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Communications in Mathematical Physics

, Volume 86, Issue 3, pp 327–336 | Cite as

More surprises in the general theory of lattice systems

  • Alan D. Sokal
Article

Abstract

I use Israel's methods to prove new theorems of “ubiquitous pathology” for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ϱ be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φ k } of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ϱ k , such that {ϱ k } converges to ϱ. In certain situations the perturbations Φ k −Φ can be chosen to lie in a cone of “antiferromagnetic pair interactions.” I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r2 interaction (Thouless effect).

Keywords

Neural Network Statistical Physic Equilibrium State Complex System General Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Israel, R.B.: Existence of phase transitions for long-range interactions. Commun. Math. Phys.43, 59–68 (1975)Google Scholar
  2. 2.
    Israel, R.B.: Convexity in the theory of lattice gases. Princeton, N.J.: Princeton University Press 1979Google Scholar
  3. 3.
    Kuratowski, K.: Topology, Vol. I. New York, London, Warsaw: Academic Press/PWN-Polish Scientific Publishers, 1966, pp. 447, 479Google Scholar
  4. 4.
    Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Teor. Veroya. Prim.13, 201–229 (1968) [Theor. Prob. Appl.13, 197–224 (1968)]Google Scholar
  5. 5.
    Dobrushin, R.L.: Gibbsian random fields for lattice systems with pairwise interactions. Funkt. Anal. Prilozhen.2, No. 4, 31–43 (1968) [Funct. Anal. Appl.2, 292–301 (1968)]Google Scholar
  6. 6.
    Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Teor. Veroya. Prim.15, 469–497 (1970) [Theor. Prob. Appl.15, 458–486 (1970)]Google Scholar
  7. 7.
    Lanford, O.E. III, Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys.13, 194–215 (1969)Google Scholar
  8. 8.
    Preston, C.: Random fields. In: Lecture Notes in Mathematics, Vol. 534. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  9. 9.
    Ruelle, D.: Thermodynamic formalism. Reading, MA: Addison-Wesley 1978Google Scholar
  10. 10.
    Daniëls, H.A.M., van Enter, A.C.D.: Differentiability properties of the pressure in lattice systems. Commun. Math. Phys.71, 65–76 (1980)Google Scholar
  11. 11.
    van Enter, A.C.D.: A note on the stability of phase diagrams in lattice systems. Commun. Math. Phys.79, 25–32 (1981)Google Scholar
  12. 12.
    van Enter, A.C.D.: Stability properties of phase diagrams in lattice systems. Proefschrift, Rijksuniversiteit te Groningen (1981)Google Scholar
  13. 13.
    Wightman, A.S.: Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics. Introduction to [2]Google Scholar
  14. 14.
    Ruelle, D.: A heuristic theory of phase transitions. Commun. Math. Phys.53, 195–208 (1977)Google Scholar
  15. 15.
    Simon, B., Sokal, A.D.: Rigorous entropy-energy arguments. J. Stat. Phys.25, 679–694 (1981)Google Scholar
  16. 16.
    Messager, A., Miracle-Sole, S.: Equilibrium states of the two-dimensional Ising model in the two-phase region. Commun. Math. Phys.40, 187–196 (1975)Google Scholar
  17. 17.
    Lebowitz, J.L.: Coexistence of phases in Ising ferromagnets. J. Stat. Phys.16, 463–476 (1977)Google Scholar
  18. 18.
    Gallavotti, G., Miracle-Sole, S.: Equilibrium states of the Ising model in the two-phase region. Phys. Rev. B5, 2555–2559 (1972)Google Scholar
  19. 19.
    Anderson, P.W., Yuval, G., Hamann, D.R.: Exact results in the Kondo problem. II. Scaling theory, qualitatively correct solution, and some new results on one-dimensional classical statistical models. Phys. Rev. B1, 4464–4473 (1970)Google Scholar
  20. 20.
    Anderson, P.W., Yuval, G.: Some numerical results on the Kondo problem and the inverse-square one-dimensional Ising model. J. Phys. C4, 607–620 (1971)Google Scholar
  21. 21.
    Cardy, J.L.: One-dimensional models with 1/r 2 interactions. J. Phys. A14, 1407–1415 (1981)Google Scholar
  22. 22.
    Fröhlich, J., Spencer, T.: The phase transition in the one dimensional Ising model with 1/r 2 interaction energy. Commun. Math. Phys.84, 87–101 (1982)Google Scholar
  23. 23.
    Thouless, D.J.: Long-range order in one-dimensional Ising systems. Phys. Rev.187, 732–733 (1969)Google Scholar
  24. 24.
    Bhattacharjee, J., Chakravarty, S., Richardson, J.L., Scalapino, D.J.: Some properties of a one-dimensional Ising chain with an inverse-square interaction. Phys. Rev. B24, 3862–3865 (1981)Google Scholar
  25. 25.
    Bishop, E., Phelps, R.R.: The support functionals of a convex set. In: Convexity. Proceedings of symposia in pure mathematics, Vol. 7. Providence RI: American Mathematical Society, 1963, pp. 27–35Google Scholar
  26. 26.
    Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc.16, 605–611 (1965)Google Scholar
  27. 27.
    Landford, O.E. III, Robinson, D.W.: Statistical mechanics of quantum spin systems. III. Commun. Math. Phys.9, 327–338 (1968)Google Scholar
  28. 28.
    Mazur, S.: Über konvexe Mengen in linearen normierten Räumen. Studia Math.4, 70–84 (1933)Google Scholar
  29. 29.
    Asplund, E.: Fréchet differentiability of convex functions. Acta Math.121, 31–47 (1968)Google Scholar
  30. 30.
    Larman, D.G., Phelps, R.R.: Gateaux differentiability of convex functions on Banach spaces. J. London Math. Soc.20, 115–127 (1979)Google Scholar
  31. 31.
    Lindenstrauss, J., Olsen, G., Sternfeld, Y.: The Poulsen simplex. Ann. Inst. Fourier (Grenoble)28, no. 1, 91–114 (1978)Google Scholar
  32. 32.
    Olsen, G.H.: On simplices and the Poulsen simplex. In: Functional Analysis: Surveys and recent results II. Proceedings of the conference on functional analysis, Paderborn, Germany, 1979. North-Holland mathematics studies #38 (eds. K.-D. Bierstedt and B. Fuchssteiner). Amsterdam, New York, Oxford: North-Holland 1980, pp. 31–52Google Scholar
  33. 33.
    Phelps, R.R.: Support cones in Banach spaces and their applications. Adv. Math.13, 1–19 (1974)Google Scholar
  34. 34.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl.47, 324–353 (1974)Google Scholar
  35. 35.
    Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc.1, 443–474 (1979)Google Scholar
  36. 36.
    Sullivan, F.: A characterization of complete metric spaces. Proc. Am. Math. Soc.83, 345–346 (1981)Google Scholar
  37. 37.
    Shih Shu-Chung: Remarques sur le gradient généralisé. C.R. Acad. Sci. Paris A291, 443–446 (1980)Google Scholar
  38. 38.
    Kenderov, P., Robert, R.: Nouveaux résultats génériques sur les opérateurs monotones dans les espaces de Banach. C.R. Acad. Sci. Paris A282, 845–847 (1976)Google Scholar
  39. 39.
    Sokal, A.D.: Unpublished manuscript (1981)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Alan D. Sokal
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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