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Dual trees and resummation theorems
1.Equlvalently, we could drop the labels on the squares and circles—thereby reducing the number of distinct trees—and calculate by dividing not by but by, the symmetry number of the tree. For example, the twelve trees in Fig. 1 would reduce to two, each having and their total contribution to τ would be the same as in our definition. That method is more convenient for evaluating τ, but the one we choose Is equivalent and seems better adapted to proving theorems. Our definitions allow trees consisting of a single square or circle and having the value or However, the theorem loses no generality in practice if it is restricted to which means that these two degenerate trees are omitted.
2.T. D. Lee and C. N. Yang, Phys. Rev. 117, 12 (1960), Appendix C.
3.It is known that but this fact is not used either here or in Ref. 2.
4.G. Jana‐Lasinio, Nuovo Cimento 34, 1790 (1964).
5.H. Ursell, Proc. Camb. Phil. Soc. 23, 685 (1927);
5.J. E. Mayer, J. Chem. Phys. 5, 67 (1937).
6.J. E. Mayer and P. F. Ackermann, J. Chem. Phys. 5, 74 (1937), especially Eqs. (9)ff, (17), and Note in Proof;
6.J. E. Mayer and S. F. Harrison, J. Chem. Phys. 6, 87 (1938), especially Appendix.
7.G. E. Uhlenbeck and G. W. Ford, Studies in Statistical Mechanics edited by DeBoer and Uhlenbeck (Interscience, New York, 1962), Vol. 1, pp. 123ff, Secs. III. 2, 3.
8.T. D. Lee and C. N. Yang, Phys. Rev. 117, 22 (1960).
9.J. M. Luttinger and J. Ward, Phys. Rev. 118, 1417 (1960).
10.C. Bloch, Studies in Statistical Mechanics edited by DeBoer and Uhlenbeck (Interscience, New York, 1964), Vol. 3, pp. 7ff, and references cited therein.
11.See Ref. 10, Sec. 4.21. The connection with the Mayer cluster expansion is indicated on p. 133.
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