The Einstein relation for the displacement of a test particle in a random environment

Abstract

Consider a stochastic system evolving in time, in which one observes the displacement of a tagged particle, X( t). Assume that this displacement process converges weakly to d-dimensional centered Brownian motion with covariance D, when space and time are appropriately scaled: X ε ( t) = εX( ε −2 t), ε→0. Now perturb the process by putting a small “force” ε a on the test particle. We prove on three different examples that under previous scaling the perturbed process converges to Brownian motion having the same covariance D, but an additional drift of the form M · a. We show that M, the “mobility” of the test particle, and D are related to each other by the Einstein formula M=( 1 2 β)D where β = 1/ kT( T being temperature and k Boltzmann's constant) is defined in such a way that the reversible state for the modified dynamics gets the correct Boltzmann factor. The method used to verify (1) is the calculus of Radon-Nikodym derivatives of measures in the space of trajectories (Girsanov's formula). Scaling simultaneously force and displacement has also a technical advantage: there is no need to show existence, under the perturbed evolution, of an invariant measure for the process “environment seen from the test particle” such that it is equivalent to the invariant measure under the unperturbed evolution.