We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic

ABSTRACT. We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that the giant component occupies a fixed fraction of the graph. One consequence is an immediate derivation of the “cavity ” formula for the fraction of sites in the giant component. As a by-product of our analysis we compute also the large-deviation rate functions for the probabilities of the event that the random graph is connected, the event that it contains no loops and the event that it contains only “small ” components. 1.

Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based model of a solvent–solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing-point depression. The limit of infinitesimal concentrations is described in a subsequent paper.

We continue our study of colligative properties of solutions initiated in ref. 1. We focus on the situations where, in a system of linear size L, the concentration and the chemical potential scale like c = ξ/L and h = b/L, respectively. We find that there exists a critical value ξt such that no phase separation occurs for ξ � ξt while, for ξ>ξt, the two phases of the solvent coexist for an interval of values of b. Moreover, phase separation begins abruptly in the sense that a macroscopic fraction of the system suddenly freezes (or melts) forming a crystal (or droplet) of the complementary phase when b reaches a critical value. For certain values of system parameters, under “frozen ” boundary conditions, phase separation also ends abruptly in the sense that the equilibrium droplet grows continuously with increasing b and then suddenly jumps in size to subsume the entire system. Our findings indicate that the onset of freezing-point depression is in fact a surface phenomenon.

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We examine some aspects of the recent results by K. Binder [1]. The equilibrium formation/dissolution of droplets in finite systems is discussed in the context of the canonical and the grand canonical distributions. Key words: phase coexistence, phase transitions, Ising model, finite-size effects, droplets

The goal of this paper is twofold. First, assuming strict convexity of the surface tension, we derive a stability property with respect to the Hausdorff distance of a coarse grained representation of the interface between the two pure phases of the Ising model. This improves the L¹ description of phase segregation. Using this result