Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation

Pavol Kalinay, Jerome K. Percus

Research output: Contribution to journalArticlepeer-review


A point-like 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 1D Fokker-Planck (Kramers) 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/γ, γ denotes the friction coefficient. The corrections 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.

Original languageEnglish (US)
Pages (from-to)1135-1155
Number of pages21
JournalJournal of Statistical Physics
Issue number6
StatePublished - Sep 2012


  • Confined diffusion
  • Inertial effects
  • Mapping
  • Smoluchowski equation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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