Unpublished Paper

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

(2014)
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. 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 the ratio K/N, the average number of particles per site, 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 where 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 restricted to the set of positive integers n satisfying n ≥ b. This LDP reveals 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. We derive the LDP via a local large deviation estimate of the probability that the number-density measures equal _ for any probability measure _ in the range of these random measures.
Disciplines

Publication Date

2014
Citation Information

Richard S Ellis and Shlomo Ta'asan. "Detailed Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution" (2014) Available at: http://works.bepress.com/richard_ellis/46/