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).
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Acknowledgements
We thank Errico Presutti for very helpful discussions. E.C., R.E. and R.M. acknowledge the kind hospitality of the Institute for Mathematical Science on the National Singapore University, where part of this work was done.
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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.
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Carlen, E.A., Esposito, R., Lebowitz, J.L. et al. Mesoscopic Analysis of Droplets in Lattice Systems with Long-Range Kac Potentials. Acta Appl Math 123, 221–237 (2013). https://doi.org/10.1007/s10440-012-9763-6
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DOI: https://doi.org/10.1007/s10440-012-9763-6