Limit Theorems for Multiparticle Systems

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

The project is devoted to the study of chaotic billiards, which includes Sinai's dispersing billiards, Bunimovich's stadium and its modifications, gases of hard balls, and Lorentz gases. The goal is to investigate the ergodic and statistical properties of such systems, focusing on bounds for correlations and probabilistic limit theorems. A special emphasis is given to gases of particles (disks) of different masses and/or sizes, where the number and/or the masses of some particles grow to infinity. The approach involves rescaling time and space in order to apply the scheme known in statistical physics as the "hydrodynamical limit." This allows one to overcome unpleasant features of the underlying dynamics such as nonuniform hyperbolicity and very slow mixing rates. By using methods borrowed from averaging theory the principal investigator hopes to establish the convergence of the orbits to certain stochastic processes. The broader impact of this project derives from its interdisciplinary character: it is motivated by problems in physics and other sciences and its main goal is the development of adequate mathematical tools for addressing such problems. In particular, the project deals with Brownian motion, which arises in a variety of natural processes whose evolution is subject to many random factors. The principal investigator also studies the behavior of a piston in a cylinder, in an attempt to model what occurs in an automobile engine. The mathematical methods developed in this project should apply to wide classes of problems coming from the natural sciences.