From ballistic to diffusive behavior in periodic potentials

TL;DR

This paper analyzes the asymptotic behavior of a particle in a periodic potential under small friction, demonstrating convergence to Brownian motion with an explicitly computed singular diffusion coefficient.

Contribution

It proves the commutation of Freidlin-Wentzell and homogenization limits and provides explicit formulas for the effective diffusion coefficient in the small friction limit.

Findings

Particle position converges to Brownian motion with a singular diffusion coefficient.

The results hold for a family of space/time rescalings.

Novel estimates on the resolvent of a hypoelliptic operator are developed.

Abstract

The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator.