Formulation of a moment method for multidimensional Fokker-Planck equations.

Abstract

A moment method for general n-dimensional (n\ensuremath{\ge}1) Fokker-Planck equations in semi-infinite domains with mixed boundary conditions is developed in this paper. Generally, time evolution equations of moments include terms with reduced distribution functions. With mixed boundary conditions in n-dimensional phase spaces, the reduced distribution functions are not explicitly known. This adds an openness to the time evolution equations of moments. We develop an auxiliary set of variables that allow the removal of this type of openness by introducing it into a general moment truncation scheme. The other openness of moment equations caused by the general phase space dependence of drift and diffusion coefficients is removed by using the conventional central moment truncation scheme. The closed set of time evolution equations of moments is numerically solved with the lsoda} package of computer programs [A. Hindmarsh, in Scientific Computing, edited by R. Stepleman et al. (North-Holland, Amsterdam, 1983), pp. 55--64]. The method is applied to three examples. The coupling of moments and reduced moments is first demonstrated by an interstitial clustering process in diatomic materials. Then, the moment equations for a one-dimensional Fokker-Planck equation in a semi-infinite domain are derived as a special case of the present method. The moment equations of the one-dimensional Fokker-Planck equation derived by Ghoniem [Phys. Rev. B 39, 11 810 (1989)] for atomic clustering are thus recovered in the second example. Finally, the moment method is also tested by applying it to a two-dimensional Ornstein-Uhlenbeck process, which can be solved analytically. Numerical calculations of the first three moments with truncation only at second-order moments are in very good agreement with the analytical results. Truncation at fourth-order moments is found to give similar results for the first three moments.