A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case (original) (raw)
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Zeitschrift f�r Physik B Condensed Matter, 1985
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We revisit the problem of the overdamped limit of the Brownian dynamics in an inhomogeneous medium characterized by a position-dependent friction coefficient and a multiplicative noise in one space dimension. Starting from the Kramers equation and analyzing it through the expansion in terms of the eigenfunctions of the quantum harmonic oscillator, we derive analytically the corresponding Fokker-Planck equation in the large friction (overdamped) limit. The result is fully consistent with the previous finding by Sancho, San Miguel, and Dürr [2], but our derivation procedure is simple and transparent. Furthermore, it would be straightforward to obtain higher-order corrections systematically. We also show that the overdamped limit is equivalent to the mass-zero limit in general. Our results are confirmed by numerical simulations for simple examples. PACS numbers: 05.10.Gg, 05.40.-a, 05.40.Jc, 66.10.C-
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arXiv: Statistical Mechanics, 1999
We present a systematic expansion of Kramers equation in the high friction limit. The latter is expanded within an operator continued fraction scheme. The relevant operators include both temporal and spatial derivatives and a covariant derivate or gauge like operator associated to the potential energy. Trivially, the first order term yields the Smoluchowsky equation. The second order term is readily obtained, known as the corrected Smoluchowsky equation. Further terms are computed in compact and straightforward fashion. As an application, the nonequilibrium thermodynamics and hydrodynamical schemes for the one dimensional Brownian motion is presented.
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Physics Letters A, 1990
An alternative to the Langevin description of Brownian motion is presented. The equations of motion are deterministic and time-reversal invariant. The friction and random forces appearing in the Langevin equation are replaced by pseudo-friction terms, which emulate the energy and momentum exchange between the Brownian particle and the medium. 122 0375-9601/90/$ 03.50 © 1990 -Elsevier Science Publishers B.V. (North-Holland) Volume 151, number 3,4 PHYSICS LETTERS A
The thermohydrodynamical picture of Brownian motion via a generalized Smoluchowsky equation
Physica A: Statistical Mechanics and its Applications, 2000
We present a systematic expansion of Kramers equation in the high friction limit. The latter is expanded within an operator continued fraction scheme. The relevant operators include both temporal and spatial derivatives and a covariant derivative or gauge like operator associated to the potential energy. Trivially, the first order term yields the Smoluchowsky equation. The second order term is readily obtained, known as the corrected Smoluchowsky equation. Further terms are computed in compact and straightforward fashion. As an application, the nonequilibrium thermodynamics and hydrodynamical schemes for the one dimensional Brownian motion is presented.
Microscopic theory of brownian motion: Mori friction kernel and langevin-equation derivation
Physica A: Statistical Mechanics and its Applications, 1975
Recei°,,'ed 28 Jammry t975 ~ derivatioa of the plaenome~o]ogical Langevif~ equadon riot d'.e momenu.m~ or a browrxian p. r~icIe from the generalized Langevin equatio~ of Mori is fresemed. This derivatio,q requires a de~aHed examination of the Mori friction kernel {or merr~ory fur~ctio~_~L it is ,demonstra[ed, on :A~e basis of prior work of Ma:zur and Oppenhei_m, that the Mc, ri ker~et doe~ ~':.m admit of a well behaved expar~sion in the ratio of bad>-a~ad brow~ianopar~icie masses. In addition, the Mori kernel is fom~d to decay o~,. t!~e siow time scaie of tI~e bro>miar>par~ide momentum° Both features, which contradict standard assumptio~, are traced to the inH~e~ce c, fcoupli~g to nonlinear powers of the momentum and preclude a Langevi~>equation derh'atier~ solety on the basis of timescale separation arguments. The Langevfi,,-equation is recovered, ho>ever, when the sinai! magnitude of s|owlzy decaying contributions is take,~ it:to accounL
Klimontovich's contributions to the kinetic theory of nonlinear Brownian motion and new developments
Journal of Physics: Conference Series, 2005
We review the concept of nonlinear Brownian motion, originally introduced by Klimontovich, and consider several applications to real systems, including e.g. atoms, molecules or ions laser cooling fields, charged grains in plasmas and interdisciplinary problems. In particular, we also discuss recent developments in the field of active Brownian particles. After summarizing the basic properties of active Brownian particle models, solutions of the corresponding Fokker-Planck equation are analyzed for free motions as well as for motions in confining fields. Furthermore, we study the distributions for finite systems of self-confined particles, interacting via Morse and Coulomb potentials. Finally, applications to clusters of atoms subject to laser cooling as well as to clusters of charged grains in dusty plasmas are discussed.
Approximate and numerically exact solutions of the Fokker-Planck equation with bistable potentials
Chemical Physics, 1989
The kinetic rate for a symmetric bistable potential is calculated from the Fokker-Planck operator on both position and momentum. Numerical results are obtained by applying the Lanczos algorithm to the matrix representation of the time evolution operator. Both the continued fraction representation of the spectral density and the first positive eigenvalue which determines the transition rate, are obtained from the computed tridiagonal matrix. The numerical results are compared with the available analytical approximations of the kinetic rate for intermediate potential barriers. In particular, the localized functions method leads to an approximation which accurately describes the approach to the intermediate friction regime from the diffusion limit. Comparison is made also with the available approximations covering all the friction range, and derived under the asymptotic condition of very large potential barriers. A satisfactory agreement is found between the Mel'nikov-Meshkov equation and the numerical results for moderately large potential barriers