Abstract
An approximation method for treating the statistical mechanics of lattice models is derived, and some of the lower-order approximations are applied to the hard-sphere lattice gas. The results are consistent with previous work on this model. The approximation is of interest because it uses information obtained from small systems—the kind of information obtainable from Monte Carlo calculations. Also, the approximation is of potential importance because it can be generalized to continuum models. The role of the boundary in establishing the long-range order in the hard-sphere lattice gas is investigated. From the approximation it appears evident that the boundary can cause preferential occupation of one sublattice over the other. Furthermore, it is shown to be rigorously true that this behavior must correspond to the largest and smallest eigenvalues (in the standard eigenvalue problem) being equal in magnitude but opposite in sign.
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