Journal of Statistical Physics

, Volume 25, Issue 4, pp 679–694 | Cite as

Rigorous entropy-energy arguments

  • Barry Simon
  • Alan D. Sokal
Articles

Abstract

We present a method for making rigorous various arguments which predict that certain situations are unstable because of a balance of energy vs. entropy. As applications, we give yet another proof that the two-dimensional plane rotor has no spontaneous magnetization and we make rigorous Thouless' arguments on the one-dimensional Ising model with couplingJ/n2.

Key words

Entropy phase transitions Ising model plane rotor spin systems one-dimensional models two-dimensional models symmetry breaking Thouless effect 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. W. Anderson, G. Yuval, and D. R. Hamann,Phys. Rev. B 1:4464 (1970); P. W. Anderson and G. Yuval,J. Phys. C 4:607 (1971).Google Scholar
  2. 2.
    J. Bricmont, J. R. Fontaine, and L. J. Landau,Commun. Math. Phys. 56:281 (1977).Google Scholar
  3. 3.
    J. Bricmont, J. L. Lebowitz, and C. E. Pfister,J. Stat. Phys. 21:573 (1979).Google Scholar
  4. 4.
    R. L. Dobrushin and S. B. Shlosman,Commun. Math. Phys. 42:31 (1975).Google Scholar
  5. 5.
    F. J. Dyson,Commun. Math. Phys. 12:212 (1969).Google Scholar
  6. 6.
    F. J. Dyson, inStatistical Mechanics at the Turn of the Decade, E. G. D. Cohen, ed. (Marcel Dekker, New York, 1971).Google Scholar
  7. 7.
    F. J. Dyson,Commun. Math. Phys. 21:269 (1971).Google Scholar
  8. 8.
    M. E. Fisher,J. Appl. Phys. 38:981 (1967).Google Scholar
  9. 9.
    R. B. Griffiths, inStatistical Mechanics and Quantum Field Theory (Les Houches 1970), C. DeWitt and R. Stora, eds. (Gordon and Breach, New York, 1971).Google Scholar
  10. 10.
    C. Herring and C. Kittel,Phys. Rev. 81:869 (1951); see footnote 8a, p. 873.Google Scholar
  11. 11.
    R. B. Israel,Convexity in the Theory of Lattice Gases (Princeton University Press, Princeton, New Jersey, 1979).Google Scholar
  12. 12.
    V. I. Kolomytsev and A. V. Rokhlenko,Theor. Math. Phys. 35:487 (1978);Sov. Phys. Dokl. 24:902 (1979).Google Scholar
  13. 13.
    H. Kunz, unpublished.Google Scholar
  14. 14.
    L. D. Landau and E. M. Lifshitz,Statistical Physics (2nd edition) (Addison-Wesley, Reading, Massachusetts, 1969), pp. 478–479.Google Scholar
  15. 15.
    J. L. Lebowitz,J. Stat. Phys. 16:463 (1977).Google Scholar
  16. 16.
    J. L. Lebowitz and A. Martin-Löf,Commun. Math. Phys. 25:276 (1972).Google Scholar
  17. 17.
    O. A. McBryan and T. Spencer,Commun. Math. Phys. 53:299 (1977).Google Scholar
  18. 18.
    N. D. Mermin,J. Math. Phys. 8:1061 (1967).Google Scholar
  19. 19.
    A. Messager, S. Miracle-Sole, and C. Pfister,Commun. Math. Phys. 58:19 (1978).Google Scholar
  20. 20.
    A. Ojo,Phys. Lett. 45A:313 (1973).Google Scholar
  21. 21.
    C.-E. Pfister, On the Symmetry of the Gibbs States in Two Dimensional Lattice Systems,Commun. Math. Phys., in press.Google Scholar
  22. 22.
    J. B. Rogers and C. J. Thompson, Absence of Long-Range Order in One-Dimensional Spin Systems, University of Melbourne preprint (1980).Google Scholar
  23. 23.
    D. Ruelle,Commun. Math. Phys. 9:267 (1968).Google Scholar
  24. 24.
    S. Sakai,J. Funct. Anal. 21:203 (1976);Tôhoku Math. J. 28:583 (1976).Google Scholar
  25. 25.
    B. Simon,The Statistical Mechanics of Lattice Gases (Princeton University Press, Princeton, New Jersey, expected 1983).Google Scholar
  26. 26.
    D. J. Thouless,Phys. Rev. 187:732 (1969).Google Scholar
  27. 27.
    G. H. Wannier,Elements of Solid State Theory (Cambridge University Press, Cambridge, 1959), Chap. 4.Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • Barry Simon
    • 1
  • Alan D. Sokal
    • 2
  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadena
  2. 2.Department of PhysicsPrinceton UniversityPrinceton

Personalised recommendations