Phase space reduction of the one-dimensional Fokker-Planck (Kramers) equation

TL;DR

This paper develops a rigorous method to reduce the one-dimensional Fokker-Planck (Kramers) equation to an effective spatial equation, the Smoluchowski equation, including systematic corrections for finite particle mass.

Contribution

It introduces a recurrence mapping procedure to derive unambiguous series corrections to the Smoluchowski equation from the Kramers equation.

Findings

Derived a series of corrections in powers of mass m

Validated the method on the harmonic oscillator model

Provided a rigorous mapping from phase space to spatial dynamics

Abstract

A pointlike particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered. We present a rigorous mapping of the Fokker-Planck equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x. The result is the Smoluchowski equation, valid in the overdamped limit, m->0, with a series of corrections expanded in powers of m. They are determined unambiguously within the recurrence mapping procedure. The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.