The Einstein relation for the displacement of a test particle in a random environment. (English) Zbl 0812.60096

Summary: 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: \(X^ \varepsilon(t) = \varepsilon X (\varepsilon^{-2}t)\), \(\varepsilon \to 0\). Now perturb the process by putting a small “force” \(\varepsilon a\) on the test particle. We prove on three different examples that under the previous scaling the perturbed process converges to Brownian motion having the same covariance \(D\), but an additional drift of the form \(M \cdot a\). We show that \(M\), the “mobility” of the test particle and \(D\) are related to each other by the Einstein formula \[ M = (1/2) \beta D, \tag{1} \] 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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
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[1] Brémaud, P., Point Processes and Queues (1981), Springer: Springer Berlin · Zbl 0478.60004
[2] Calderoni, P.; Dürr, D., The Smoluchowski limit for a simple mechanical model, J. Statist. Phys., 55, 695-738 (1989) · Zbl 0713.60110
[3] De Masi, A.; Ferrari, P.; Goldstein, S.; Wick, D., An invariance principle for reversible Markov processes. Applications to random motion in random environments, J. Statist. Phys., 55, 787-856 (1989) · Zbl 0713.60041
[4] Einstein, A., Die von der molekularkinetichen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 4. Folge, 17, 549-560 (1905) · JFM 36.0975.01
[5] Einstein, A., Investigations on the Theory of the Brownian Movement, (Fürth, R. (1956), Dover: Dover New York), (English translation of preceding reference.) · JFM 53.0876.09
[6] Ferrari, P.; Goldstein, S.; Lebowitz, J. L., Diffusion, mobility and the Einstein relation, (Fritz, J., Statistical Physics and Dynamical Systems (1985), Birkhaeuser: Birkhaeuser Boston), 405-441
[7] Hájek, J.; Šidák, Z., Theory of Rank Tests (1967), Academic Press: Academic Press New York · Zbl 0161.38102
[8] Jakubowski, A.; Mémin, J.; Pagés, G., Convergence en loi des suites d’intégrales stochastiques sur l’espace D de Skorokhod, Probab. Theory Related Fields, 81, 11-137 (1989) · Zbl 0638.60049
[9] Kipnis, C.; Varadhan, S. R.S., Central limit theorem for additive functionals of reversible processes and applications to simple exclusion, Comm. Math. Physics, 104, 1-19 (1986) · Zbl 0588.60058
[10] Künnemann, M., The diffusion limit for reversible jump processes in \(Z^d\) with ergodic bond conductivities, Comm. Math. Phys., 90, 27-68 (1983)
[11] Lang, R., Unendlich-dimensionale Wienerprozesse mit Wechselwirkung, Teil I, Z. Wahrscheinlichkeitstheorie verw, Geb, 38, 55-72 (1977) · Zbl 0349.60103
[12] Nelson, E., The Dynamical Theories of Brownian motion (1967), Princeton University Press: Princeton University Press Princeton · Zbl 0165.58502
[13] Papanicolaou, G. C.; Varadhan, S. R.S., Ornstein-Uhlenbeck process in a random potential, Comm. Pure Appl. Math., 38, 819-834 (1985) · Zbl 0617.60078
[14] Stroock, D. W.; Varadhan, S. R.S., Multidimensional Diffusion Processes (1979), Springer: Springer Berlin · Zbl 0426.60069
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