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Brownian MotionFluctuations, Dynamics, and Applications$
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Robert M. Mazo

Print publication date: 2008

Print ISBN-13: 9780199556441

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199556441.001.0001

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STOCHASTIC EQUATIONS FROM A STATISTICAL MECHANICAL VIEWPOINT

STOCHASTIC EQUATIONS FROM A STATISTICAL MECHANICAL VIEWPOINT

Chapter:
(p.138) 11 STOCHASTIC EQUATIONS FROM A STATISTICAL MECHANICAL VIEWPOINT
Source:
Brownian Motion
Author(s):

Robert M. Mazo

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199556441.003.0011

This chapter begins with a heuristic derivation of the Langevin equation using the Mori projection operator as the main tool. The random force and the friction constant are identified. A similar procedure is carried out for the Fokker–Planck equation using the Zwanzig projection operator. The assumptions of rapid decay of certain correlations, made in the derivations, are criticized and the existence of slow decay modes is pointed out. The slow decay modes are studied using the technique of elimination of fast variables. The effects on the validity of the Langevin and Fokker–Planck equations in their classical forms are discussed.

Keywords:   Mori projection operator, Zwanzig projection operator, Langevin equation, Fokker–Planck equation, slow variable, mode coupling, non Markovian, hydrodynamic modes

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