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 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(θ∣ρ∗), where θ is a possible asymptotic configuration of the number-density measures and ρ∗ is a Poisson distribution with mean c, restricted to the set of positive integers n satisfying n≥b. This LDP implies that ρ∗ is the equilibrium distribution of the number-density measures, which in turn implies that ρ∗ is the equilibrium distribution of the random variables that count the droplet sizes.
Highlights
This paper is motivated by a natural question for a basic model of a droplet
Under the assumption that K/N = c and that each site is occupied by a minimum of b particles, what is the equilibrium distribution, as N → ∞, of the number of particles per site? We prove in Corollary 3 that this equilibrium distribution is a Poisson distribution, with mean c, restricted to the set of positive integers n satisfying n ≥ b
For the droplet model we prove the large deviation principle (LDP) for a sequence of random probability measures, called number-density measures, which are the empirical measures of a sequence of dependent random variables that count the droplet sizes
Summary
This paper is motivated by a natural question for a basic model of a droplet. Given b ∈ N and c > b, K distinguishable particles are placed, each with equal probability 1/N, onto the N sites of a lattice Λ N = {1, 2, . . . , N}. We end the Introduction by expanding on a comment made at the beginning of this section This comment concerns one of the main applications of large deviation theory in statistical mechanics, which is to identify the equilibrium distribution or distributions of a model as the minimum point(s) of the rate function in an LDP for the model. This procedure is useful to study phase transitions in the model, which concern how the structure of the set of equilibrium distributions changes as the parameters defining the model change. Details of this analysis are given in [31]
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