Deterministic and time-reversal invariant description of Brownian motion (original) (raw)
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Microscopic theory of brownian motion
Physica A: Statistical Mechanics and its Applications, 1975
In this article nonlinear Langevin equations for a brownian (B) particle are derived and analyzed. Attention is focussed on the role of nonlinear B particle momentum (P) modes (powers of P). The multimode Mori formalism is used to derive equations of motion for P(t) for different numbers n of modes included in the description. The well-known linear equation of Mori corresponds to the case n = 1. Friction kernels and random forces in these equations exhibit slow decay and mass ratio (2) expansion anomalies due to mode coupling. The nonlinear Langevin equation obtained for a complete mode set (n = co) is free of these difficulties and is used to examine the first correction [0(24)] to standard 0(22) results. Although no closed set of nonlinear Langewin equations exists at order ;t*, a truncated set extends standard momentum correlation function predictions. * In the present notation, the one mode quantities of sec.2 are Km(e)-= K~l(e) and A~l(e) Axxx(e).
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
From Lagrangian to Brownian motion
Journal of Statistical Physics, 1988
We present a Lagrangian describing an idealized liquid interacting with a particle immersed in it. We show that the equation describing the motion of the particle as a functional of the initial conditions of the liquid incorporates noise and friction, which are attributed to specific dynamical processes. The equation is approximated to yield a Langevin equation with parameters depending on the Lagrangian and the temperature of the liquid. The origin of irreversibility and dissipation is discussed.
Overdamped limit and inverse-friction expansion for Brownian motion in an inhomogeneous medium
Physical Review E, 2015
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-
Firstly, this review studies Brownian motion to establish the existence of atomic nature of matter. Motion of brownian particles is investigated in two situations- parallel and perpendicular to the direction of gravity. Then, to understand better the dynamics that leads to equilibrium, Fluctuation Dissipation Theorem is discussed.
Journal of Statistical Mechanics: Theory and Experiment, 2009
The non-linear dissipation corresponding to a non-Gaussian thermal bath is introduced together with a multiplicative white noise source in the phenomenological Langevin description for the velocity of a particle moving in some potential landscape. Deriving the closed Kolmogorov's equation for the joint probability distribution of the particle displacement and its velocity by use of functional methods and taking into account the well-known Gibbs form of the thermal equilibrium distribution and the condition of 'detailed balance' symmetry, we obtain the exact master equation: given the white noise statistics, this master equation relates the non-linear friction function to the velocitydependent noise function. In particular, for multiplicative Gaussian white noise this operator equation yields a unique inter-relation between the generally nonlinear friction and the (multiplicative) velocity-dependent noise amplitude. This relation allows us to find, for example, the form of velocity-dependent noise function for the case of non-linear Coulomb friction.
An Efficient Method to Study Nondiffusive Motion of Brownian Particles
The experimental access to short timescales has pointed to the inadequacy of the standard Langevin theory of the Brownian motion (BM) in fluids. The hydrodynamic theory of the BM describes well the observed motion of the particles; however, the published approach should be improved in several points. In particular, it leads to incorrect correlation properties of the thermal noise driving the particles. In our contribution we present an efficient method, which is applicable to linear generalized Langevin equations describing the BM of particles with any kind of memory and apply it to interpret the experiments where nondiffusive BM of particles was observed. It is shown that the applicability of the method is much broader, allowing, among all, to obtain efficient solutions of various problems of anomalous BM.
Notes on Brownian motion and related phenomena
Eprint Arxiv Physics 9903033, 1999
In this article we explore the phenomena of nonequilibrium stochastic process starting from the phenomenological Brownian motion. The essential points are described in terms of Einstein's theory of Brownian motion and then the theory extended to Langevin and Fokker-Planck formalism. Then the theory is applied to barrier crossing dynamics, popularly known as Kramers' theory of activated rate processes. The various regimes are discussed extensively and Smoluchowski equation is derived as a special case. Then we discuss some of the aspects of Master equation and two of its applications.
Physica A: Statistical Mechanics and its Applications, 1978
The motion of a Brownian particle in an external field can be described on two levels: by a Fokker-Planck equation for the joint probability distribution of position and velocity, and by a Smoluchowski equation for the distribution in position space only. We derive the second description, with corrections, from the first by means of a systematic expansion procedure of the Chapman-Enskog type in terms of the inverse friction coefficient. We also derive equations describing the initial period, when the Smoluchowski description is not yet valid; in particular we find formulae connecting the initial value to be used for the Smoluchowski equation with that of the full Fokker-Planck equation. The special case of an harmonically bound Brownian particle can be solved exactly; the results are used to check and to illustrate our expressions for general potential.