Stochastic modeling of generalized Fokker-Planck equations. I.

TLDR: In this paper, a relatively simple method is developed whereby the manybody features of a typical generalized Fokker-planck equation (GFPE) for a diffusing molecule are first replaced by stochastic bath variables that are assumed to be Markovian.

Abstract: A relatively simple method is developed whereby the many‐body features of a typical generalized Fokker–Planck equation (GFPE) for a diffusing molecule are first replaced by stochastic bath variables that are assumed to be Markovian. Then the combined molecular and bath variables are characterized as a multidimensional Markov process obeying a stochastic–Liouville equation, which is, in general, incomplete, because it ignores the back reaction of the molecule on the bath variables. In the final step, the equation is completed by subjecting it to the appropriate constraints required for detailed balance. In this form the augmented Fokker–Planck equation (AFPE) properly describes relaxation to thermal equilibrium, and, for the appropriate limiting conditions, it reduces to the classical Fokker–Planck equation. This procedure for stochastic modeling of GFPE is both an improvement on and a generalization of a method previously outlined by Hwang, Mason, Hwang, and Freed (HMHF). Detailed illustrations of AFPE’s ...