Numerical method for the nonlinear Fokker-Planck equation

A practical method based on distributed approximating functionals (DAFs) is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate $K$-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo [Phys. Rev. E 54, 931 (1996)] to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the “flat” region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schrödinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem.