit is much simpler than the decoupling approxima-tion and also because it seems to indicate that thesystem of static approximations described abovecan be generalized to solve the corresponding dy-namic problems.

%'e start out with the general Glauber equationfor a system of any dimensionality in Sec. II. In

Sec. III, we generalize Bethe's method to approxi-mate the time-dependent distribution functions.The latter are then used to obtain a dynamic equa-tion for the one-spin average. Although the equa-tion is too complicated to be solved analytically (ex-cept for the one-dimensional case), its approximatesolution in the pretransitional region can be studiedin detail, which we do. We also compare some ofour results with the zeroth-order approximations.Section IV is devoted to the time-dependent as wellas the equilibrium two-spin correlation functions.Again we find Bethe's method can be generalizedto treat these nonlocal functions if they are aver-aged over all directions. It turns out that the first-order approximation of the static pair-correlationfunctions has been discussed by Elliott and Mar-shall' and Fisher and Burford. Our method isessentially the same as Elliott and Marshall's butalgebraically simpler. In Sec. V we discuss thefluctuation-dissipation theorem where the two-spincorrelation functions should be properly related tothe dissipative coefficient calculated from the field-dependent one-spin average. In general the first-order approximation does not change the characters(such as static and dynamic critical-point exponents)of the mean-field results, but one expects the for-mer to be closer to the exact solution quantitatively.

II. GLAUBER'S MODEL

Consider an n-dimensional Ising model (n= 1, 2, 3)with spin variables S; (i= 1, 2, . . . , N), each of which

takes on the values + 1. A given set of number

(S)= S„S„... , S„specifies a configuration of thewhole system, The Hamiltonian of the system is

given by

a=-OPS, S, fIQS, (2. 1)

where {ij)denotes a nearest-neighbor pair of spins,G the coupling constant, and B the external mag-netic field. For simplicity, we have assumed themagnetic moment associated with the spin to be one.The spin system also interacts with a heat bath oftemperature T, which induces spontaneous flips ofspins in a stochastic fashion. Let P((S); t) denotethe probability of finding the spins in the configura-tion IS) at time t. Then if a fine enough time scaleis chosen, one can assume that there is only onespin flip at a time and the master equation~ can bewritten as

+ Q au, (- S,)P((S)„—S,; t), (2. 2)

where (S); denotes the configuration {S)excludingthe spin on the jth site, and zoz(S;) is the transitionprobability that the jth spin flips from the value S,to —S&, while the other spins remain temporarilyfixed. Glauber used the detailed balancing condi-tion at equilibrium to determine the ratio of wo(S~)to u)o(- SQ):

n 0(SO) exp —bSo -gSo S160(- So/ s=i

exp bSo+ gSo S;&=i

where the sites 1 to o are the nearest neighbors ofan arbitrarily chosen site, the 0th, and 5 = B/kT,g= G/kT. o is called the coordination number.Thus one can write

(2. 4)

While n is quite arbitrary (see Ref. 11), we makethe simplest choice, i. e. , n = const. SubstitutingEq. (2. 4) into Eq. (2. 2), one gets the general Glau-ber equation which describes spin relaxation of theIsing model. Unfortunately this equation in its gen-eral form is not very useful, except in the one-di-mensional case. For instance, if one multipliesEq. (2. 2) by an arbitrary spin and sum over allpossible spin values, the equation for the one-spinaverage will contain terms proportional to multiple-spin correlation functions. By the same method,the equation for a two-spin correlation function willcontai~ another set of correlation functions, and soforth. In fact, one gets an infinite series of equa-tions; each relates various multiple-spin correla-tion functions, for which a solution simply seems

HUEY %. HUANG

impossible. The zeroth-order approximations ofsuch equations have been published by Suzuki andKubo. The fsrst-order approx1matxons are dis-cussed in the following sections.

III. ONE-SPIN AVERAGE

where C, (t) and C (t) are the normalization factorsand z(t) represents, in a mean-field fashion, theeffect of the background formed by the rest of thelattice. We are now ready to calculate the timerate of change of q(t):

The expectation value of a spin, for example,SO, at time t is defined as

——=- PS, +w, (S,)P({S),, S,.;t)fS}

(So(t)) -=P S,f'({S];t). (3. 1) + P S, gw, (-S,.)J({S),, -S,.;t)

Since we are mainly concerned with homogeneoussystems, we shall use the notation q(t) for the sin-gle- spin average.

To calculate q(t), we follow the method previ-ously developed to treat a one-dimensional mod-

el. "'.6 For a given site, say the 0th, let p(SO, n; t)be the probability at time t that n of the 0 nearestneighbors have spin + 1 while the 0th site itself hasa spin value SO. 0 the molecular collision time ismuch shorter than the macroscopic relaxation time,one can assume the hydrodynamicslike local equi-librium (equilibrium with a given value of q). Thenthe distribution function p(SO, n; t) takes the follow-ing form in the first-order approximation"':

(3.3)

With the distribution functions given by (3.2), itis most convenient to rewrite the transition proba-bility in the following form:

wo(SO) = —,o. '(1 —pSO) ] [(1—8SOS,.), (s.4)

dqddt

= (e '+ ze')'(C. e ' —C e'), (s. 5)

where n'= ncoshbcosh'g, p=tanhb, and 8 =tanhg.Then Eq. (3.3) can be reduced to the following sim-ple form:

(s. 2)in which the explicit time dependence of q(t), C, (t),C (t), and z(t) have been omitted for simplicity. Toobtain Eq. (3. 5) one uses the following formulasfor the expectation values of m-spin products:

SS. . .SS(+1 n 1)=C( )(-e"+ee'") (e'' ee")'

S 1t~ o o tS fy & t2 t ~ ~ o t l= 1(i&j& ~ ~ '&l )

SS.. SS(-( n;()=C ( )(-e e ee ) (e eee )'~

~

~

~ ~ ~

S 19e ~ ~ tSQ $ t j ~ o ~ etl-1 m(i&j& ~ "&l )

We now need to express z, C, and C as func-tions of q(t). The relations are found in the follow-ing self- consistency conditions:

C, = e'~co„co, = 2(1+ q)(e s+ zoes)

C = e' C(, , C() ——s'(1 —q)(es+zoe ~)

(3. 10)

[P(+1,n;t)+P(- l, n;t)]=1, (s. t)where the quantities evaluated in the absence ofmagnetic field are denoted by a subscript 0. Equa-tion (3. 5) can now be written as

p(+ 1, n; t) = — n[p(+ 1,n; t) + p(- 1,n; t) (, (3.8)1

n=0 0' „ q =(1+z,)'(C„e' —C, e').ddt (s. 11)

[P(+ l, n;t) —P(-1, n;t)]= q(t). (3.9)

Substituting the distribution functions (3.2) intothem, we find

2gz = e-"z„z,= {q+[q'+ (1 —q')e" ]"');09 0

y g

Equations (3. 10) and (3. 11) form a complete set ofdynamic equations for the single-spin average q(t).For a one-dimensional system (o = 2), they can bereduced to the form of Eq. (4. 10) in Ref. 15, whichis integrable. For systems in two or three dimen-sions, however, the equations may have to be in-

FIRST-ORDER APPROXIMATION FOR THE TIME-DEPENDENT. . . 1147

tegrated numerically.The function Q(t) is according to the ordinary

definition the long-range order. The short-rangeorder I'(f), defined as the fraction of the nearest-neighbor pairs with both spins up, is given by

r(f) =- nP(+ i, n; f)1o n=

1+Q Q —1i.Ip" (i - p'&"'&'" ' ) '

Unlike the zeroth-order approximation whichgives a spurious critical point in one dimension,the first-order approximation has the exact equi-librium limit' in this simple model which as iswell known has no phase transition. Despite itslack of critical phenomena, the one-dimensionalsolution has many interesting applications. ' Thetwo- and three-dimensional solutions possess acritical point, and it is of considerable interest tostudy their dynamic behavior in the pretransitionalregion. First consider the case of no externalmagnetic field. At a temperature above the criti-cal point T, = 2G/kin[a/(o —2)], the equilibrium val-ue of Q equals to zero, so we may expand the equa-tion in a power series of Q.

dQ 1= —Q+tcQ +''',Qdt (s. is)

with

1 o'2 o —2

ev 2 cosh'g o.

("2c/kT e 2c/&(Tq)

2 cosh'g (s. i4)

(1 —e ")'[(a+4)e "—(o —2)]. (3. 15)48 cosh'g

For comparison, the zeroth-order approximationscorresponding to the transition probability (2. 4)are

1/rp=(i-~g)=(i- T,'/T),

t(p = -', (og)' (1 ——,'og),

(s. 18)

(s. iv)

where the critical temperature of the zeroth-orderapproximation is T, = oG/&&, . Both methods predicta critical slowdown, ~- as T- T,. As a result,when temperature approaches the critical point,the cubic term becomes more and more importantand the decay process is no longer describable bya single relaxation time.

~2 (2 n+1)/2&-(2 n+1)a.t /T

n! 2" Qp+ (7&()' (3. 18)

The decoupling approximation" (see Sec. I) alsopredicts a similar critical slowdown. Perhaps itis not surprising that all methods based on theprinciple of the mean-field theory arrive at thesame critical-point exponents —both static and dy-namic ones —because in the immediate vicinity ofthe critical point the inclusion of limited local fluc-tuations (such as Bethe's approximation) becomestotally inadequate. However, a higher-order ap-proximation does give a better critical-point tem-perature and a better account of critical phenomenaoutside that immediate vicinity. (See the end ofthis section. ) The relaxation time below the crit-ical point is one-half of that above the critical point

dQ = (1+zp)' (Cp+ Cp-)

(1+z', )' (C;, + C; ) . (s. 2i)

For T& T„ the equation takes the simple form

Its solution is simplified considerably if we assumethat the system is in thermal equilibrium to zerothorder in B, i. e. , that the field induces only smalldepartures from equilibrium. In that case the co-efficient of B can be replaced by its equilibriumvalue which will be specified by an index e:

~'(T' & T,) 1if T, —T' = T —T, « T, . (3. 19)7'(T & T~) 2

This relation is independent of models.On the other hand, in the limit of weak magnetic

fields, B«/tT, we consider the equation for Q(t)to first order in B:

dQ Q BcY df, T kT cosh g

1 t

kT cosh'ge-(a/t&(t-t '

&B(f t)& df t

Its solution

Q(f) Q(f )e-(alt&(t tP&-(3. 22)

(3.23)

=(1+zp) (Cp Cp )dQ0 Cft

B (1+z(&)' (C(&, + C(& ). (3.20)

is useful for studying the dynamic magnetic sus-ceptibility.

The magnetization is defined as

HUEY W. HUANG

m(t) = g S,. (t); (3. 24) +3080v +' ' (3. 30)

therefore the expectation value of the magnetizationequals to NQ(t), where N is the total number ofspins in the system. Since B(t) as a time-depen-dent field can always be resolved into a set ofmonochromatic components, for each componentI3oe '"', we may define a dyna, mic magnetic suscep-tibility y(&u) via the relation (M(t)) = y(&u)Boe '"'. Inorder to study the effect of a magnetic field, we letthe initial time in Eq. (3.23) to be —~ to make surethe influence of the initial condition negligible.Thus

&~(t))=,— e ""'"-"B(t-')n dt' (3.25)AT cosh'g

and the susceptibility is given by

Nn7.

kT cosh'g(n —ta& ~) '

similar to the mean field result

g((0) = Nn To/kT(n —XQ) To).

(3.26)

(3.27)

(3.23)

have been carried out by Yahata and Suzuki. Inparticular, they obtained

-- lim -- —= 1+Bv+44v + 200vnuT . Imp((o)

N

+ 894v + 2984v'+

where v =- tanhg. The first-order approximationwith the transition probability (3. 28) does not leadto a simple closed form like (3. 11). However, itis not difficult to calculate its high-temperatureseries expansions. The first-order approximationcorresponding to (3.29) is

nkT . Imp((o)lim = 1+Bv+44v +200v +812vSethe

The temperature and frequency dependence [see(3. 14)] are in qualitative agreement with the elec-tric susceptibility of some ferroelectrics such asCa&sr(C, H, CO2)6; the data show that the real partof susceptibility dips at T, while the imaginary partpeaks at T„and both the dips and peaks are higherfor lower frequencies. 2' However, since the ferro-electric interactions are probably long-rangeforces, the nearest-neighbor Ising model is notsuitable for quantitative studies of ferroelectricity.

Finally it is of interest to compare the zeroth-and first-order approximations with the exact high-temperature series expansion. The series expan-sions of Glauber's equation with the transitionprobability

accurate to order 1/T'. In comparison, the zeroth-order approximation is

lim — = 1+Bv+48v + 2583 v'c

nuT . Imp(co)

MFA

+ 1312v + 6401-', v +

(3. 31)accurate only to order 1/T.

IV. PAIR-CORRELATION FUNCTIONS

If one applies Bethe's method to evaluate a gen-eral pair-correlation function (S;(t )S,(t'+ t)), itwould require multiple mean-field parameters.This is, of course, because the function possessesno spherical symmetry. Consider the nearestneighbors of the jth site. The mean field acting oneach of them depends on its position relative to thei-j axis. Such a formalism has been worked outby Elliott and Marshall. . ' To first order in devia-tion from the mean field of (3. 2), the geometricasymmetry is averaged out and one gets a simpledifference equation for the correlations. A com-parison of Elliott and Marshall's result with thehigh-temperature expansion has also been dis-cussed by Fisher and Burford. '

Instead of (S,.(t')S, (t + t)), we shall consider itsaverage over all directions. The formalism isthen greatly simplified. Take an arbitrary spin,say the 0th, to be the center. We assume that thesurrounding spins reside on successive sphericalshells of increasing radius x. For example, thenearest neighbors reside on the x=1 shell, thesecond-nearest neighbors on the ~= 2 shell, etc.I et S„denote the sum of the spins on the shell ofradius y. We now ask: If a spin variation occursat the center, what is its effect on the polarizationat a radius z after a finite interval of time'P Thisinformation is described by the expectation valueof the product of the spin So evaluated at time t'and the sum of spins S„(y 2) at time t + t or(S,(t ')S„'(t '+ t)).

Note that (So(t ')Sr(t '+ t)) is equivalent to(S„(t )So(t + t)), which measures the effect of thespins at a radius y on the spin at the center aftera time delay, and we find that it is more conve-nient to discuss the latter. The equivalence is dueto the fact that the microscopic equations of motionare symmetrical under time reversal if the signsof spins and of the magnetic field are simultaneous-ly changed. Thus one has

&s,(t ')s„'(t '+ t)), = &s,(t ')s„'(t ' - t)),=(S,(t ')S„'(t '- t)&. ,

where the invariance under translation of the originof time has been used and the last equality is due

F IRST-ORDE R AP P ROXIMATION FOR THE TIME -DE P ENDENT. . . 1149

to the fact that the Hamiltonian is invariant underthe transformations S-—S and B-—B.

According to its own definition, the correlationfunction can be written as

p„(so, n) be the probability that n of the nearestneighbors of the 0th site have spin+ 1, while the0th site itself has a spin value Sp. GeneralizingBethe's method to this case, we assume

&s„'(o)s,(f)& = g ~(&s};o)s„"p(&s}I1S'};t)s,', (4. 1)

f~) f~'}p (+ 1 n) Q en eb (2n-v)+t(2n-s)o&

r+ & r(4. 4)

z P(1S}l(s'}; f)sl = (s.(t)) (4. 2)

where P(1S}I {S'};f) is the conditional probabilityfor finding the configuration 1S } at time t if thesystem had a configuration (S) t time equal tozero. Obviously we have

p ( 1 n) g &n eb(2 )(- ((&+ t((-(2n &

r & r- + r 7

where C„„C„-,and z„have the same meaning asin the previous case, Eq. (3.2), except now thebackground has a definite value of S„. The self-consistency conditions now become

with the initial condition (So(0)& = So. Thus in orderto evaluate the correlation function, one shouldsubstitute the explicit solution of Eq. (3. 11) into(4. 1) and then carry out the summation over thevariables (S}.

For temperatures above the critical point wherethe linearization is justified, we get

&S,'(0)So(f)& = &S„'(0)So(0)&e '"

[p„(+ 1, n) + p„(- 1, n) ]= 1,n=

(S„~~S,) = g (2n —a)[t, (+ (, m) ~ p, (- (, n)],

[p, (+ l, n) —p„(-1,n)]=(S„' iS,),n=

where

(4. 5)

(4. 8)

(4. V)

)) e-e(t-t' &IT/(tt)ATcosh'g Q

(4. 3)The problem now becomes that of evaluating theinstantaneous correlation function (S„(0)So(0)).This again can be solved to first-order approxima-tion by properly generalizing Bethe's method. Forthis purpose, we shall assume that the spin systemis initially in thermal equilibrium at a certain tern-perature. For a given value of N(t ~2), let

(s„'~s,&

= g p(s„'~s,)s„

Sp

(s„'~

s', ) = +~(s„'~s, )s,".

1

The equilibrium condition is given by

C,e ' = C„e'.

(4. 8)

(4. 9)

(4. 10)

Eliminating C„„C„andz„among the above condi-tions, one easily obtains the following differenceequation for y™2:

g(S„~Sr& = (2cosh2g)(S ~so&+e ' '(1, —(S„~so&)' " '(1+ (S„~so&)' '

—e"'(1+(S'~S,&)' ""(1—(S'~s,&)"' (4. 11)

If this equation is expanded as a power series of

T, =(s,'ls') —&s')

and

T„.= (s,' ls,&- &s,),

the linear equation

(2 sinh2g) T„,—(2o cosh2g) T„o= —cr~

e2' + —o e 'b T„o (4. 12)1 —So ) 1+ So + Q

can be approximated by a differential equation andgives rise to the Ornstein-Zernike result. " Itsboundary condition can be found at x= 2. For tem-peratures above T, and in the absence of magnetic

field, the next-lowest-order terms in the power-series expansion are cubic in T„o [because the equa-tion (4. 11) is odd in (Sr IS, ) and (S„[so&], and there-fore are negligible to second order. These are

1150 HUEY W. HUANG

Elliott and Marshall's results.When the spin-up and spin-down have different

chemical potential, it is equivalent to having anexternal magnetic field. In the presence of a field,the second-lowest-order terms in the power ex-pansion of (4. 11) are quadratic in T„o. Thus theymay have to be retained for small z. Possible ap-plications of the general formula (4. 11) to criticalscattering in a field will be discussed somewhereelse.

Note that

If (S„So) and (S„Si ) are functions of l rl only, wehave

(S„so)= N„(S„S())

and

(S„'s', ) = N„(S„S', ),

where N„ is the number of sites on the shell ofradius z. Introducing the notation

I'„,= (s„s,) —(s)'Z p(s„')S„T„,= & s„s', ) —&s„') &s', ),

r

g I'(s„')s„'T„,= (s„'s,) —(s„')(sg.$'Tr

(4. iS)

(4. 14)

and

I „', = (S„S', ) —o (S)',

we obtain from (4. 12),

(2 sil1112g)I'„, —(2e cosh2g)1„o= ~—e

Ie

1 S ~ +I —(Sy i 1+ (So) 1+ qSy

(4. 15)

Like (4. 12), this equation can be approximated bya differential equation when x is large and its solu-tion is the well-known Qrnstein- Zernike correla-tion function. For x= 0 and 1, I'„, and 1"„,can becalculated exactly within the Bethe formalism, andit turns out (4. 15) is an exact relation for r= i.The explicit solutions and their application to criti-cal scattering in the absence of an external fieldhave been given by Elliott and Marshall. In Sec. Vwe will use (4. 12) to calculate the correlation func-tion for the magnetization.

Imp(o)) =~0

(iaaf(O) m(t)), e*«dt, (5. i)

(1+o sinhog) (SrSo),

which is a well-established theorem in statisticalmechanics. Perhaps the exercise may be regardedas a test of consistency of our methods.

Note that the correlation function on the right-hand side of the fluctuation-dissipation relation(5. 1) is evaluated in the absence of magnetic field.In that case, Eq. (4. 12) gives for T& T„

V. FLUCTUATION-DISSIPATION THEOREM= sinhg coshg(srsri ), (y ~ 2). (5. 2)

The foregoing expressions for the correlationfunctions and for the single-spin average shouldreproduce the fluctuation-dissipation relation, ' '

If we sum the equations over the index r from 2 toinfinity and make use of the following relations:

half = QS.'+Si+Sor=a

(iliS', ) = e(iliS,),

(SnS, ), = g QS;(S,(+(, n) —S,(-l, n)]=ntnnhn (T T, ),S' y ~ ..~ s S fI I, = l

(5. s)

(5.4)

(5. 5)

(S,S, ), = o+ 2 p $$,.[p, (+ 1, n) + p, (- 1, n) ]= o + a (o —1) tanh2g (T & T ),S ye. ..&ST i ~ "-1

(i&& )

(5. 5)

where S;, S& (i,j = 1, . . . , o) are the spin values ofthe sites on the x= 1 shell and p, (+ I, n) are the equi-librium limits of P(+ 1,n;t), it can be shown that

Hence, the time-dependent correlation function forthe magnetization can be written as

(Mso), = ~cosh 'g. (5. V) (M(0)M(t)), =N7 cosh 'ge ~'" ' (5.8)

F IRST-ORDER APP ROXIMATION FOR THE TIME - DE P ENDE& T. ~ ~ 1151

This expression is valid for T & T„B= 0, and

t ~~0. Thus we have2Nn 2kT

( )cosll g[(Q'r ) + (0 ] (d

according. to Eq. (3.26).

*Work supported in part by PHS Research Grant No. GM21721-1 from National Inst; of Gen. Med. Sciences and

by the Office of Naval Research Contract No. N00014-67-A-0145-0007.

~C. Domb, Adv. Phys. 9, 149 (1960). Other methodswhich have been applied to dynamic statistics includeseries expansions [C. Domb, see above; Ref. 2, and

H. Yahata, J. Phys. Soc. Jpn. 30, 657 (1971)], com-puter simulation [K. Binder and H. Muller-Krumbhaar,Phys. Rev. B 9, 2328 (1974) and references cited there-in], and renormalization-group method [B. I. Halperin,P. C. Hohenberg, and S. K. Ma, Phys. Rev. B 10,139 (1974)].

Phase Transitions and Cmtical Phenomena, edited by C.Domb and M. S. Green (Academic, London, 1972),Vol. II; and ibid. , Vol. III (to be published).

See, for example, P. G. de Gennes, Supe~conductivityof Metals and Alloys (Benjamin, New York, 1966).

4V. W. Maier and A. Saupe, Z. Naturforsch A 14, 882(1959).

~T. W. Stinson, J. D. Litster, and N. A. Clark, J. Phys.(Paris) 33, Cl-69 (1972).

V. L. Ginzburg, Fiz. Tverd. Tela 2, 2031 (1960) [Sov.Phys. -Solid State 2, 1824 (1961)].

J. A. Qonzalo, Phys. Rev. B 1, 3125 (1970).H. A. Bethe, Proc. R. Soc. Lond. A150, 522 (1935); seealso R. Peierls, Proc. Cambridge Phil. Soc. 32, 471

(1936); sometimes it is called the Bethe-Peierls method.R. Kikuchi, Phys. Rev. 81, 988 (1951).D. M. Burley, in Ref. 2, Vol. II, p. 329.R. J. Glauber, J. Math. Phys. 4, 294 (1963).

'2For a more general discussion of the master-equationapproach, see K. Kawasaki in Ref. 2, Vol. II, p. 443.

'3B. U. Felderhoff, Rep. Math. Phys. 1, 215 (1971); 2,151 (1971).

4M. Suzuki and R. Kubo, J. Phys. Soc. Jpn. 24, 51- (1968).

'5H; W. Huang, Phys. Rev. A 8, 2553 (1973).6A preliminary result was reported in H. W. Huang,Phys. Lett. A 48, 395 (1974).

'VN. Matsudaira, Can. J. Phys. 45, 2091 (1967); J.Phys. Soc. Jpn. 23, 232 (1967).R. J. Elliott and W. Marshall, Rev. Mod. Phys. 30, 75(1958).

' M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583O.967).W. Pauli, Festschrift zum 60 Gebuxtstage A. Sommex-feld (Hirzel, Leipzig, 1928), p. 30.E. Nakamura, N. Takai, K. Ishida, T. Nagai, Y. Shio-zaki, and T. Titsui, J. Phys. Soc. Jpn. Suppl. 26, 174(1969).H. Yahata and M. Suzuki, J. Phys. Soc. Jpn. 27, 1421(1969); see also H. Yahata, Ref. 1.

3R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).