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Correlation Functions and the Coexistence of Phases
1.By the thermodynamic limit in the canonical and grand canonical ensembles for a system in a domain Ω we understand the limit in which the volume approaches infinity at constant temperature while the specific volume v or activity z, respectively, approaches a finite limit. (See Refs. 2–4.)
2.D. Ruelle, Helv. Phys. Acta 36, 183, 789 (1963).
3.M. E. Fisher, Arch. Ratl. Mech. Anal. 17, 377 (1964).
4.A similar analysis of the microcanonical ensemble for quantum mechanical systems has been given recently by R. B. Griffiths (to be published).
4.Griffiths has also discussed spin systems [J. Math. Phys. 5, 1215 (1964)].
5.J. E. Mayer and E. W. Montroll, J. Chem. Phys. 9, 2 (1941).
6.G. E. Uhlenbeck, P. C. Hemmer and M. Kac, J. Math. Phys. 4, 229 (1963).
7.D. Ruelle, Ann. Phys. (N.Y.) 25, 109 (1963);
7.D. Ruelle, Rev. Mod. Phys. 36, 580 (1964);
7.D. Ruelle, J. Math. Phys. 6, 201 (1964);
7.O. Penrose, J. Math. Phys. 4, 1312, 1488 (1963).
8.J. Ginibre, J. Math. Phys. 6, 238, 252 (1965).
9.If the pair potential does not have a hard core but satisfies instead the stability condition as the total potential will satisfy A and B for positive three‐ and more‐body potentials but may not do so if these are negative unless diverges more strongly. Indeed if is finite at one can’prove that A cannot be satisfied if some of the many‐body potentials are negative in certain regions ( will diverge to as ). In such a case the corresponding l‐particle correlation functions cannot be “well denned,” in the sense explained below. I am indebted to D. Ruelle for observations on this point.
10.See, for example, G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (Cambridge University Press, New York, 1952), 2nd ed.
11.See, for example, G. E. Uhlenbeck and G. W. Ford, Studies in Statistical Mechanics I, edited by J. de Boer and G. E. Uhlenbeck (North‐Holland Publishing Company, Amsterdam, 1962), Part B.
12.See, for example, M. E. Fisher and J. Stephenson, Phys. Rev. 132, 1411 (1963), where the effects of the boundary on singlet and pair correlation functions in a lattice filled with hard “dimers” are derived.
13.See, for example, F. Riesz and B. Sz.‐Nagy, Functional Analysis (Fredrick Ungar Publishing Company, New York, 1955), p. 61 et seq.
14.See, for example, N. N. Bogoliubov, Studies in Statistical Mechanics I, edited by J. de Boer and G. E. Uhlenbeck (North‐Holland Publishing Company, 1962), Part A;
14.and J. L. Lebowitz and J. K. Percus, J. Math. Phys. 4, 116 (1963).
15.If one starts from the usual quantum mechanical definitions in terms of the diagonal elements of the reduced density matrices it is only necessary to recall that even when A and B do not commute.
16.R. Peierls, Phys. Rev. 54, 918 (1938).
16.See also Ref. 3 and K. Huang, Statistical Mechanics (John Wiley & Sons, Inc., New York, 1963), p. 220.
17.This may fail for large enough z in a Bose system with insufficiently strong repulsions between the particles, but such values of z are essentially without physical significance (see the discussion in Ref. 3).
18.We use the adjective “hypercritical” in contradistinction to “critical” since in normal usage a critical point is one at which certain double derivatives of the thermodynamic potentials become continuously infinite (e.g., the compressibility at the gas‐liquid critical point or the specific heat at a lambda transition) but their integrals remain continuous. In this sense the usual first‐order transition points (e.g., an ideal ferromagnet below its Curie point in zero magnet field) are examples of hypercritical points.
19.Notice that for a system with pair interactions only, the energy is equal to a correlation functional evaluated with a proportional to the pair potential A change in η is then precisely equivalent to a change in β.
20.Note, however, that Ruelle (Ref. 2) has proved that the pressure is continuous in v for a classical system of particles interacting through pair potentials which are bounded above (in addition to satisfying conditions A and B). It seems probable that the pressure should remain continuous under much weaker restrictions.
21.M. E. Fisher, The Theory of Condensation, Lecture on the Centennial Conference on Phase Transformation, University of Kentucky, March, 1965, to be published in the Proceedings.
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