### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 1135-1155 |

Number of pages | 21 |

Journal | Journal of Statistical Physics |

Volume | 148 |

Issue number | 6 |

DOIs | |

State | Published - Sep 2012 |

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### Keywords

- Confined diffusion
- Inertial effects
- Mapping
- Smoluchowski equation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*148*(6), 1135-1155. https://doi.org/10.1007/s10955-012-0570-2

**Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation.** / Kalinay, Pavol; Percus, Jerome.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 148, no. 6, pp. 1135-1155. https://doi.org/10.1007/s10955-012-0570-2

}

TY - JOUR

T1 - Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation

AU - Kalinay, Pavol

AU - Percus, Jerome

PY - 2012/9

Y1 - 2012/9

N2 - 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.

AB - 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.

KW - Confined diffusion

KW - Inertial effects

KW - Mapping

KW - Smoluchowski equation

UR - http://www.scopus.com/inward/record.url?scp=84866563049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866563049&partnerID=8YFLogxK

U2 - 10.1007/s10955-012-0570-2

DO - 10.1007/s10955-012-0570-2

M3 - Article

VL - 148

SP - 1135

EP - 1155

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 6

ER -