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

Joel L. Lebowitz and Hermann Rost

Stochastic Processes and their Applications, 1994, vol. 54, issue 2, 183-196

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[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.

Keywords: Girsanov; formula; Interacting; particle; system; Random; environment; Central; limit; theorem; Boltzmann; factor; Einstein; relation (search for similar items in EconPapers)
Date: 1994
References: Add references at CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/0304-4149(94)00015-8
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:54:y:1994:i:2:p:183-196

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().