High-accuracy discrete path integral solutions for stochastic processes with noninvertible diffusion matrices. II. Numerical evaluation

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Abstract
We present a fast, high precision and easily implementable path integral method for numerically solving Fokker–Planck equations. It is based on a generalized Trotter formula, which permits one to attain an adequate description of dynamical and equilibrium properties even though the time increment τ=t/N is rather large. A remarkable property of the symmetric Trotter splitting is used to systematically eliminate the lower-order errors resulting from time discretization. This means a significant reduction of the number of time steps that are required to retain a given accuracy for a given net increment t=Nτ, and, therefore, significantly increasing the feasibility of path integral calculations. Yet another attractive feature of the present technique is that it allows for equations with singular diffusion matrices that are known 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 readjusts the distribution function every time step with a mild increase in cost and with no loss of precision. These developments substantially improve the path integral method and extend its applicability to various time-dependent problems which are difficult to treat by other means. One can even afford to extract information on eigenvalues and eigenfunctions from a time-dependent solution thanks to the numerical efficiency of the present technique. This is illustrated by calculating the propagator and the lowest eigenvalues of a one-dimensional Fokker–Planck equation. The method is also applied to a two-dimensional Fokker–Planck equation, whose diffusion matrix does not possess an inverse (a so-called Klein-Kramers equation). The numerical applications show our method to be a dramatic improvement over the standard matrix multiplication techniques available for evaluating path integrals in that it is much more efficient in terms of speed and storage requirements.