Ideal Mixture Approximation of Cluster Size Distributions at Low Density

Abstract

We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture. The present paper improves our earlier results by taking into account the mixing entropy.

The crystallization conjecture: a review

“…Recent works [59,131,133] investigate the exponentially small region of convergence of the Mayer series close to T = ρ = 0, where things can be proved in any dimension. Namely, they considered the limit T → 0 and ρ → 0 with the constraint that T log ρ → ν (recall that ρ ∼ z = e µ/T at small activity, so this is similar to fixing µ).…”

The Crystallization Conjecture: A Review

“…Given that there is no crystalline order in one and two-dimensional systems at positive temperature, but that crystallization is expected at T = 0, a natural question is to ask what is happening when T → 0. Recent works [59,131,133] investigate the exponentially small region of convergence of the Mayer series close to T = ρ = 0, where things can be proved in any dimension. Namely, they considered the limit T → 0 and ρ → 0 with the constraint that T log ρ → ν (recall that ρ ∼ z = e µ/T at small activity, so this is similar to fixing µ).…”

Thermodynamics of a Hierarchical Mixture of Cubes

“…Fisher proved that the phase transition for ideal droplet models subsists for a class of onedimensional models [Fis67,FF70]; the one-dimensional model serves as a counter-example to the strict convexity of the pressure as a function of interaction potentials when the class of potentials is chosen too large [Fis72], compare [Isr79, Chapter V.2]. For particles in R d with attractive interactions, errors in the ideal mixture approximation are bounded in [JK12,JKM15], however the bounds do not allow for a proof of phase transitions.…”