Large deviations for cluster size distributions in a continuous classical many-body system

An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable LennardJones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature β ∈ (0,∞) and particle density ρ ∈ (0, ρcp) in the thermodynamic limit. Here ρcp > 0 is the close packing density. While in general the rate function is an abstract object, our second main result is the Γ-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit β → ∞, ρ ↓ 0 such that −β−1 log ρ → ν for some ν ∈ (0,∞). The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter ν. Under additional assumptions on the potential, the Γ-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle. MSC 2010. Primary 82B21; Secondary 60F10, 60K35, 82B31, 82B05.