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

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 eective velocity and the eective diusion tensor that are valid for arbitrary tilts. Furthermore, we obtain power series expansions for the velocity and the diusion coecient 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 ecient and accurate calculation of the eective velocity and the eective diusion tensor.