Skip to main content
Log in

On the correlation for Kac-like models in the convex case

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The aim of this paper is to stu the behavior asm tends to ∞ of a family of measures exp[-Φ (m)(x)]dx (m) on ℝm, whereΦ (m) is a potential on ℝm which is a perturbation “in a suitable sense” of the harmonic potential Σ j x 2j .

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.

Similar content being viewed by others

References

  1. J. Avron, I. Herbst, and B. Simon, Schrödinger operators with magnetic fields, III Atoms in homogeneous magnetic fields,Commun. Math. Phys. 79:529–572 (1981).

    Google Scholar 

  2. P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).

    Google Scholar 

  3. H. J. Brascamp and E. Lieb, On extensions of the Brunn-Minkovski and Prékopa-Leindler Theorems, including inequalities for log concave functions, and with an application to diffusion equation,J. Funct. Anal. 22 (1976).

  4. E. Brézin, Course, ENS (1989).

  5. M. Brunaud and B. Helffer, Un problème de double puits provenant de la théorie statistico-mécanique des changements de phase (ou relecture d'un cours de M. Kac), Preprint (March 1991).

  6. P. Cartier, Inégalités de corrélation en mécanique statistique,Séminaire Bourbaki 25ème année, 1972–73, No. 431 (Lecture Notes in Mathematics, No. 383; Springer-Verlag, Berlin).

  7. R. S. Ellis,Entropy, Large Deviations, and Statistical Mechanics (Springer, New York, 1985).

    Google Scholar 

  8. C. Fortuin, P. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets,Commun. Math. Phys. 22:89–103 (1971).

    Google Scholar 

  9. J. Glimm and A. Jaffe,Quantum Physics (A Functional Integral Point of View), 2nd ed. (Springer-Verlag, Berlin).

  10. F. Guerra, J. Rosen, and B. Simon, TheP(φ)2 Euclidean quantum field theory as classical statistical mechanics,Ann. Math. 101:111–259 (1975).

    Google Scholar 

  11. B. Helffer, On Schrödinger equation in large dimension and connected problems in statistical mechanics, inDifferential Equations with Applications to Mathematical Physics, W. F. Ames, E. M. Harrell II, and J. V. Herod, eds. (Academic Press, New York, 1992), pp. 153–166.

    Google Scholar 

  12. B. Helffer, Around a stationary phase theorem in large dimension,J. Funct. Anal., to appear.

  13. B. Helffer, Problèmes de double puits provenant de la théorie statistico-mécanique des changements de phase, II Modèles de Kac avec champ magnétique, étude de modèles près de la température critique, Preprint (March 1992).

  14. B. Helffer and J. Sjöstrand, Multiple wells in the semi-classical limit, I,Commun. PDE 9(4):337–408 (1984); II,Ann. Inst. H. Poincaré Phys. Theor. 42(2):127–212 (1985).

    Google Scholar 

  15. B. Helffer and J. Sjöstrand, Semiclassical expansions of the thermonamic limit for a Schrödinger equation,Astérisque 210:135–181 (1992).

    Google Scholar 

  16. B. Helffer and J. Sjöstrand, Semiclassical expansions of the thermonamic limit for a Schrödinger equation II,Helv. Phys. Acta 65:748–765 (1992).

    Google Scholar 

  17. M. Kac, Statistical mechanics of some one-dimensional systems, inStudies in Mathematical Analysis and Related Topics: Essays in Honor of Georges Polya (Stanford University Press, Stanford, California, 1962), pp. 165–169.

    Google Scholar 

  18. M. Kac,Mathematical Mechanisms of Phase Transitions (Gordon and Breach, New York, 1966).

    Google Scholar 

  19. D. Robert, Propriétés spectrales d'opérateurs pseudo-différentiels,Commun. PDE 3(9):755–826 (1978).

    Google Scholar 

  20. D. Ruelle,Statistical Mechanics (Benjamin, New York, 1969).

    Google Scholar 

  21. B. Simon,The P(φ)2 Euclidean Quantum Field Theory (Princeton University Press, Princeton, New Jersey, 1974).

    Google Scholar 

  22. B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).

    Google Scholar 

  23. B. Simon,The Statistical Mechanics of Lattice Gases (Princeton University Press, Princeton, New Jersey, 1993).

    Google Scholar 

  24. I. M. Singer, B. Wong, S. T. Yau, and S. S. T. Yau, An estimate of the gap of the first two eigenvalues of the Schrödinger operator,Ann. Scuola Norm. Sup. Pisa (4) 12:319–333 (1985).

    Google Scholar 

  25. J. Sjöstrand, Potential wells in high dimensions I,Ann. Inst. H. Poincaré Phys. Theor. 58(1) (1993).

  26. J. Sjöstrand, Potential wells in high dimensions II, more about the one well case,Ann. Inst. H. Poincaré Phys. Theor. 58(1) (1993).

  27. J. Sjöstrand, Exponential convergence of the first eigenvalue divided by the dimension, for certain sequences of Schrödinger operator,Astérisque 210:303–326 (1992).

    Google Scholar 

  28. J. Sjöstrand, Schrödinger in high dimensions, asymptotic constructions and estimates, inProceeding of the French-Japanese Symposium on Algebraic Analysis and Singular Perturbations (CIRM, Luminy, France, 1991).

    Google Scholar 

  29. J. Sjöstrand, Evolution equations in a large number of variables,Math. Nachr., to appear.

  30. A. D. Sokal, Mean field bounds and correlation inequalities,J. Stat. Phys. 28:431–439 (1982).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Helffer, B., Sjöstrand, J. On the correlation for Kac-like models in the convex case. J Stat Phys 74, 349–409 (1994). https://doi.org/10.1007/BF02186817

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02186817

Key Words

Navigation