Variational Schemes in the Fokker-Planck Equation

TL;DR

This paper explores variational methods for approximating solutions to the Fokker-Planck equation, especially when detailed balance is absent, by constructing Hermitian operators or using solvable approximations, tested on quantum and toy models.

Contribution

It introduces and compares two classes of variational schemes for the Fokker-Planck equation, extending their application to non-equilibrium stochastic processes.

Findings

Methods successfully tested on quantum-mechanical problems.

Effective in modeling particles in potentials with non-white noise.

Provides a framework for non-equilibrium Fokker-Planck solutions.

Abstract

We investigate variational methods for finding approximate solutions to the Fokker-Planck equation, especially in cases lacking detailed balance. These schemes fall into two classes: those in which a Hermitian operator is constructed from the (non-Hermitian) Fokker-Planck operator, and those which are based on soluble approximations to this operator. The various approaches are first tested on a simple quantum-mechanical problem and then applied to a toy Fokker-Planck equation. The problem of a particle moving in a potential and subject to external non-white noise is then investigated using the formalism developed earlier on in the paper.