Random walk and diffusion in a stochastic lattice gas model

Abstract

The time evolution of the stochastic lattice gas with simple exclusion interaction is shown to converge in the thermodynamic limit and it is studied in the asymptotic regime of large time. The diffusion equation applies to the bulk transport of matter in an appropriate scaling limit. As regards the one-dimensional model, a conjecture by Spitzer is proved, stating that the distance travelled by a tagged particle is of the order of the fourth root of the elapsed time. Asymptotic exponents β and α are defined for more general models as giving the power law in time for the distances travelled by density fluctuations and tagged particles respectively. It is argued that α = β/2 should be valid for all one-dimensional models with exclusion.