Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Interfaces and typical Gibbs configurations for one-dimensional Kac potentials
Download PDF
Download PDF
  • Published: March 1993

Interfaces and typical Gibbs configurations for one-dimensional Kac potentials

  • Marzio Cassandro1,
  • Enza Orlandi2 &
  • Errico Presutti3 

Probability Theory and Related Fields volume 96, pages 57–96 (1993)Cite this article

  • 179 Accesses

  • 33 Citations

  • Metrics details

Summary

We consider a one dimensional Ising spin system with a ferromagnetic Kac potential γJ(γ|r|),J having compact support. We study the system in the limit, γ»0, below the Lebowitz-Penrose critical temperature, where there are two distinct thermodynamic phases with different magnetizations. We prove that the empirical spin average in blocks of size δγ−1 (for any positive δ) converges, as γ»0, to one of the two thermodynamic magnetizations, uniformly in the intervals of size γ−p, for any given positivep≧1. We then show that the intervals where the magnetization is approximately constant have lengths of the order of exp(cγ −1),c>0, and that, when normalized, they converge to independent variables with exponential distribution. We show this by proving large deviation estimates and applying the Ventsel and Friedlin methods to Gibbs random fields. Finally, if the temperature is low enough, we characterize the interface, namely the typical magnetization pattern in the region connecting the two phases.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bricmont, J., Fontaine, J.R., Speer, E.: Perturbation about the mean field critical point. Commun. Math. Phys.86, 337–362 (1982)

    Google Scholar 

  2. Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behaviour of stochastic dynamics: a pathwise approach. J. Stat. Phys.35, 603–634 (1984)

    Google Scholar 

  3. Cassandro, M., Olivieri, E., Picco, P.: Small random perturbations of infinite dimensional dynamical systems and nucleation theory. Ann. Inst. H. Poincaré B44 343–396 (1986)

    Google Scholar 

  4. Comets, F.: Nucleation for a long range magnetic model. Ann. Inst. H. Poincaré23, 135–178 (1987)

    Google Scholar 

  5. Comets, F., Eisele, T., Schatzman, M.: On secondary bifurcation for some non linear convolution equations. Trans. Am. Math. Soc.296, 661–702 (1986)

    Google Scholar 

  6. Dal Passo, R., de Mottoni, P.: The heat equation with a nonlocal density dependent advection term. (Preprint 1991)

  7. Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theory Probab. Appl.15, 458–486 (1970)

    Google Scholar 

  8. Dobrushin, R.L., Kotercký, R., Shlosman, S.: Wulff construction: a global shape from local interactions. Providence, RI: Am. Math. Soc. (to appear)

  9. Eisele, T., Ellis, R.S.: Symmetry breaking and random walks for magnetic systems on a circle. Z. Wahrscheinlichkeitstheor. Verw. Geb.63, 297–348 (1983)

    Google Scholar 

  10. Feller, W.: An introduction to probability theory and its applications, vol. I. vol. II, 1966 New York: Wiley 1957

    Google Scholar 

  11. Gates, D.J., Penrose, O.: The van der Waals limit for classical systems. I A variational principle. Commun. Math. Phys.15, 255–276 (1969); Existence and continuity of the canonical pressure. Commun. Math. Phys.17, 194–276 (1970)

    Google Scholar 

  12. Georgii, H.O.: Gibbs measures and phase transitions. (De Gruyter Stud. Math., vol. 9) Berlin New York: Walter de Gruyter 1988

    Google Scholar 

  13. Hemmer, P.C., Lebowitz, J.L.: Systems with long range potentials. In: Domb, C., Green, M.S., (eds.) Phase transitions and critical phenomena 5b, pp. 107–203. London New York San Francisco: Academic Press 1976

    Google Scholar 

  14. Kac, M., Uhlenbeck, G., Hemmer, P.C.: On the van der Waals theory of vapor-liquid equilibrium. I. Discussion of a one dimensional model. J. Math. Phys.4, 216–228 (1963); II. Discussion of the distribution functions. J. Math. Phys4, 229–247 (1963); III. Discussion of the critical region. J. Math. Phys.5, 60–74 (1964)

    Google Scholar 

  15. Lebowitz, J., Penrose, O.: Rigorous treatment of the Van der Waals Maxwell theory of the liquid vapour transition. J. Math. Phys.7, 98 (1966).

    Google Scholar 

  16. Minlos, R.A., Sinai, Ya.G.: Some new results on first order phase transition in lattice gas models. Trans. Mosc. Math. Soc.17, 237–267 (1967); The phenomenon of “phase separation” at low temperature in some lattice models of a gas. I. Math. USSR, Sb.2, 335–395 (1967); II. Trans. Mosc. Math. Soc.19 121–196 (1967).

    Google Scholar 

  17. Penrose, O., Lebowitz, J.: Rigorous treatment of metastable states in the Van der Waals Maxwell theory. J. Stat. Phys.3, 211–236 (1971)

    Google Scholar 

  18. Pfister, C.E.: Large deviations and phase separation in the two dimensional Ising model. Helv. Phys. Acta (to appear)

  19. Pisani, C., Thompson, C.J.: Generalized classical theory of magnetism. J. Stat. Phys.46, 971–982 (1987)

    Google Scholar 

  20. Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys.18, 127 (1970); Probability estimates for continuous spin systems. Commun Math. Phys.50, 189–194 (1976)

    Google Scholar 

  21. Schonmann, R., Tanaka, N.J.: One dimensional caricature of phase transition. (Preprint 1990)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Università di Roma La Sapienza, P. Le A. Moro, I-00185, Roma, Italy

    Marzio Cassandro

  2. Dipartimento di Matematica, Università di L'Aquila, I-67100, L'Aquila, Italy

    Enza Orlandi

  3. Dipartimento di Matematica, Università di Roma Tor Vergata, Via Fontanile di Carcaricola, I-00133, Roma, Italy

    Errico Presutti

Authors
  1. Marzio Cassandro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enza Orlandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Errico Presutti
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research has been partially supported by CNR, GNFM, GNSM and by grant SC1CT91-0695 of the Commission of European Communities

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cassandro, M., Orlandi, E. & Presutti, E. Interfaces and typical Gibbs configurations for one-dimensional Kac potentials. Probab. Th. Rel. Fields 96, 57–96 (1993). https://doi.org/10.1007/BF01195883

Download citation

  • Received: 20 February 1992

  • Revised: 16 November 1992

  • Issue Date: March 1993

  • DOI: https://doi.org/10.1007/BF01195883

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (1980)

  • 60K35
  • 82A05
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature