Phase Transitions For Dilute Particle Systems with Lennard-Jones Potential

Abstract

We consider a classical dilute particle system in a large box with pair- interaction given by a Lennard-Jones-type potential. The inverse temperature is picked proportionally to the logarithm of the particle density. We identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity. Loosely speaking, the particle system separates into spatially distant components in such a way that within each phase all components are of the same size, which is the larger the lower the temperature. The main tool in our proof is a new large deviation principle for sparse point configurations

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).…”

Ideal Mixture Approximation of Cluster Size Distributions at Low Density

“…(1.15) Then e ∞ := lim k→∞ E k /k exists in (−∞, 0), and ν * := inf k∈N (E k − ke ∞ ) is positive [3,9]; note that Assumption (V)(4) is needed for proving the positivity of ν * .…”

“…In the thermodynamic limit N, |Λ| → ∞ such that N/|Λ| → ρ, under μ ideal β,N,Λ , the vector (N k /|Λ|) k∈N satisfies a large deviations principle with speed β|Λ| and rate function f ideal (β, ρ, ·). On the other hand, under μ CKMS β,N,Λ , the vector (kN k /N ) k∈N satisfies a large deviations principle with speed βN and rate function q → g ν (q) − 1 β k∈N q k k , which differs from the approximate functional g ν from [3,9] only by a vanishing term. Hence, from (1.21) we see that g ν (·) arises from f ideal (β, ρ, ·) by two simplifications: Z cl k (β) ≈ exp(−βE k ) and the omission of the multinomial coefficient, that is, the cluster free energy is approximated by the ground state energy, and the mixing entropy is neglected.…”

“…We note that both f ideal (β, ρ, ·) and g ν considered in [3,9] appear as exact large deviations rate functions for simple random partitions models. We consider vectors (N k ) 1≤k≤N ∈ N N 0 with N k=1 kN k = N as integer partitions, and look at the (not normalised) measures on partitions given by N,Λ (N 1 , .…”

“…A first quick consistency check concerns the comparison of the critical line ρ = exp(−βν * ) from [3,9] with the saturation density of the ideal mixture; this strengthens Lemma 3.4.…”