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

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

2012-04-05

arxiv.org/abs/1204.1145v1

Physics

Condensed Matter

Statistical Mechanics

13 pages, 1 figure

Scientific paper

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.

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