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Decay of Correlations in Linear Systems
1.L. S. Ornstein and F. Zernike, Physik. Z. 19, 134 (1918);
1.L. S. Ornstein and F. Zernike, 27, 761 (1926); , Phys. Z.
1.see also M. E. Fisher, J. Math. Phys. 5, 944 (1964).
2.B. Widom, J. Chem. Phys. 41, 74 (1964).
3.J. E. Enderby, T. Gaskell, and N. H. March, Proc. Phys. Soc. (London) 85, 217 (1965).
4.J. Groeneveld, (a) “Proceedings of the I.U.P.A.P. Conference on Statistical Mechanics and Thermodynamics, Copenhagen, July 1966,” Statistical Mechanics—Foundations and Applications, T. A. Bak, Ed. (W. A. Benjamin, Inc., New York, 1967);
4.(b) (private communication).
5.Though widely known as an exact result, it seems never to have been so stated in any published work. F. Zernike and J. A. Prins, Z. Physik 41, 184 (1927), who first derived g(r) for the one‐dimensional hard‐sphere fluid, showed its oscillatory character by plotting the function over a finite range of r.
5.In calculations based on the superposition and related approximations, oscillatory decay of G(r) in the three‐dimensional hard‐sphere fluid was found by J. G. Kirkwood, E. K. Maun, and B. J. Alder, J. Chem. Phys. 18, 1040 (1950),
5.and by G. A. Martynov, Zh. Eksp. Teor. Fiz. 45, 656 (1963)
5.[G. A. Martynov, Sov. Phys.‐JETP 18, 450 (1964)]. The latter author also noted that within the framework of his approximation scheme G(r) decays monotonically when attractive forces are dominant.
6.T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952).
7.T. L. Hill, Statistical Mechanics (McGraw‐Hill Book Co., New York, 1956), Chap. 7.
8.A Bethe lattice is an infinite Cayley tree (with no closed circuits); see, e.g., C. Domb, Advan. Phys. 9, 245 (1960).
9.B. Widom (unpublished).
9.It should be noted that this definition of “distance” r on a Bethe lattice is not always the most appropriate one in comparisons with real lattices; see M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967), Sec. 4.2.
10.B. Widom, Science 157, 375 (1967).
11.M. E. Fisher, Physica 28, 172 (1962).
12.M. E. Fisher, unpublished work reported at the Second Eastern Theoretical Physics Conference, Chapel Hill, N.C., 25 October 1963 and at Institut des Hautes Etudes Scientifiques, Bures‐sur‐Yvette, Paris, 26 April 1965.
13.F. Gürsey, Proc. Camb. Phil. Soc. 46, 182 (1950).
14.(a) Z. W. Salsburg, R. W. Zwanzig, and J. G. Kirkwood, J. Chem. Phys. 21, 1098 (1953);
14.(b) R. Kikuchi, J. Chem. Phys. 23, 2327 (1955); , J. Chem. Phys.
14.(c) J. L. Lebowitz, J. K. Percus, and I. J. Zucker, Bull. Am. Phys. Soc. 7, 415 (1962);
14.(d) S. Katsura and Y. Tago, J. Chem. Phys. 48, 4246 (1968).
15.Appendix A of Ref. 14(a) contains the most appropriate expressions for analyzing the correlation functions. [See especially Eqs. (A6)–(A8).]
16.This convenience is easily discarded but it has no effect on the results for a thermodynamic (infinitely large) system.
17.M. Kac, G. E. Uhlenbeck, and P. Hemmer, J. Math. Phys. 4, 216, 299 (1963);
17.J. L. Lebowitz and O. Penrose, J. Math. Phys. 4, 248 (1963)., J. Math. Phys.
18.Compare with M. E. Fisher, Phys. Rev. 136, A1626 (1964).
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