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

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