ACCURATE PATH-INTEGRAL REPRESENTATIONS OF THE FOKKER-PLANCK EQUATION WITH A LINEAR REFERENCE SYSTEM - COMPARATIVE-STUDY OF CURRENT THEORIES

This paper presents an application of new discrete path integral solut ions recently introduced for Fokker-Planck dynamics with the aim to co mpare their relative efficacy in giving precise numerical results. The basic idea used in the derivation of these solutions is to model a co mplex Fokker-Planck equation with a general drift coefficient by a lin ear (Ornstein-Uhlenbeck) process, which is solved exactly, and to then employ an iterative technique to quantify what is missing from the re ference description. We reexamine and analyze two different approaches to realize the above strategy. These are an operator decoupling techn ique and a power series expansion method. Both approaches allow one to construct higher-order propagators valid to any desired precision in a time increment tau. Their use in a path integral means that many few er time steps N are required to achieve a given accuracy for a given n et increment t=N tau. Our comparison also includes results from standa rd path integral representations. The relative efficacy of the various different methods is illustrated by means of two problems, namely, th e dynamics of an overdamped Brownian particle in a potential field and the Kramers model of chemical reaction. The former process can be mod eled by a one-dimensional Fokker-Planck equation for the position coor dinate only, while the latter is governed by a two-dimensional Fokker- Planck equation where the relaxation over velocity is taken into accou nt. The numerical applications clearly demonstrate that the new repres entations are superior in the sense that they yield much more accurate results with less computational effort than the best alternative path integral method now in use.