Generalized Fokker-Planck equation, Brownian motion, and ergodicity

TL;DR

This paper derives a generalized Fokker-Planck equation and Langevin dynamics for Brownian particles beyond the lowest order, revealing non-Maxwellian stationary distributions and including finite collision time effects.

Contribution

It extends microscopic Brownian motion theory to include nonlinear and higher-order derivatives, providing explicit coefficients and analyzing finite collision time corrections.

Findings

Derived generalized Fokker-Planck and Langevin equations with higher-order derivatives.

Identified corrections to stationary distribution making it non-Maxwellian.

Explicitly evaluated coefficients for a generalized Rayleigh model.

Abstract

Microscopic theory of Brownian motion of a particle of mass $M$ in a bath of molecules of mass $m\ll M$ is considered beyond lowest order in the mass ratio $m/M$. The corresponding Langevin equation contains nonlinear corrections to the dissipative force, and the generalized Fokker-Planck equation involves derivatives of order higher than two. These equations are derived from first principles with coefficients expressed in terms of correlation functions of microscopic force on the particle. The coefficients are evaluated explicitly for a generalized Rayleigh model with a finite time of molecule-particle collisions. In the limit of a low-density bath, we recover the results obtained previously for a model with instantaneous binary collisions. In general case, the equations contain additional corrections, quadratic in bath density, originating from a finite collision time. These corrections survive to order $(m/M)^2$ and are found to make the stationary distribution non-Maxwellian. Some relevant numerical simulations are also presented.