Journal of Statistical Physics

, Volume 32, Issue 1, pp 141–152 | Cite as

The thermodynamic limit for long-range random systems

  • A. C. D. van Enter
  • J. L. van Hemmen


Long-range spin systems with random interactions are considered. A simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape, provided there is no external field. The argument is valid for both classical and quantum spin systems, and can be applied to (a) spins randomly distributed on a lattice and interacting via dipolar interactions; and (b) spin systems with potentials of the formJ(x1,x2)/|x1 -x2| αd , where theJ(x1,x2) are independent random variables with mean zero,d is the dimension, and α > 1/2. The key to the proof is a (multidimensional) subadditive ergodic theorem. As a corollary we show that, for random ferromagnets, the correlation length is a nonrandom quantity.

Key words

Subadditive ergodic theorem random interactions longrange interactions dipolar coupling free energy correlation length 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1. (a)
    P. A. Vuillermot,J. Phys. A: Math. Gen. 10:1319–1333 (1977).Google Scholar
  2. 1. (b)
    F. Ledrappier,Commun. Math. Phys. 56:297–302 (1977).Google Scholar
  3. 1. (c)
    J. L. van Hemmen and R. G. Palmer,J. Phys. A: Math. Gen. 15:3881–3890 (1982).Google Scholar
  4. 2.
    P. Walters,Ergodic theory—Introductory lectures, Springer Lecture Notes in Mathematics 458 (Springer, Berlin, 1975) Chap. I, §5; N. Wiener,Duke Math. J. 5:1–18 (1939), in particular Theorem II″Google Scholar
  5. 3.
    M. A. Akcoglu and J. Krengel,J. reine angew. Math. 323:53–67 (1981).Google Scholar
  6. 4.
    K. M. Khanin and Ya. G. Sinai,J. Stat. Phys. 20:573–584 (1979).Google Scholar
  7. 5. (a)
    D. Ruelle,Statistical mechanics (Benjamin, New York, 1969);Google Scholar
  8. 5. (b)
    R. B. Israel,Convexity in the theory of lattice gases (Princeton University Press, Princeton, New Jersey, 1979).Google Scholar
  9. 6.
    R. B. Griffiths,Phys. Rev. 176:655–659 (1968).Google Scholar
  10. 7.
    S. Goulart Rosa,J. Phys. A: Math. Gen. 15:L51–54 (1982).Google Scholar
  11. 8.
    O. E. Lanford and D. W. Robinson,J. Math. Phys. 9:1120–1125 (1968).Google Scholar
  12. 9.
    N. M. Hugenholtz,Proc. Symp. Pure Math. 38, part 2 (Amer. Math. Soc., Providence, Rhode Island, 1982) pp. 407–465; note in particular the discussion of entropy in Sec. 4.Google Scholar
  13. 10.
    Develop log〈exptJ(i, j)〉 in the power series aroundt = 0. This gives (2.13). Cf. Yu. V. Prohorov and Yu. A. Rozanov,Probability Theory (Springer-Verlag, New York, 1969) §VI.3Google Scholar
  14. 11.
    E. H. Lieb,Commun. Math. Phys. 31:327–340 (1973).Google Scholar
  15. 12.
    M. E. Fisher and D. Ruelle,J. Math. Phys. 7:260–270 (1966).Google Scholar
  16. 13.
    J. F. C. Kingman,Ann. Prob. 1:883–909 (1973) and Springer Lecture Notes in Mathematics 539 (Springer, Berlin, 1976), pp. 168–223; Y. Derrienic,C. R. Acad. Sci. Paris 281A:985–988 (1975).Google Scholar
  17. 14.
    R. Fisch,J. Stat. Phys. 18:111–114 (1978);Phys. Rev. B 24:5410–5412 (1981).Google Scholar
  18. 15.
    R. B. Griffiths,J. Math. Phys. 8:478–483, 484–489 (1967); R. Graham,J. Stat. Phys. 29:177–183 (1982).Google Scholar
  19. 16.
    B. Simon,J. Stat. Phys. 26:53–58 (1981).Google Scholar
  20. 17.
    B. Simon,Commun. Math. Phys. 77:111–126 (1980); D. Iagolnitzer and B. Souillard, in:Coll. Math. Soc. Janos Bolyai 27, Random fields, Esztergom, 1979 (North-Holland, Amsterdam, 1982), pp. 573–592.Google Scholar
  21. 18.
    An example has been given by C. Grillenberger and U. Krengel, inProc. Conf. “Ergodic theory and related topics”, Vitte/Hiddensee, GDR, 1981 (Akademie-Verlag, Berlin, 1982) pp. 53–57.Google Scholar

Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • A. C. D. van Enter
    • 1
  • J. L. van Hemmen
    • 1
  1. 1.Universität HeidelbergHeidelberg 1Germany

Personalised recommendations