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

TL;DR

This paper applies large deviation theory to rigorously derive the equilibrium distribution of droplet sizes in a lattice model, showing it converges to a Poisson distribution under certain limits.

Contribution

It establishes a large deviation principle for the droplet model, identifying the Poisson distribution as the equilibrium distribution of droplet sizes.

Findings

Poisson distribution is the equilibrium for droplet sizes.

Large deviation principle is proven for the model.

Relative entropy characterizes the rate function.

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 $K \rightarrow \infty$ and $N \rightarrow \infty$ 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 $R(\theta | \rho^*)$, where $\theta$ is a possible asymptotic configuration of the number-density measures and $\rho^*$ is a Poisson distribution restricted to the set of positive integers. This LDP reveals that $\rho^*$ is the equilibrium distribution of the number-density measures, which in turn implies that $\rho^*$ 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.