WIAS Preprint No. 1632, (2011)

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



Authors

  • Jansen, Sabine
  • König, Wolfgang
  • Metzger, Bernd

2010 Mathematics Subject Classification

  • 82B21 60F10 60K35 82B31 82B05

Keywords

  • Classical particle system, canonical ensemble, equilibrium statistical mechanics, dilute system, large deviations

Abstract

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 Lennard-Jones-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 $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. 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 $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

Appeared in

  • Ann. Appl. Probab., 25 (2015) pp. 930--973.

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