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

Abstract

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 ∈ N and > , distinguishable particles are placed, each with equal probability 1/ , onto the sites of a lattice, where / equals . We focus on configurations for which each site is occupied by a minimum of particles. The main result is the large deviation principle (LDP), in the limit → ∞ and → ∞ with / = , 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 ( | * ), where is a possible asymptotic configuration of the number-density measures and * is a Poisson distribution with mean , restricted to the set of positive integers satisfying ≥ . 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.