26 August 1996

ELSEYIER

PHYSICS LETTERS A

Phycics Letters A 2 19 ( 1996) 309-3 12

Kinetic Ising model with non-thermal noise

Michael Schulz, Steffen Trimper Fachbereich Physik. Martin-Luther-Universitiir Halle, 06099 Halle. Germany

Received 14 May 1996; accepted for publkation 22 May 1996

Communicated by V.M. Agranovich

Abstract

We study conserved (Kawasaki) spin-exchange processes at finite temperature under external non-thermal noise. The dynamics of the total system based on the master equation is formulated in terms of Pauli operators. The result is a nonlinear diffusion equation for the magnetisation density. Already in mean-field approximation the correlation function reveals a stretched exponential behaviour. The critical exponents resulting from a one-loop renormalization group approach will be supported by simulations in one dimension.

PACS: 05.4O.+j; 05.5O.+q; 82.2O.Mj

Recently there have been many studies in non- equilibrium growth phenomena. Various discrete ki-

netic growth models and continuous equations for the

growth processes have been introduced and studied analytically and numerically [ I]. Applications are discussed in relation to other seemingly disparate problems, such as crystal growth, vapour deposition, bacterial colony growth, fire front motion, flux lines

in superconductors, and directed polymers in ran- dom media [2]. The ideas fuelling the interest are universality and dynamic scaling behaviour.

We study in this Letter the non-linear diffusion equation for the magnetisation density m( x, t),

8,rn= DV2m+AV(m2Vm) +v(x,t). (1)

The noise q( x, t) has zero mean and is Gaussian,

(rl(x,t)v(x’,t’)) =/J&X-XXI)&?-f’), (2)

where ,u specifies the noise amplitude. ( 1) describes a growth process of the magnetisation density in a d-

dimensional system. Such an equation can be derived from a master equation, see for details Ref. [ 33.

This equation has the form &m = -V - j + 7, Various Langevin equations of this type have been studied in the literature [4,5], for an overview see Ref. [ 61. The nontrivial scaling behaviour of such an equation is similar to that of the KPZ [7,2] which is widely accepted as describing the large-distance,

long- and short-time dynamics of a variety of different growth processes. This scaiing relation is manifested in the interface width w( L, t) N Laf( t/L") where L is the lateral size of the system and the scaling function f(x) - xp for x < 1, and f(x) + const for x >> 1. At an early stage of surface the width behaves as w( L, T) N tP where the scaling exponent p is related to the dynamic exponent z by j? = (Y/Z. However ( 1) is not an equation for surface growth. The latter have to be invariant under the transformation m + m + const.

Here we want to sketch the procedure which enables us to get the basic equation ( 1). To this aim we start from a master equation [ 81 on a lattice which can be

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310 M. Schulz, S. Trimper/Physics Letters A 219 (1996) 309-312

transformed into a Fock space representation where the dynamical process is written in terms of second

quantised operators [9-I I]. Usually that approach is formulated in terms of Bose operators. Recently, the

method has been extended to include the exclusion principle [ 12,131, compare also Ref. [ 141; for a recent review consult Ref. [ 151.

We have recently extended the formalism [ 31 to in- clude also thermal effects as a heat bath. In particular,

it has been demonstrated that weighted spin-exchange rates can be introduced in accordance with the prin-

ciple of detailed balance where the weight function is simply given by the Hamiltonian of the Ising model.

The dynamic process is realized by Kawasaki spin-

exchange dynamics [ 161 which will be explicitly ex- pressed in terms of Pauli operators. The algebraic

properties of those operators determine the dynamic properties of the classical system with a stochastic dy- namics with exclusion. This fact allows a straightfor-

ward generalisation of the method to more compli-

cated systems as viscous liquids and glasses [ I7- 191. To be specific let us analyse a spin lattice of unit

size 1. Each state of the system is characterised by the set of spin states s = {s;} where the s; = fl denotes the spin on the lattice site i. The time evolution of the

probability P (s, t) for a certain configuration s at time

t follows the master equation written in the form

d,P(s,t) = c L’( s,S) P(S,r) + fj(l). 5

(3)

The linear operator L’ is determined by the micro-

scopic dynamical processes of the system whereas e(t) corresponds to a general external noise. Contrary

to L’ the noise term is not controlled by temperature. For instance, such a behaviour can be realized for a kinetic Ising model with spin diffusion by a pulsed strong external stochastic field, i.e. the spins flip after

each pulse independent of the local configuration. Due to Doi [ 91 the probability distribution P( s, t)

can be related to a vector lF( t)) in a Fock space with the basic vectors denoted by Is). Under this transfor- mation one gets an equivalent equation [ 9,121

&/F(r)) = iIF(f (4)

The special expression for e in our model will be presented in (7), ( 13).

The relation between the quantum formalism and the probability approach is realized by expanding the

vector IF(t)) with respect to the basic vectors Is) of the Fock space,

IF([)) = c P(s,t) Is). S

(5)

As was shown by Doi [9], the average of a physi- cal quantity R(s) is given by the mean value of the corresponding operator i? = ((o/f?lF(r)), with the no-

tations (cp] = (01 exp(jJdi) and (qplF(f)) = I. Here, di is an annihilation operator. This rule remains valid

also in the case of Fermi operators [ 121. To any phys- ical quantity R(s) we can assign an operator ti which obeys the kinetic equation

a,(/?) = (&X/F(r)). (6)

If the dynamics of the systems is additionally con- trolled by thermal activations, the method should be

extended [ 31 in the sense that the hopping rates will be weighted with the equilibrium Hamiltonian. Tak-

ing into consideration a (Kawasaki) spin-exchange (Model B) dynamics, the evolution operator L in (4) is written as [3]

(7)

where the sum is restricted to all pairs of nearest neigh- bours. H is the Hamiltonian of the Ising model in terms of kBT without an external field,

& = JO C SiS,j.

(id

(8)

The evolution operator (7) describes the spin- exchange process within the underlying master equa- tion. The spin-exchange rates are weighted with the equilibrium Hamiltonian (8). It is easy to show that the evolution operator i,, fulfils the principle of detailed balance at any temperature.

Choosing as fi the particle number operator D; = dtd, we get the exact evolution equation (6) in the form

M. Schulz, S. Trimper/Physics Letters A 219 (1996) 309-312 311

&(D;) = -v * j, with j = y( (B;VA; - AiVB;)), (9)

with y = koZ* exp(2Je) and the following notations,

A;=(1 -O,,exp(-2Jo~&).

4 = DiexP(2JogDk). ( 10)

Furthermore, V2 is the discrete version of the usual differential operators and k(i) means summation over all lattice sites k adjacent to the ith. It is obvious that

(9) reflects the conservation of the total spin. Eq. (9) is analysed in mean-field approximation with theresult

d,n = -yV. j, withj=-Vn-n(l-n)j,, (II)

and

.i”, = (exP(-2Jo~Dx))V(enp(2Jo~Dx))

- (exP(2Jo~Dk))v(exp(-2Jo~Dk)),

where n(x. t) plays the role of (0;) in the continu-

ous limit. The conventional current (-Vn) has been supplemented by an additional term in which the re-

stricted occupation numbers are explicitly taken into account by the factor n( I - n).

A further expansion of the exponential terms in

( IO) in terms of the relevant field II and Vn yields the following equation for the magnetisation field m(x,t) = I -2n(x,r),

d,m = y( I + Jot)V2m - Jo~tV(m2Vm), (12)

with the coordination number t. Introducing the two independent constants D and A, respectively, the last equation is written in the form already given by ( I), however without the noise term.

The additive noise term can be originated within the used second quantised formulation. To this aim the

evolution operator i (4) is defined by i = i,, + i, with

t.=C7);(f)(dJ-di-d,dj+dtd;). (13)

This operator corresponds to an external non-thermal noise in (3). Note that the normalisation condition (c, P(s, t) = I, (+IF(t)) = I ) is not violated by ( 13). Here, we assume a Gaussian distribution for q_ The external noise breaks both detailed balance

at finite temperatures and spin conservation of model B. Numerical evidence is given in Ref. [ 31. Remark

that an important class of similar evolution processes,

molecular beam epitaxy, is conservative at least in an ideal sense. The basic equation discussed in relation to the mentioned problem is different from ( I ), compare Refs. [4-61. Finally, the parameter A in ( I ) reveals different signs depending on the kind of interaction represented by JO.

Eq. ( I ) will be studied in lowest order in A > 0. After Fourier transformation we get

&m(k,t) = -k*Dedt) m(k,t) +v(k,t). (14)

Here, the effective time-dependent diffusivity is intro-

duced by

with C(q,tjq= (m(q,r) nz(-q,r)). (15)

Without the non-relevant solution of the homogeneous

equation the correlation function reads

,

C(q,t) = $ /dtt exp{-2q2[b(r) - b(tt )I}.

0

with the notation B(t) = $ D,~T( t’) dt’. ( 16) can be

solved by the ansatz D,ff(t) = At” with an unknown exponent (Y. We obtain LY = (2 - d)/( 2 + d) for sufficiently large times (d < 2). Using ( 16) we find

the correlation function

( y( a, x) : incomplete gamma function). The exponent

(T characterises the stretched exponential behaviour ~=a+l=4/(d+2).Using(m2)=~d”qC(q,t), we get (m*(t)) - t2P, where the exponent p is given by 20 = 1 - ud/2 = (2 - d)/(2 + d). For one dimension it results p = l/6 which is yet in reasonable agreement with renormalization group result presented

below.

312 M. Schulz, S. Trimper/ Physics Lerters A 219 (1996) 309-312

Let us return to the basic equation ( 1 1. The per- turbation series is reorganised into a renormalization

group calculation [ 71. The renormalised coefficients

D, A, J_L obey in lowest order the how equations

dD -==(z-2+gK,,), $ dl

~(z -d-2a),

dA --=A(2 +2cu-2-3gK,,), dl

(17)

with g = puA/2D2 and Kd = SC//( 27~)” ($1 is the sur-

face of the d-dimensional unit shell). Note that due to the violation of Galilean invariance of ( I ) nontriv-

ial vertex renormalization is taken into account which

leads to the violation of the scaling relation c +2cu = 2

expected from naive scaling arguments. There appears no correction for the noise amplitude in the one-loop expansion. From here we confirm the general scaling

relation [4] of the form z - 2a = d. Eq. ( 17) can be converted into one equation for the dimensionless

coupling constant g.

dg z =g(c-5gK,l), withE=2-d (18)

(critical dimension: d, = 2). The exponents p and cy follow from the fixed points of ( 17) and ( 18). We find

z = 2 - E/S and Q = 2615. The exponent p describes the early stage of the growth of the magnetisation.

Using the relation p = CY/Z we get

(m’) N r2P, with p = &.

Our calculation is supported by a numerical simulation of(l) ind= I [3].TheexponentP=0.235f0.01

is in very good agreement with the analytical result. We have modelled the dynamics of the master equa-

tion in terms of second quantised operators which en- ables us to consider different dynamical processes [ 31 where in the present Letter only spin exchange pro- cesses are considered. The dynamics is determined by

the evolution operator and the algebraic properties of the Pauli operators. As a new aspect our method proves to include thermal simulated processes too. This ap- proach leads to a nonlinear evolution equation. Due to the conservation of the total spin component and the non-conservation of the noise term the model does not belong to the universality class of model B with d, =4.

A further application will be realized in a sys- tem with highly internal cooperativeness and strong neighbour-neighbour interactions. In this case there

appears a large number of possible neighboured con- figurations (with respect to the phase space) and

therefore transitions between different configurations. This fact includes the existence of a large number of different kinetic coefficients. An interesting applica-

tion seems the analysis of spin glasses [ 191 and com-

mon glasses within the Fredrickson model [ 17,201.

We thank Gunter Schiitz and David Mukamel for

useful discussions.

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