Mesoscopic Analysis of Droplets in Lattice Systems with Long-Range Kac Potentials
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Mesoscopic Analysis of Droplets in Lattice Systems with Long-Range Kac Potentials E.A. Carlen · R. Esposito · J.L. Lebowitz · R. Marra
Received: 26 April 2011 / Accepted: 24 May 2012 / Published online: 13 June 2012 © Springer Science+Business Media B.V. 2012
Abstract We investigate the geometry of typical equilibrium configurations for a lattice gas in a finite macroscopic domain with attractive, long range Kac potentials. We focus on the case when the system is below the critical temperature and has a fixed number of occupied sites. We connect the properties of typical configurations to the analysis of the constrained minimizers of a mesoscopic non-local free energy functional, which we prove to be the large deviation functional for a density profile in the canonical Gibbs measure with prescribed global density. In the case in which the global density of occupied sites lies between the two equilibrium densities that one would have without a constraint on the particle number, a “droplet” of the high (low) density phase may or may not form in a background of the low (high) density phase. We determine the critical density for droplet formation, and the nature of the droplet, as a function of the temperature and the size of the system, by combining the present large deviation principle with the analysis of the mesoscopic functional given in Nonlinearity 22, 2919–2952 (2009). Keywords Lattice systems · Kac potential · Critical droplet Mathematics Subject Classification (2000) 49S05 · 52A40 · 82B26 E.A. Carlen work partially supported by U.S. National Science Foundation grant DMS 0901632. J.L. Lebowitz work partially supported by U.S. National Science Foundation grant DMR 08-02120 and AFOSR Grant FA 9550-10-1-0131. R. Marra work partially supported by MURST and INDAM-GNFN. E.A. Carlen Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA R. Esposito () MEMOCS, Università dell’Aquila, Cisterna di Latina, 04012 Latina, Italy e-mail: [email protected] J.L. Lebowitz Departments of Mathematics and Physics, Rutgers University, Piscataway, NJ 08854, USA R. Marra Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Rome, Italy
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1 Introduction The mathematical study of the behavior of a lattice gas of particles (or spins) interacting via a long range Kac potential, both in equilibrium and in non-equilibrium, has been the subject of many works: a recent book [7] provides a comprehensive treatment of the subject as it has developed so far. The long-range Kac potential introduces a third length scale between the microscopic scale of the lattice spacing and the macroscopic scale of the size of the domain. This third scale is referred to as the mesoscopic scale. As we show here, one can determine the geometric nature of “typical” microscopic particle configurations for such systems through the analysis of a mesoscopic free energy functional that serves as a large deviations functional for the system. We do this in a scaling regime that is critical for droplet for
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