Does a Brownian particle equilibrate?

TL;DR

This paper investigates the derivation of Brownian motion equations from first principles, revealing that higher-order non-Markovian effects lead to deviations from classical Boltzmann-Gibbs equilibrium.

Contribution

It extends the derivation of Brownian motion equations to order , highlighting the impact of non-Markovian corrections on equilibrium properties.

Findings

Standard equations are consistent at order 

Order  corrections introduce non-Markovian effects

Stationary solutions deviate from Boltzmann-Gibbs statistics at higher order

Abstract

The conventional equations of Brownian motion can be derived from the first principles to order $\lambda^2=m/M$, where $m$ and $M$ are the masses of a bath molecule and a Brownian particle respectively. We discuss the extension to order $\lambda^4$ using a perturbation analysis of the Kramers-Moyal expansion. For the momentum distribution such method yields an equation whose stationary solution is inconsistent with Boltzmann-Gibbs statistics. This property originates entirely from non-Markovian corrections which are negligible in lowest order but contribute to order $\lambda^4$.