Approximation to a Fokker-Planck equation for the Brownian motor

Abstract

An approximate method is proposed and numerically confirmed for a time-dependent Fokker-Planck equation (FPE) modeling the Brownian motor. The situation addressed in this paper is that the potential is spatially asymmetric and multiplicatively modulated. The approximate solution is composed of Sturm-Liouville eigenfunctions. While some time-independent partial differential equations and those perturbed around them have been known to be reduced to ordinary differential equations with a few degrees of freedom, the time-dependent FPE dealt with in this paper is also found to be reduced to an ordinary differential equation.