Volume-Dependent Cluster Sums for Lattice Gases. I : Calculation of the Temperley Coefficients

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A systematic method is proposed for the calculation of the volume-dependent cluster sums for three-dimensional (simple cubic), two-dimensional (square) and one-dimensional lattice gases with nearest neighbour interactions, the interaction energy being ε. The method is based on Temperley's simplification of the cluster series. Cluster diagrams are formed by appending some particles to Temperley diagrams, which are sets of networks connected together by short-lines (i.e., links between particles on the same site, corresponding to the factor-1); here networks are sets of particles connected together by ling-lines (i.e., links between particles on neighbouring sites, corresponding to the factor ƒ≡exp[-ε/kT]-1). Patterns are introduced to express the essential properties of Temperley diagrams. Prototypes are introduced to express the essential shape and magnitude of patterns. Formal diagrams (of various shapes) derived from a prototype are defined, and the numbers of their configurations in the lattice, divided by the volume (i.e., the number of sites), are called their values. The Temperley coefficients c_m, which are related to the volume-dependent cluster sums, are expressed in the form Σ_μΩ_<mμ>ƒ^μ. The , Ω_<mμ> are obtained from the characteristic factors of patterns (which depend on the structural properties of patterns) and the values of prototypes (which are derived from the values of formal diagrams). Hyper complex numbers are used to represent the values for the three-, two- and one-dimensional cases simultaneously. It is shown that the coefficient of the highest power of ƒ in every cluster sum is positive. By the above method the Temperley coefficients up to c_6 are calculated.
理論物理学刊行会の論文
1975-02-25

Volume-Dependent Cluster Sums for Lattice Gases. I : Calculation of the Temperley Coefficients

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