A series of approximations for the statistical mechanics of order-disorder, proposed by Bethe, Takagi, Yang-Li-Hill, Kikuchi, and others, are investigated in detail in two ways. (1) A new interpretation of the method for constructing the combinatory factor is presented in order to give a better understanding of the nature of approximations. (2) The partition functions with approximate combinatory factors are expanded to compare with the rigorous expansion and the discrepancies between them are investigated in detail. One of the conclusions is that in order to obtain a higher approximation, it is necessary to use the basic figure ``closed'' with respect to the cluster of the preceding approximation. In appendices, an improved treatment of the body-centered cubic lattice (Ising model) is given, and Bethe's fundamental assumptions are derived from our scheme.
Some contributions are made to the refining of the lattice model theory of Flory of chain polymer solutions by introducing the idea used in the recent theory of regular assembly that combinatory factor constructed in terms of multiple site clusters can be properly used to take long-range correlations between a given site and its far distant neighbors into account.
In Part I, the combinatory factors of Flory and others in current use are reconstructed to exemplify the pseudo-assembly method which has been presented by Kikuchi and the present authors for an easy construction of combinatory factors of higher-order approximation for regular assembly, and to give a thorough understanding of the succeeding uses in Part II.
In Part II, a refined theory of chain polymer solutions is developed. The final expressions for the entropy of dilution and the osmotic pressure depend on the flexibility and the degree of branching of the polymer chain, and give a better agreement with observed data even at high dilution.
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