HIGH-ACCURACY DISCRETE PATH-INTEGRAL SOLUTIONS FOR STOCHASTIC-PROCESSES WITH NONINVERTIBLE DIFFUSION MATRICES .2. NUMERICAL EVALUATION

We present a fast, high precision and easily implementable path integr al method for numerically solving Fokker-Planck equations. It is based on a generalized Trotter formula, which permits one to attain an adeq uate description of dynamical and equilibrium properties even though t he time increment tau= t/N is rather large. A remarkable property of t he symmetric Trotter splitting is used to systematically eliminate the lower-order errors resulting from time discretization. This means a s ignificant reduction of the number of time steps that are required to retain a given accuracy for a given net increment t=N tau, and, theref ore, significantly increasing the feasibility of path integral calcula tions. Yet another attractive feature of the present technique is that it allows for equations with singular diffusion matrices that are kno wn to present a special problem within the scope of the path integral formalism. The favorable scaling of the fast Fourier transform is used to numerically evaluate the path integral on a grid. High efficiency is achieved due to the Stirling interpolation which dynamically readju sts the distribution function every time step with a mild increase in cost and with no loss of precision. These developments substantially i mprove the path integral method and extend its applicability to variou s time-dependent problems which are difficult to treat by other means. One can even afford to extract information on eigenvalues and eigenfu nctions from a time-dependent solution thanks to the numerical efficie ncy of the present technique. This is illustrated by calculating the p ropagator and the lowest eigenvalues of a one-dimensional Fokker-Planc k equation. The method is also applied to a two-dimensional Fokker-Pla nck equation, whose diffusion matrix does not possess;an inverse (a so -called Klein-Kramers equation). The numerical applications show our m ethod to be a dramatic improvement over the standard matrix multiplica tion techniques available for evaluating path integrals in that it is much more efficient in terms of speed and storage requirements. (C) 19 97 American Institute of Physics.