Corrections to Einstein's relation for Brownian motion in a tilted periodic potential

TL;DR

This paper revisits Brownian motion in a tilted periodic potential, deriving systematic corrections to Einstein's relation and linear response theory using homogenization, supported by efficient spectral numerical simulations.

Contribution

It provides general formulas and power series expansions for effective velocity and diffusion in tilted potentials, extending classical results with systematic corrections.

Findings

Derived formulas for velocity and diffusion tensor for arbitrary tilts.

Provided power series expansions for transport coefficients.

Validated theoretical results with efficient spectral numerical methods.

Abstract

In this paper we revisit the problem of Brownian motion in a tilted periodic potential. We use homogenization theory to derive general formulas for the effective velocity and the effective diffusion tensor that are valid for arbitrary tilts. Furthermore, we obtain power series expansions for the velocity and the diffusion coefficient as functions of the external forcing. Thus, we provide systematic corrections to Einstein's formula and to linear response theory. Our theoretical results are supported by extensive numerical simulations. For our numerical experiments we use a novel spectral numerical method that leads to a very efficient and accurate calculation of the effective velocity and the effective diffusion tensor.