APPLICATION OF A STOCHASTIC REPRESENTATION IN NUMERICAL STUDIES OF THE RELAXATION FROM A METASTABLE STATE

Abstract

Abstract. We study the relaxation from a metastable state using a stochastic process which is related to the generating function of the system by means of Feynman-Kac formula. The results of such representation are compared with direct numerical simulations of the stochastic differential equations describing system's evolution. We have found that the stochastic representation is more efficient from computational point of view then the direct simulations. The problems related to its numerical implementation are discussed. INTRODUCTION In this paper we are concerned with numerical simulations of the escape from a metastable state caused by the presence of an external noise. Such problem appears in many areas of physics and the literature on this subject is very large (see for example [1], [2]). The process of escape can be rigorously described by a Fokker-Planck equation, which gives the probability distribution of finding the system at different points of its phase space as a function of time [1]. Alternatively one may introduce a set of stochastic differential equations for the process such that the ensemble of states evolving according to these equations is characterized by the time dependent probability distribution obtained from the given by the Fokker-Planck equation. In the following we shall consider a very special case in which the stochastic differential equation describing the time evolution of our system has the form: