### Abstract

A generalized diffusion equation is derived from the Mori-Kubo generalized Langevin for a brownian oscillator subject to gaussian random but in general non-markovian noise. This equation involves a time-dependent diffusion function rather than a phenomenological diffusion constant. For long times the diffusion function approaches a constant for overdamped markovian oscillators; only in the limit of extreme overdamping is the phenomenological theory recovered. A previously derived generalized phase space Fokker-Planck equation for the brownian oscillator is shown to have incorrect short-time behaviour. The difficulty is traced to a transient systematic component of the Mori random force which is non-vanishing for classical lattices at 0 K. Fokker-Planck and diffusion equations for the brownian oscillator are derived from a generalized Langevin representation equivalent to, but distinct from, that of Mori and Kubo. The random force in this representation lacks the systematic transient component. The Fokker-Planck and diffusion equations obtained from this alternative Langevin representation are thus correct at all times.

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

Pages (from-to) | 1671-1681 |

Number of pages | 11 |

Journal | Molecular Physics |

Volume | 33 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 1977 |

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### All Science Journal Classification (ASJC) codes

- Biophysics
- Molecular Biology
- Condensed Matter Physics
- Physical and Theoretical Chemistry

### Cite this

*Molecular Physics*,

*33*(6), 1671-1681. https://doi.org/10.1080/00268977700101391

}

*Molecular Physics*, vol. 33, no. 6, pp. 1671-1681. https://doi.org/10.1080/00268977700101391

**Non-markovian diffusion and Fokker-Planck equations for brownian oscillators.** / Adelman, S. A.; Garrison, Barbara Jane.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Non-markovian diffusion and Fokker-Planck equations for brownian oscillators

AU - Adelman, S. A.

AU - Garrison, Barbara Jane

PY - 1977/1/1

Y1 - 1977/1/1

N2 - A generalized diffusion equation is derived from the Mori-Kubo generalized Langevin for a brownian oscillator subject to gaussian random but in general non-markovian noise. This equation involves a time-dependent diffusion function rather than a phenomenological diffusion constant. For long times the diffusion function approaches a constant for overdamped markovian oscillators; only in the limit of extreme overdamping is the phenomenological theory recovered. A previously derived generalized phase space Fokker-Planck equation for the brownian oscillator is shown to have incorrect short-time behaviour. The difficulty is traced to a transient systematic component of the Mori random force which is non-vanishing for classical lattices at 0 K. Fokker-Planck and diffusion equations for the brownian oscillator are derived from a generalized Langevin representation equivalent to, but distinct from, that of Mori and Kubo. The random force in this representation lacks the systematic transient component. The Fokker-Planck and diffusion equations obtained from this alternative Langevin representation are thus correct at all times.

AB - A generalized diffusion equation is derived from the Mori-Kubo generalized Langevin for a brownian oscillator subject to gaussian random but in general non-markovian noise. This equation involves a time-dependent diffusion function rather than a phenomenological diffusion constant. For long times the diffusion function approaches a constant for overdamped markovian oscillators; only in the limit of extreme overdamping is the phenomenological theory recovered. A previously derived generalized phase space Fokker-Planck equation for the brownian oscillator is shown to have incorrect short-time behaviour. The difficulty is traced to a transient systematic component of the Mori random force which is non-vanishing for classical lattices at 0 K. Fokker-Planck and diffusion equations for the brownian oscillator are derived from a generalized Langevin representation equivalent to, but distinct from, that of Mori and Kubo. The random force in this representation lacks the systematic transient component. The Fokker-Planck and diffusion equations obtained from this alternative Langevin representation are thus correct at all times.

UR - http://www.scopus.com/inward/record.url?scp=0347239884&partnerID=8YFLogxK

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U2 - 10.1080/00268977700101391

DO - 10.1080/00268977700101391

M3 - Article

AN - SCOPUS:0347239884

VL - 33

SP - 1671

EP - 1681

JO - Molecular Physics

JF - Molecular Physics

SN - 0026-8976

IS - 6

ER -