Brownian Motion in an N-scale periodic Potential

TL;DR

This paper investigates Brownian motion in multiscale periodic potentials with N+1 scales, deriving an effective Langevin equation with multiplicative noise and providing a method to compute the diffusion tensor.

Contribution

It introduces a homogenization framework for nonseparable multiscale potentials, resulting in a Langevin equation with multiplicative noise and a systematic way to compute the diffusion tensor.

Findings

Effective Langevin equation with multiplicative noise derived

Homogenization method for N-scale potentials developed

Explicit computation of diffusion tensor via coupled Poisson equations

Abstract

We study the problem of Brownian motion in a multiscale potential. The potential is assumed to have N+1 scales (i.e. N small scales and one macroscale) and to depend periodically on all the small scales. We show that for nonseparable potentials, i.e. potentials in which the microscales and the macroscale are fully coupled, the homogenized equation is an overdamped Langevin equation with multiplicative noise driven by the free energy, for which the detailed balance condition still holds. The calculation of the effective diffusion tensor requires the solution of a system of N coupled Poisson equations.