Abstract
It has been proved that a tracer particle in a reversible lattice gas converges to Brownian motion. However, only in a few particular cases has a strictly positive self-diffusion coefficientD been established. Here we supply the missing piece and show thatD>0 in general. The exceptions are one-dimensional lattice gases with nearest neighbor jumps only, for whichD=0. The proof establishes a variational formula forD which could be used to obtain realistic bounds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Perrin,Atoms (Van Nostrand, New York, 1916).
K. W. Kehr and K. Binder, Simulation of diffusion in lattice gases and related kinetic phenomena, inApplications of the Monte Carlo Method in Statistical Physics, K. Binder, ed. (Springer, Berlin, 1984), pp. 181–221.
C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion,Commun. Math. Phys. 106:1–19 (1986).
A. DeMasi, P. A. Ferrari, S. Goldstein, and W. D. Wick, An invariance principle for reversible Markov processes. Applications to random motions in random environments,J. Stat. Phys. 55:787–855 (1989).
R. Arratia, The motion of a tagged particle in the simple symmetric exclusion process on ℤ,Ann. Prob. 11:362–373 (1983).
T. M. Liggett,Interacting Particle Systems (Springer, Berlin, 1985).
H. O. Georgii,Canonical Gibbs Measures (Springer, Berlin, 1979).
H. van Beijeren, Mode-coupling theory for purely diffusive systems,J. Stat. Phys. 35:399–112 (1984).
H. van Beijeren, Transport properties of stochastic Lorentz models,Rev. Mod. Phys. 54:195–234 (1982).
H. van Beijeren and H. Spohn, Transport properties of the one-dimensional stochastic Lorentz model. I velocity autocorrelation,J. Stat. Phys. 31:231–254 (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Spohn, H. Tracer diffusion in lattice gases. J Stat Phys 59, 1227–1239 (1990). https://doi.org/10.1007/BF01334748
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01334748