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

Abstract

One of the main contributions of this paper is to illustrate how large deviation theory can be used to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and statistical mechanics. The model is simply defined. $K$ distinguishable particles are placed at random onto the $N$ sites of a lattice, where the ratio $K/N$, the average number of particles per site, equals a constant $c\in (1,\infty )$. We focus on configurations for which each site is occupied by at least one particle. The main result is the large deviation principle (LDP), in the limit where $K\to \infty$ and $N\to \infty$ 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 $R(\theta |{\rho }^{\ast })$, where $\theta$ is a possible asymptotic configuration of the number-density measures and ${\rho }^{\ast }$ is a Poisson distribution restricted to the set of positive integers. This LDP reveals that ${\rho }^{\ast }$ is the equilibrium distribution of the number-density measures, which in turn implies that ${\rho }^{\ast }$ is the equilibrium distribution of the random variables that count the droplet sizes. We derive the LDP via a local large deviation estimate of the probability that the number-density measures equal $\theta$ for any probability measure $\theta$ in the range of these random measures.