The partition function of a one-dimensional system of particles with all interactions active is formulated in terms of Ising ferromagnetic spin states. It is shown how the partition function for two- and three-dimensional systems can be obtained from the one-dimensional one by restricting the number of bonds per particle. After writing the partition function in matrix form, the operator of interest is diagonalized and its trace expressed in the form of a convenient infinite series. The series is shown to be absolutely convergent and its analytic properties are briefly investigated. It is then applied to one-, two-, and three-dimensional systems with nearest-neighbor interactions. The validity of the model is established by summing the one-dimensional series and comparing it with the known solution obtainable by other methods. Two- and three-dimensional series are determined by algebraic techniques identical to the one-dimensional series. These are seen to agree with the low-temperature expansions obtained by other authors. The method of this article is seen to have the advantage of simplicity and uniformity regardless of the dimension of the system. A subsequent article is devoted to the thermodynamics of the model.



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Publication Date

01 Jan 1968

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