Journal of Statistical Physics

, Volume 150, Issue 4, pp 776–803

Corrections to Einstein’s Relation for Brownian Motion in a Tilted Periodic Potential


DOI: 10.1007/s10955-013-0692-1

Cite this article as:
Latorre, J.C., Pavliotis, G.A. & Kramer, P.R. J Stat Phys (2013) 150: 776. doi:10.1007/s10955-013-0692-1


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.


Homogenization theory Linear response theory Einstein’s relation Spectral methods for PDEs 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • J. C. Latorre
    • 1
  • G. A. Pavliotis
    • 1
    • 2
  • P. R. Kramer
    • 3
  1. 1.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany
  2. 2.Department of MathematicsImperial College LondonLondonUK
  3. 3.Mathematical Sciences DepartmentRensselaer Polytechnic InstituteTroyUSA