On the Ising Model with Long‐Range Interaction
1.R. Brout, Phys. Rev. 118, 1009 (1960).
2.Our γ is essentially equivalent to the parameter used by Brout, z being the effective number of neighbors. In some of the literature γ is used to denote the reciprocal range of interaction and corresponds to our where D is the dimensionality of the model.
3.See Ref. 1, the discussion of Eqs. (4.2) and (4.17).
4.G. Horwitz and H. B. Callen, Phys. Rev. 124, 1757 (1961).
5.B. Mühlschlegel and H. Zittartz, Z. Physik 175, 553 (1963).
6.M. Coopersmith and R. Brout, Phys. Rev. 130, 2539 (1963), the last paragraph in Sec. II;
6.R. Brout, Phys. Rev. 122, 469 (1961), Eq. (2.1)ff.
7.M. Kac, in Applied Probability, L. A. MacColl, Ed. (McGraw‐Hill Book Co., New York, 1957), Vol. VIII, pp. 73–85.
8.M. Kac, Phys. Fluids 2, 8 (1959);
8.and “Statistical Mechanics of Some One‐dimensional Systems” in Studies in Mathematical Analysis and Related Topics, Gilbarg et al., Eds. (Stanford University Press, Stanford, Calif., 1962).
9.G. A. Baker, Phys. Rev. 122, 1477 (1961).
10.A short survey of the method of random fields in equilibrium statistical mechanics, in which some of the present calculations are sketched as examples, was given by one of us (A. J. F. S.) in Analysis in Function Space, W. T. Martin and I. Segal, Eds. (MIT Press, Cambridge, Mass., 1964), Chap. 9;
10.also in Statistical Physics, Vol. III of Brandeis Summer Institute Lectures in Theoretical Physics (W. A. Benjamin, Inc., New York, 1963).
11.M. Kac and E. Helfand, J. Math. Phys. 4, 1078 (1963).
12.The existence of metastable states for the finite Ising model with nearest neighbor interaction was shown by A. J. F. Siegert, Phys. Rev. 97, 1456 (1955).
13.L. van Hove (personal communication). However, the existence of metastable states, together with certain other conditions, can be shown to be a sufficient condition for condensation. See
13.A. J. F. Siegert, Phys. Rev. 96, 243 (1954).
14.Actually, is sufficient for all the proofs, provided that for some positive integer This means that for any k and l there is a chain of positive elements connecting k and l. This excludes the case of a lattice composed of two noninteracting sublattices.
15.H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, N.J., 1951), p. 118.
16.Some examples were pointed out to us by Dr. George Moore. These known examples result in maxima small compared with the two largest maxima.
17.If we consider instead the inverted solution
18.This does not mean that the contribution of the other maxima to the integral is negligible, but only that their contribution to the free energy per particle is irrelevant. See the argument preceding Eq. (2.4) in G. F. Newell and E. Montroll, Rev. Mod. Phys. 25, 353 (1953).
19.B. Kahn and G. E. Uhlenbeck, Physica 5, 399 (1938).
19.See also B. Kahn in Studies in Statistical Mechanics, J. de Boer and G. E. Uhlenbeck, Eds. (North‐Holland Publishing Co., Amsterdam, 1965), Vol. III.
20.G. Stell et al., J. Math. Phys. 7, 1532 (1966).
21.T. H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952);
21.see also Newell and Montroll, Ref. 18.
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