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Variational Method for Classical Statistical Mechanics
1.R. E. Peierls, Phys. Rev. 54, 918 (1938).
2.N. N. Bogolubov, unpublished work cited in Footnote 4 of V. V. Tolmachev, Dokl. Akad. Nauk SSSR 134, 1324 (1960)
2.[English transl.: V. V. Tolmachev, Soviet Phys.‐Doklady 5, 984 (1961).
3.B. Mühlschlegel, Sitzber. Bayer. Akad. Wiss. München, Math.‐Naturw. Kl. 10, 123 (1960), Sec. 3.
4.M. D. Girardeau, J. Math. Phys. 3, 131 (1962).
5.H. Koppe, in Werner Heisenberg und die Physik unserer Zeit (Verlag Friedr. Vieweg und Sohn, Braunschweig Germany, 1961).
6.J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
7.The function is “convex downward” in the sense of having nonnegative second derivative; it follows (e.g., from Taylor’s theorem with remainder) that it is convex in the sense (1).
8.Some recent work on the Ising model by B. Mühlschlegel and H. Zittartz [Z. Physik (to be published), Sec. V],
8.may be regarded as a special case of such a method. Also, Kirkwood’s exact formulation of the cell theory [J. G. Kirkwood, J. Chem. Phys. 18, 380 (1950)] is closely related, although it is not formulated in terms of the Bogolubov variational principle.
9.For example, one might treat mixtures by taking to be the Hamiltonian of a pure component with masses and interactions intermediate between those of the different components of the mixture.
10.In the application to mixtures suggested in Footnote 9, the masses and interactions occurring in might both depend on the same variational parameters (e.g., the number of particles in the fictitious pure component.)
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