Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution

[摘要] In this paper we use large deviation theory to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and statistical mechanics. Given b?N and c>b, K distinguishable particles are placed, each with equal probability 1/N, onto the N sites of a lattice, where K/N equals c. We focus on configurations for which each site is occupied by a minimum of b particles. The main result is the large deviation principle (LDP), in the limit K?? and N?? with K/N=c, for a sequence of random, number-density measures, which are the empirical measures of dependent random variables that count the droplet sizes. The rate function in the LDP is the relative entropy R(????), where ? is a possible asymptotic configuration of the number-density measures and ?? is a Poisson distribution with mean c, restricted to the set of positive integers n satisfying n?b. This LDP implies that ?? is the equilibrium distribution of the number-density measures, which in turn implies that ?? is the equilibrium distribution of the random variables that count the droplet sizes.

[学科分类] 应用数学