Approximate solution for Fokker-Planck equation

TL;DR

This paper introduces an approximate method for solving a class of Fokker-Planck equations by leveraging their connection to Schrödinger-type equations, validated through examples involving harmonic and polynomial potentials.

Contribution

It presents a novel approximation approach based on Schrödinger equations for Fokker-Planck problems, including error estimation and analysis of temperature effects.

Findings

Effective approximation for truncated harmonic potential

Error estimation method for the solution

Analysis of temperature dependence in system behavior

Abstract

In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schr\"{o}dinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are analyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of temperature is also discussed.