WIAS Preprint No. 1632, (2011)

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



Authors

  • Jansen, Sabine
  • König, Wolfgang
    ORCID: 0000-0002-4212-0065
  • Metzger, Bernd

2010 Mathematics Subject Classification

  • 82B21 60F10 60K35 82B31 82B05

Keywords

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

DOI

10.20347/WIAS.PREPRINT.1632

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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