Abstract
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(x 1,x 2)/|x 1 -x 2|αd, where theJ(x 1,x 2) 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. A. Vuillermot,J. Phys. A: Math. Gen. 10:1319–1333 (1977).
F. Ledrappier,Commun. Math. Phys. 56:297–302 (1977).
J. L. van Hemmen and R. G. Palmer,J. Phys. A: Math. Gen. 15:3881–3890 (1982).
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″
M. A. Akcoglu and J. Krengel,J. reine angew. Math. 323:53–67 (1981).
K. M. Khanin and Ya. G. Sinai,J. Stat. Phys. 20:573–584 (1979).
D. Ruelle,Statistical mechanics (Benjamin, New York, 1969);
R. B. Israel,Convexity in the theory of lattice gases (Princeton University Press, Princeton, New Jersey, 1979).
R. B. Griffiths,Phys. Rev. 176:655–659 (1968).
S. Goulart Rosa,J. Phys. A: Math. Gen. 15:L51–54 (1982).
O. E. Lanford and D. W. Robinson,J. Math. Phys. 9:1120–1125 (1968).
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.
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.3
E. H. Lieb,Commun. Math. Phys. 31:327–340 (1973).
M. E. Fisher and D. Ruelle,J. Math. Phys. 7:260–270 (1966).
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).
R. Fisch,J. Stat. Phys. 18:111–114 (1978);Phys. Rev. B 24:5410–5412 (1981).
R. B. Griffiths,J. Math. Phys. 8:478–483, 484–489 (1967); R. Graham,J. Stat. Phys. 29:177–183 (1982).
B. Simon,J. Stat. Phys. 26:53–58 (1981).
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.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van Enter, A.C.D., van Hemmen, J.L. The thermodynamic limit for long-range random systems. J Stat Phys 32, 141–152 (1983). https://doi.org/10.1007/BF01009426
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01009426