Skip to main content
Log in

Mesoscopic Analysis of Droplets in Lattice Systems with Long-Range Kac Potentials

Acta Applicandae Mathematicae Aims and scope Submit manuscript

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Alberti, G., Bellettini, G., Cassandro, M., Presutti, E.: Surface tension in Ising systems with Kac potentials. J. Stat. Phys. 82, 743–796 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlen, E.A., Carvalho, M.C., Esposito, R., Lebowitz, J.L., Marra, R.: Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional. Nonlinearity 22, 2919–2952 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gates, D.J., Penrose, O.: The van der Waals limit for classical systems. I. A variational principle. Commun. Math. Phys. 15, 255–276 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lebowitz, J.L., Penrose, O.: Rigorous treatment of the van der Waals Maxwell theory of the liquid vapor transition. J. Math. Phys. 7, 98 (1966)

    Article  MathSciNet  Google Scholar 

  7. Presutti, E.: Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, Berlin (2009)

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Esposito.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-012-9763-6

Keywords

Mathematics Subject Classification (2000)

Navigation