New method for studying the dynamics of disordered spin systems without finite-size effects

Abstract

A new method is presented which allows the determination of dynamical properties of disordered spin systems avoiding finite-size effects. The method is based on exact dynamical mean-field equations for the infinite large system. The resulting single-spin dynamics is solved by Monte Carlo simulations. We outline the formalism for the parallel dynamics of a fully connected model with random couplings. The decay of remanent magnetization of the model is studied. We find a power-law decay: m(t) -m(oo) oc/ ~"^ with a =0.474 and m{oo) =0.184 for the infinite system, PACS numbers: 87.10.+e Networks composed of spinlike elements which interact via long-range random interactions are of growing interest. Such models served as approximations for disordered materials as, for example, the Sherrington-Kirkpatrick (SK) model for spin glasses [l]. More recently similar systems have become popular as models for neural networks [2].The equilibrium behavior of such random spin systems is now well understood by applying the replica approach to determine thermal averages for disordered systems [3].Because of the lack of ergodicity these models show many interesting features that are essentially of nonequilibrium nature. Usually the dynamics at low temperatures will reach a state which will strongly depend on its initial conditions. This property made networks of spins useful for the models of associative memories [2].To study the dynamics of fully connected spin models, there are mainly two different approaches. The first one is to calculate the time evolution of dynamic quantities analytically using the so-called dynamical functional method [4]. This method has been successfully applied to the behavior of disordered spin systems at long time scales being able to recover and understand the equilibrium results mentioned above from a purely dynamical viewpoint [5][6][7].Unfortunately the situation is less satisfactory for the nonequilibrium, transient behavior. Apart from approximate treatments [8,9] exact calculations are only possible for very few time steps, e.g., up to four time steps for the SK model and two time steps for the Hopfield model [10].So most of the studies on dynamics of disordered spin systems rest on numerical simulations, which is the other possible approach. However, there are strong finite-size effects, which for example do not even allow a decision if there is a finite remanent magnetization for the infinite [Z(/)]y = spin system. In this Letter we propose a new method to avoid these finite-size problems. Our method combines the dynamical functional method, which allows us to perform the limit TV-• <» exactly, and a Monte Carlo simulation of the resulting stochastic one-particle equations. The method is demonstrated for a disordered spin system with synchronous dynamics.This model consists of TV Ising spins 5*, = ± 1, where every spin St is connected to all other spins Sj by couplings Jij, which are Gaussian random variables with distribution PUij) =VTV/2;rexp(-jNJ^j).Additionally the symmetry of the couplings ...

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