Dynamical relaxation in finite size systems

Nuclear Physics B285 [FS19] (1987) 519-534 North-Holland, Amsterdam

DYNAMICAL RELAXATION IN FINITE SIZE SYSTEMS Non-linear and linear decay of the magnetization Yadin Y. GOLDSCHMIDT

Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA Received 15 January 1987

We calculate analytically the non-linear time-dependent equation of state for the magnetization in a finite size system. The calculation is done to one-loop order in the q ~ 0 modes. The equation is obtained in an expansion in powers of e1/2 where e = 4 - d up to o(e). The stochastic equation of state is solved numerically at the critical point and the results are compared with a recent Monte Carlo simulation. The solution displays the full crossover from the bulk power law decay to the finite size exponential relaxation. We also present more explicitly the results obtained for the linear relaxation time in an earlier paper, in a form which enables a direct comparison with simulations. To simplify the calculation we exploit the supersymmetry of the corresponding quantum mechanical hamiltonian.

I. Introduction A quantitative understanding of finite size effects is very useful for an efficient interpretation of data obtained by numerical Monte Carlo simulations, which are typically performed on small systems. Finite size scaling (FSS) theory has been initiated by Fisher [1] and has been the subject of many subsequent investigations [2]. Br~zin and Zinn-Justin [3] showed how to calculate the size-dependent universal functions in an e expansion, which is singular about four dimensions [4]. Thus the series generated are in powers of E1/2 o r e1/3 depending upon the geometry of the sample. Rudniek et al. [5] reported on theoretical and numerical results for the free energy. Recently I showed [6] how to extend the finite size calculation to the dynamics of Ising and vector spin systems, and I calculated the universal scaling function associated with the linear relaxation time. This relaxation time governs the exponential decay of the autocorrelation function for the magnetization. The calculation was performed for the cubic geometry with periodic boundary conditions in 4 - e dimensions in powers of e 1/z, and in 2 + e dimensions using the non-linear o-model. Related work has been reported by Niel and Zinn-Justin [7] and more recently by Diehl [8]. 0619-6823/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

520

Y. Y. Goldschmidt / Dynamical relaxation

The essence of the method which I have used in ref. [6] (to be denoted by (I) in this sequel) is as follows: In a finite size scaling calculation, it is necessary to treat the q = 0 mode of the order parameter, say the magnetization separately from the q 4:0 modes [3]. The q 4:0 modes (q is discrete for a system in a finite box) can be treated perturbatively using the loop expansion. They can be traced over to yield an effective equation of motion, or an effective action for the q = 0 mode: e-S~"(*q-°) = Tr e -S(q~q=°'q~q*°) .

(1.1)

g'q.0

The action appearing on the r.h.s, of eq. (1.1) is that derived from the generating functional for the dynamical correlation functions [9]. The effective dynamics for the q = 0 mode is then reduced to a solution of a quantum-mechanical problem with h c~ L -d, where L is the size of the box. The quantum mechanical hamiltonian associated with the dynamics of the q = 0 mode is shown in (I) to be supersymmetric. A non-perturbative treatment of the q = 0 mode is essential to avoid infrared divergences, as opposed to the perturbative treatment of the q 4= 0 modes. Thus in (I) and in sect. 2 of this paper, we proceed by solving for the gap of the effective hamiltonian. In this paper our aim is twofold: First, we present the results of the theory for the linear relaxation time, based on the formalism developed in (I), in a more explicit way which enables a direct comparison with Monte Carlo simulations. We calculate explicitly the linear relaxation time in d = 3 (e = 1 in 4 - e expansion) as a function of the variable y = tL 1/" (where t = ( T - T c ) / T c , and L is the system size). The supersymmetry of the effective hamiltonian is used to simplify the calculation. This is discussed in detail in sect. 2. Second, we derive for the first time an expression for the finite-size time-dependent non-linear equation of state for the magnetization to one loop order. The "surprise" is that for d < 4, unlike the linear case, it is not enough, in the one-loop calculation, to just replace the coupling constants by their L dependent counterparts, but the equation of state turns out to be of a form which allows both for the correct non-classical behavior in the non-linear regime where the magnetization is large, and in the linear regime where the magnetization is small. We continue to solve the equation numerically. For e = 1 (three dimensions), the results are compared with recent Monte Carlo simulations by Kikuchi and Okabe [10] who simulated the critical relaxation of the magnetization of an Ising system. Our results display the full crossover from the bulk non-linear relaxation behavior M - "r-/~/vz (where M is the magnetization and • is the time) to a finite size relaxation behavior M-exp(-r/~R(L)} with " r R ( L ) - L z being the linear relaxation time. Our numerical solution of the non-linear equation for the magnetization uses only a very small fraction of the computer time that is needed for the MC simulation.

Y.Y. Goldschmidt/ Dynamicalrelaxation

521

2. The linear relaxation time The linear relaxation time, governs the long time behavior of the autocorrelation function of the magnetization ( M ( 0 ) M ( ¢ ) ) - exp( - ~'/ZR ( L ) ) -

(2.1)

For a system of the cubic geometry with linear size L and periodic boundary conditions, I have shown in ref. [6] that for T >~ T~, d = 4 - e

----TL = 2---~4---~e-1/zf(x(y))(1 + o ( e ) ) ,

(2.2)

with f i x ) being the inverse gap of the quantum-mechanical hamiltonian ~,o= ~p 1 z + Xq2(x + 1 q 2 ) 2

~q2 -

(2.3)

1

and x is a function of the scaling variable y = tL 1/~ (with t = ( T - T~)/T~) through the relation [3, 6, 7]:

x ( y ) = ~-~

y-¼eY-~eylny+4~raeyF2(y)+~r2eF1(y)),

(2.4)

with

Fn(Y) =

A(u)=

fo duun-1 A a ( u ) - 1 -

~

exp -

exp(-n2u).

,

n = 1,2,

(2.5)

The gap of the harniltonian (3) for a few discrete values of x can be extracted from ref. [11] (one has to divide their parameters K and the gap #, by 2 ~ in order to compare with our conventions). Nevertheless in order to obtain the value of the gap for many more values of x which is needed in particular for plotting the relaxation time as a function of y = tL 1/~ we found it useful to use the supersymmetry of the hamiltonian (2.3) which as discussed in ref. [6] can be written in the form ..~___ ½p2+ ½W2(q) _ 1W,(q),

p2=

dz dq2 '

W(q)=q(x

+

lq2).

(2.6)

(2.7)

Y.Y. Goldschmidt / Dynamicalrelaxation

522

5._ 6`2 C~ (._9 "-- 4.20

2.20

0.20 , ........

-2,00

, , _

- 100,

.......

, .........

0.00

, .........

1,00

, .........

2,00

,

, .........

3,00

4.00

X

Fig.

1. A

plot of the inverse gap of the hamiltonianversus x .

The supersymmetry implies [12] that the ground state of the hamiltonian has an energy E 0 = 0 and the wave function is given exactly by

~po(q)=Noexp{-~(q2+ 12x)2) •

(2.8)

we have chosen a variational wave function for the first excited state of the form

~l( q) = Nlq e-~(qZ+#)2,

(2.9)

where a, fl are variational parameters. This function has one node and is automatically orthogonal to the exact ground state. Calculating ( # l l H l + l ) / ( ~ k 1 1 + l ) and minimizing it by varying the parameters a and/3 using a two variable minimization routine, we obtained a very good estimate of the first excited state energy E 1, which is the value of the gap. For the few discrete values calculated in ref. [11] the results agree very well. The inverse gap is plotted versus the variable x in fig. 1. For large values of x it is clear from eq. (2.3) that the term !x~qZ in the potential dominates 2 and hence the gap of the hamiltonian is proportional to x and f ( x ) - x -1. Inspection of eq. (2.4) shows that for large y, x ( y ) ~ y1-~/1~ (since F2(y ) - _y-2 and Fx(y ) - _y-1 for large y). But reexamining eq. (2.2) reveals that for large y,

523

Y. Y. Goldschmidt / Dynamical relaxation

which differs from the correct asymptotic behavior y-~L We can obtain the correct behavior by including the o(e) terms originating from the shift in the coupling constant to order one loop. Eq. (2.2) originated from the expression [6] ,rR _ y - l + e / 1 2

q'R ( L ) / L 2--~- ( 2 X ) - l g - 1 / 2 Z - ~ / 2 f ( x )

(2.10)

evaluated at the fixed point, where ~ is the shifted dimensionless coupling constant to order one loop: 3 = g + 3 - ~ 2 g2(1 + In t) - ~g2F2(tL2 ) ,

(2.11)

with F2(Y ) defined by eq. (2.5). Hence at the fixed point g* = ~rr2e

g-1/2.-~/21Ifixedpoint =(g*)-l/2(1-¼e-¼eln y+e47r2F2(y))

(2.12)

Thus eq. (2.10) reads to order one loop ~'R(L) L2

1 7~-e_a/2( 1 2X 4~r

~e_¼elny+e4~r2F2(y))f(x(y))

(2.13)

and now for large y, ' r R - - y - l - E / 6 which is the desired behavior to o(e). Note that corrections to the effective potential involving higher order operators, like a contribution proportional to qS to W(q) of eq. (2.7), would lead to corrections of o(e 3/z) which are higher than those included in eq. (2.13). The same is true for contributions to the effective action involving derivatives of the field, which also enter only at o(e3/z). Note also that eq. (2.13) has a finite limit as y ~ 0, i.e. at the bulk critical temperature. Certainly f i x ) is finite at the point

x c - x ( y = O) = ffTrFl(O)e 1/z,

FI(0 ) = -0.14045.

(2.14)

uy2 }) exp{ - ~--~

(2.15)

By substituting lny=

f~° -~-\ dU (expI - ~-~2~ u2 } -

and using the expression (2.5) for F2(y ), it is readily found that as y ~ 0

zR(L ) .

L2 .

.

1 Vc3e_~/Z(l_ ~ .

with

22~ 4~r

du[ ( C=fo

--u exp - ~ 2

¼ c)f(~3

u2

FI(0) 1/2 )

u2]

- ~yA4(u) + ~-2 •

(2.16)

(2.17)

Y. Y. Goldschmidt / Dynamical relaxation

524 10

I

\

k, \,

I 0 -i

1 0 -~ 1 0 -=

I

I

IIIIII1

10 -~

I

I

[lllll[

I

I I11111

1

I

10

I

[

Iiiiii

\ I

10 2

I

I

I[llll

I

10 ~

tL I/v

Fig. 2. A plot of the linear relaxation time versus reduced temperature in scaling units.

In fig. 2 we display a log-log plot of k- ' r R ( L ) / L 2 versus the variable y = t L 1/~ as obtained from eq. (2.11) with e = 1 using our variational results for the inverse gap and the function x ( y ) as given by (2.4) and (2.5). In plotting eq. (2.11) we have omitted the coefficient x/3-e-1/E/(8~r) which contributes only to a vertical shift of the graph in the log-log plot. We have also replaced the other prefactor by the expression exp( _l~e - ¼eln y + 47r2eF2(y) },

which is the same to o(e). The linear part of the graph corresponds to a bulk-like behavior ~'R -- t - ~ , whereas when y ~ 0, ~'R(L) ~ A L z with z = 2 to first order in e. So far we are aware only of results of MC simulations in two dimensions by Miyoshita and Takano [13] who obtained the linear relaxation time away from Tc and plot it against the variable tL 1/'. Our results in three dimensions behave qualitatively as theirs, but a direct comparison is impossible, since e = 2 is too big to substitute in the 4 - e expansion. In three dimensions we are aware only of simulations at Tc [14]. Thus the situation calls for more MC results in three dimensions for the behavior of the linear relaxation time in the vicinity of Tc.

Y. Y. Goldschmidt / Dynamical relaxation

525

3. The non-linear relaxation time We now turn to the non-linear relaxation of the magnetization at Tc. In this case it is preferable to consider directly the equation of motion for the q = 0 mode, rather than the corresponding effective hamiltonian. Starting with Langevin equation of motions for a field ~,(x, ¢) for model A dynamics (a non-conserved order parameter), we expand the field in Fourier modes as discussed in ref. [6]:

~ ( x , ~-) = Eeia'X~q(T),

(3.1)

q

where the components of q are quantized in units of 2~r/L. We denote ~(~-)= ~q=O(¢). I consider first the case d > 4, the contribution of the q ~ 0 modes can be neglected [6] and the equation for qJ reads 1 3 a~=-~0(r0O+ ~u0~ )+~(¢),

(3.2) (3.3)

Q/(~')r/(C)) = 2XoL-aS(~ - C). Upon making the rescaling transformation dp -'-) U O 1 / 4 L - d / 4 ~ ,

1 ,f --~ ~ o

71 ~ 2~ouV4L-3d/4~I,

UO1/2Ld/2~" '

(3.4)

we obtain the equation o~ = -x~-

1~ 3 + ~(~),

(3.5)

with

(n(~)n(~')) = ~(~- ~'), x = ~1 u o-- 1 / 2 L d / 2 r o .

(3.6) (3.7)

We have solved eq. (3.5) numerically by the following procedure: We divided the interval (0, T) into K subintervals of length A. Starting with an initial value ~0 we iterated the discrete difference equation ('¢'~+1 - ~ ' i ) / A

=

-x,~-

~1( ~ i 3 ~- T~i,

(3.8)

where ~i are random variables with a normal (gaussian) distribution satisfying

1

(3.9)

526

Y. Y. Goldschmidt / Dynamicalrelaxation

The solution vector (~/i} was stored, and the procedure was repeated N times starting from the same initial value ~o. The average solution 1

N

k=l

was then calculated, and consisted the numerical solution of eq. (3.5) in the interval (0, T ) which is a function g~(q~0, "r). The values we have chosen for the different parameters will be discussed later. For d > 4 we see (using the inverse of the transformation (3.4)) that the time dependent magnetization will behave like

M('r) = Uol/4L-d/4gx(MoLa/4Ulo/4,2XoU~o/2L d/zr).

(3.10)

We will discuss below the dependence of g~ on the initial value M 0. Eq. (3.10) is consistent with the breakdown of finite size scaling above four dimensions. From the renormalization group point of view [15, 4] the magnetization in dimension d < 4 ought to scale according to the formula

M( h, t, ~, L) = L-t~/~G( hL t~/~, tL 1/~, ~L-Z) ,

(3.11)

where h is the value of the magnetic field which induces the initial magnetization, and is then turned off to zero. [16] At T = Tc (t = 0) one has h = Mo~, hence

m ( mo, q, L ) = L-O/UGl( moLfl/~ , 'rt -z) =,l--fl/PZG2( moLB/U , ~ t - z ) .

(3.12)

Notice that formally our expression (3.12) differs from eq. (3.2) of ref. [10] by the appearance of the argument MoL t~/" in G2. We would like to comment here on the dependence of G 2 (or G1) on the parameter M o. If Mo L¢/" >> 1 there exists a region for which

Mo z~l~ << z < L z .

(3.13)

In this region we expect G 2 to behave like a constant independent of the initial magnetization M0, and

m - A'r -t3/~z ,

(3.14)

which is the bulk non-linear relaxation at Tc. On the other hand if Mo L¢/~ - 1, then condition (3.13) never holds and by the time that r >> M o ~/~ it turns out that "r >> L ~, hence one expects to see only the linear relaxation M - B exp{ _ ~/~.~t) ( L ) }.

(3.15)

Thus the existence of a universal scaling function

M('r, L) = L-~/~GI ('rL-Z ) = T-~/(~Z)Gz(~'L-Z ) ,

(3.16)

Y. Y. Goldschmidt / Dynamical relaxation

527

10

° "°*,,°

. %°

>

-,,

\ \ %

I0

-I

1 0 -'

I

~

I

I

I

I

I

~ I

1

*

I

I

[

[

I

I

f ]

10

Fig. 3. A plot of the magnetization versus time in scaling units for d >/4 (mean field).

which do not depend on M 0 and crosses over from the non-linear dependence (3.14) to the linear dependence (3.15) can be achieved provided the condition

MoLp/~ >> 1

(3.17)

holds. This can be accomplished by taking M 0 --1 and L - 20-60 as was used practically in the Monte Carlo simulation [10]. Before we turn to the discussion of the stochastic equation in the case of d < 4 let us first display the mean-field results for d > 4, as obtained from the numerical solution of eqs. (3.8) and (3.9). The function gx=o(~0, 1-), is plotted versus z in a log-log plot, in fig. 3, using an initial value ~0 = 20. One can see clearly the initial power law decay g - 1"-1/2 which appears as a straight line with slope - ½ in the figure. The function then crosses over to an exponential decay as can be seen more clearly from fig. 4, where a semi-log plot of the same function is depicted. The time interval (0, 10) has been divided into 1000 segments, thus A = 0.01. The number of times that the solution vector has been obtained for averaging has been N = 1600. When we repeated the calculation using an initial value q~o= 1 only the exponential decay regime has been observed as is clear from the discussion above. When starting with q~0 = 10 the function nearly overlapped with that obtained with q~o= 20 when the time interval was slightly shifted.

528

Y. Y. Goldschmidt / Dynamical relaxation 10

%_J

1

*° *° "°*%%° **% %********°**%**°, **%% °%** *°*%°** "*%* *% %*%,

10 -' ,, 0.00

....2.001. . . . . . . . .4-.001. . . . . . . . .6,00'. . . . . . . . .8.001. . . . . . . .1()100. 7 _ _ k -z

Fig. 4. A plot of the magnettzationversus time on a semilog scale which emphasizes the region of linear relaxation. Again d >/4 (mean field).

We turn now to the case d < 4. It turns out that in order to obtain the correct non-linear relaxation when d < 4 it is not enough to simply fix the value of x in eq. (3.5) or (3.8) to the value x c as given by eq. (2.14). The form of the stochastic equation of motion or the q = 0 mode should be such that it will describe correctly the non-linear regime where M is large, and where the bulk relaxation behavior M - ~-a/(~z) should emerge with the correct non-mean-field exponent. The non-linear time-dependent equation of state for the bulk, to one-loop order, has been derived by Bausch et al. [17]. Their equation is deterministic, whereas our equation for the q = 0 mode of the order parameter in the finite size system is stochastic and the solution has yet to be averaged over the random noise. We only outline here the steps of the calculation, the details being presented in the appendix. We write down the action appearing in the generating functional for the dynamical correlation function, using the Martin-Siggia-Rose field [9]. We separate the action into three parts and distinguish the q - 0 component of the field, denoted by ~(~-) from the q = 0 components ~q(r). Thus

s = So + s, + s2,

(3.18)

K Y. Goldschmidt / Dynamicalrelaxation

529

where

so= fd~-,~(,)(O, + Xo(ro + ~Uo,t,=(r)),~(r) - f d~',t;(~)n(~') +f

~_q(r)

d~" E ' [ - X 0 ~ q ( r ) ~ _ q ( r ) - ] q

× ( O, + X0(q 2 + r0 + 21-Uo~2(T))~q(r)] ,

J

S l = 1)koU0 dr~('l"

,[

,(T)E q

(3.19)

( r ) ~ _ q ( r ) q- 1 Z dpq,(r)~q2(r)~_q,_qz(r qiq2

,]

,

s2= ~Xouof d, E'~;-q(,) q X[~k(q')Ztll)ql(r)~kq-ql(r)q-lEt~kql(r)'q2(q')~kq-ql-q2(r)] " q ql l q2 The prime denotes summation over q ~ 0. ~/(r) is the q = 0 Fourier component of the noise ~(x, r). The q ~ 0 components of the noise have been integrated out. Factors of L -a associated with summations over momenta have been suppressed. From SO we obtain the free response and correlation functions for the q 4=0 modes

in the background of the q = 0 mode: Gq(T,r')=O(,--r')exp{--)tof, Td'c"[q2+ro+½uoeo(,")l},

(3.20)

, r " ) G _ q ( , , r") . Cq(r, r') = 21~0f~ d r " "tlqt'g '

(3.21)

The equation of motion for the magnetization is given by

("/ 8~;(,)

= o,

(3.22)

where the average is taken only with respect to the q ~ 0 modes. To one-loop order we obtain 1 2( , ) ] q ~ ( r ) - ~(3 - ~ / ( r ) , O=O~,(.r)+Xo[ro+~Uoep (3.23) q~O with

•q~0 -o = -~XoUo+(,)2Xof- d r '

~E'exp(-2h0(r-r')q q

X e x p ( - 2~of~dr"[r0 + q52(r")]) ; the rest of the details are given in the appendix.

2

(3.24)

530

Y.Y. Goldschmidt / Dynarnicalrelaxation

Upon the introduction of renormalized parameters g instead of u 0 and t instead of r 0 and making the change of variables 1

~ ~--£g-1/2LZ'c d~ ~

g-1/4t-B/vd~

with z = 2 + o(e2),

(3.25)

with ]3/1, = 1 - I e + o(e2),

(3.26)

~ 2Xgl/4L-2-¢/~71,

(3.27)

we can express the equation of motion at the fixed point g* = 16~r2e/3 in terms of variable y = tL 1/v in the form:

0-----0rqb+ I ( g * ) - 1 / 2 y ~ +

~t~3- ~/('r)+ ¼gl/~-~bFl(y + i g~/~-.-q~2)

+ 641r2~b(Y + ½g / / ~ 2 ) l n ( y + I gl/rgi-~2)

q- ¼g~dp('r)f0°° d'r'[A4(4'/r2'r')-

1]

x [exp( - f0~' d~-"(y + I gl/~q~2(1" - gl/~-~")) - e x p ( - ~-'(y + I g~/-~-,2(~)} ],

(3.28)

with the random noise satisfying (~(~)7/(r')) = 3(, - r'). The function Fl(y ) has been defined in eq. (2.5). The last term in (3.28) is a non-markovian (memory) term. At the bulk critical point y = 0, we finally obtain: 0 = 0,q5 + 22, 3 -71(~-)+ ~el/2eoFl(2~rrel/2, 2) + ~eq53ln(2~-3 ~rd/2q~z) + ¼V~qS(r)f0°° d , ' [ A'(v~-vr,')- 1 l × [exp{- lvCe-f0" d , " q}2(~"- v~-r")}- exp { - IvTe~"q~2(~-)} ] .

(3.29)

Let us discuss briefly some limiting situations associated with eq. (3.29). First if the initial value ~0 is very large, there is a region where the bulk relaxation is observed. This is because for large argument Fl(y ) - _y-X and hence the term including Fx(~ 2) is negligible compared to the other terms. The random noise and the non-markovian term are also small compared to the other terms. It is then clear

531

Y. Y. Goldschmidt / Dynamical relaxation

that ~ - r-p/(,z) is an approximate solution to the equation in this region with = { -

+

On the other hand at much later times when ~ is already very small we can use the expansion 1

1

rl(y)=rx(o ) - 1--~2Yln y - 1-~2(l +C)y+o(y2),

(3.30)

where C is the constant defined in eq. (2.17), to expand the term containing F 1 in eq. (3.29). It is easy also to verify that the non-markovian term is equal to

¢'3e3/2

d

o~ (3.31)

144 ~r3 d?( ~") "~-~#p2('r) f 0 d~"[A4(~-') - 1] I-'2 + o ( e 2 ) ,

and thus this term is of higher order and can be discarded in a calculation to o(e). We thus find in that region eq. (3.29) becomes 0 = 0,¢~ + f~-rrrl(0) e'/2q, + ~(1 L ½(1 +

C)e)eO3 - ~(~) + o(e3/2). (3.31)

Eq. (3.31) is just the one expected for linear relaxation from the formalism developed in (I), and the linear relaxation time is given by the inverse gap of the corresponding quantum mechanical hamiltonian. (Notice that our rescaling of the 10

\ I

I 0 -~ 0-'

,

I

~

I

I

I

1

~ ["

I

I

1

i

i

I

1

~ 11

10

q-L-"

Fig. 5. A plot of the magnetization versus time for d = 3 (e = 1).

Y.Y. Goldschmidt / Dynamicalrelaxation

532

field q) in (3.26) differs from that used in (I) by terms of o(~); that's the reason for the appearance of the o(e) correction to the coefficient of (/)3 in eq. (3.31)). Thus for a large initial value ~0 the solution of eq. (3.29) is expected to display the full crossover from the bulk power law non-linear relaxation to the finite size linear relaxation which is exponential with characteristic time ~R(L). After finding the solution of eq. (3.29) in the form q)('r) = g(q)0, T),

(3.32)

the final result for the non-linear relaxation of the magnetization, taking into account the change of variables (3.25) and (3.26) is

M( Mo,'r, L ) = ( g* )-XI4L-#/~F( ( g* )X/4MoL tJ/~,ZX( g* )I/2TL-~).

(3.33)

The function F(q, 0, ~') has been calculated numerically from eq. (3.29) using the method outlined in the beginning of the section. We verified that the memory term (last term in eq. (3.29)) is negligible even in the non-linear regime compared to other terms and thus it was not retained in the numerical calculation. The function F(q,o, ~-) is plotted versus • in fig. 5 using a log-log scale and in fig. 6 using a semilog scale. The initial value chosen was ~0 = 20. One can observe clearly the 10

*% "%%,.°, ~_..A

10

-~

0,10

. . . . . .

.1' 6 . . . .

......

.

........

4-.10 * .........

q - L -~ Fig. 6. A plot of the magnetization versus time for d ~ 3, which emphasizes the region of linear relaxation.

533

Y. Y. Goldschmidt / Dynamicalrelaxation

crossover from the power law decay to an exponential decay at large time. This figure should be compared with fig. 5 of the MC simulation by Kikuchi and Okabe [10]. For this calculation we have used A = 0.005 and N = 400.

4. Summary and discussion In this paper we have shown how to obtain analytically the stochastic equation of state for the non-linear relaxation in a finite size system. Below four dimensions the equation is written in a form which fulfills the correct finite size scaling behavior. The solution at the bulk critical point displays the full crossover from the bulk power law decay to the finite size exponential relaxation as a function of time. Thus at long times the decay is governed by the linear relaxation time. We also solve explicitly for the linear relaxation time by solving for the gap of the associated quantum mechanical hamiltonian. The results are obtained in the vicinity of Tc and show a crossover from the bulk power law increase of the relaxation time to the finite size saturation at T~. The results are cast in a form which enables comparison with Monte Carlo simulation. Our results agree nicely with results of non-linear critical relaxation in three dimensions. More simulations in the vicinity of T~ are needed for comparison with the shape of the linear relaxation time. Using our results it is straightforward to obtain the relaxation of a system of any size, starting with different values of the initial magnetization. Besides obtaining the linear relaxation time it is also possible to use our methods to obtain an approximate expression for the full autocorrelation function of the magnetization. This will require solving for higher energy levels and eigenfunctions of the associated supersymmetric hamiltonian. This work has been supported in part by the National Science Foundation.

Appendix In this appendix we explain some of the intermediate steps used in the derivation of eq. (3.28). We start with eqs. (3.23) and (3.24). Using the function A(u) defined in eq. (2.5) we find 2

*-0 =-~°U°Lq-~O

-d

L 2

r~

eO(¢) 8--~o Jo d¢'[Ad('c') - l] "'d~'" ro + ½Uo*2

X exp - 47r----7 ~o

~"

8~r2)to

~'"

.

(A.1)

From the last integral subtract and add the term

fo ~°d'r' ~'r')-'-,~rd/2(

-7---4" (L2

ro+

lu o t~2/'l'))) ,

•

(A.2)

534

Y. Y. Goldschmidt / Dynamical relaxation

We substitute the result in eq. (3.23). We then use the identity

fo°~U2_e/2exp du { - _~2}

e

1 - ~e

(a.3)

and proceed to carry out the usual bulk renormalizations within the framework of minimal subtraction. Upon introducing renormalized coupling constants t instead of ro and g (dimensionless) instead of u0, and carrying out the rescaling transformations (3.25) through (3.27) we obtain eq. (3.28). References [1] M.E. Fisher, in Critical phenomena, Proc. 51st Enrico Fermi Summer School, Varena, ed. M.S. Green (Academic Press, NY, 1972); M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516. [2] M.N. Barber, in Phase transitions and critical phenomena, vol. VIII, ed. C. Domb and J. Lebowitz (Academic Press, NY, 1984) and references therein [3] E. Br6zin and J. Zinn-Justin, Nucl. Phys. B257 (1985) 868 [4] E. Br~zin, J. de Phys. 43 (1982) 15 [5] J. Rudnick, H. Guo and D. Jasnow, J. Stat. Phys. 41 (1985) 353 [6] Y.Y. Goldschmidt, Nucl. Phys. B280 [FS18] (1987) 340 [7] J. Niel and J. Zinn-Justin, Nucl. Phys. B280 [FS18] (1987) 355 [8] H. Diehl, to be published [9] P.C. Martin, E.D. Siggia, H.A. Rose, Phys. Rev. A8 (1973) 423; R. Bausch et al., Z. Phys. B24 (1976) 113; A. Munoz Sudupe and R.F. Alvarez-Estrada, J. Phys. A16 (1983) 3049 [10] M. Kikuchi and Y. Okabe, J. Phys. Soc. Japan, 55 (1986) 1359 [11] H. Dekker and N.G. Van Kampen, Phys. Lett. 73A (1979) 374 [12] E. Witten, Nucl. Phys. B188 (1981) 513; F. Cooper and B. Freedman, Ann. of Phys. 146 (1983) 262 [13] S. Miyoshita and H. Takano, Prog. Theor. Phys. (Japan) 73 (1985) 1122 [14] M.C. Yalabik and J.D. Gunton, Phys. Rev. B25 (1982) 534; N. Jan et al., J. Stat. Phys. 33 (1983) 1; C. Kalle, J. Phys. A17 (1984) L801; R.B. Pearson et al., Phys. Rev. B31 (1985) 4472; S. Wansleben and D.P. Landau, Univ. of Georgia preprint (1986) [15] M. Suzuki, Prog. Theor. Phys. 58 (1977) 1142 [16] J.M. Sancho et al., J. Phys. A13 (1980) L443 [17] R. Bauch and H.K. Janssen, Z. Phys. B25 (1976) 275; R. Bauch et al., Z. Phys. B36 (1979) 179