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

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[var epsilon](t) = [var epsilon]X([var epsilon]-2t), [var epsilon]-->0. Now perturb the process by putting a small "force" [var epsilon]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 where [beta] = 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.

[1]  P. Sen,et al.  Theory of rank tests , 1969 .

[2]  Rolf Künnemann,et al.  The diffusion limit for reversible jump processes onZd with ergodic random bond conductivities , 1983 .

[3]  R. Lang.  Unendlich-dimensionale Wienerprozesse mit Wechselwirkung , 1977 .

[4]  A. Einstein.  Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [AdP 17, 549 (1905)] , 2005, Annalen der Physik.

[5]  D. Dürr,et al.  The Smoluchowski limit for a simple mechanical model , 1989 .

[6]  Edward Nelson.  Dynamical Theories of Brownian Motion , 1967 .

[7]  S. Varadhan,et al.  Ohrnstein—uhlenbeck process in a random potential , 1985 .

[8]  S. Varadhan,et al.  Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions , 1986 .

[9]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[10]  J. Lebowitz,et al.  DIFFUSION, MOBILITY AND THE EINSTEIN RELATION , 1985 .

[11]  P. Ferrari,et al.  An invariance principle for reversible Markov processes. Applications to random motions in random environments , 1989 .

[12]  Hyperfinite methods applied to the Critical Branching Diffusion , 1989 .

[13]  J. Mémin,et al.  Convergence en loi des suites d'intégrales stochastiques sur l'espace $$\mathbb{D}$$ 1 de Skorokhod , 1989 .

[14]  H. L. Dryden,et al.  Investigations on the Theory of the Brownian Movement , 1957 .

[15]  P. Brémaud.  Point Processes and Queues , 1981 .