A Brownian particle in a microscopic periodic potential

TL;DR

This paper analyzes a model of a massive particle in a microscopic periodic potential interacting with a light particle reservoir, showing convergence to an Ornstein-Uhlenbeck process and highlighting the perturbative role of the potential.

Contribution

It demonstrates the convergence of the reduced dynamics to an Ornstein-Uhlenbeck process in the small mass ratio limit, accounting for the perturbative effects of the periodic potential.

Findings

Convergence to Ornstein-Uhlenbeck process as mass ratio tends to zero

Periodic potential acts as a perturbation in the dynamics

Bounded the effect of the potential on momentum fluctuations

Abstract

We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle's momentum resulting from collisions typically outweigh the shifts in momentum generated by the periodic force, and so the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that there is convergence to the Ornstein-Uhlenbeck process under the standard normalizations for the test particle variables. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle's momentum.