The linear Fokker-Planck equation for the Ornstein-Uhlenbeck process as an (almost) nonlinear kinetic equation for an isolated N-particle system

TL;DR

This paper reveals that the Fokker-Planck equation, traditionally modeling Brownian motion, also describes the kinetic evolution of an isolated N-particle system with stochastic interactions when coefficients depend on the solution.

Contribution

It introduces a novel interpretation of the Fokker-Planck equation as an (almost) nonlinear kinetic equation for isolated particle systems.

Findings

Fokker-Planck equation with solution-dependent coefficients models N-particle systems.

Provides detailed discussion of this new interpretation.

Connects classical Brownian motion equations to many-particle kinetic theory.

Abstract

It is long known that the Fokker-Planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a Brownian test particle immersed in a heat bath of fixed temperature. Apparently, it is not so well known that the same partial differential equation, but now with constant coefficients which are functionals of the solution itself rather than being prescribed, describes the kinetic evolution (in the infinite particle limit) of an isolated N-particle system with certain stochastic interactions. Here we discuss in detail this recently discovered interpretation.