A three parameter isotropic distribution of atoms and the hard-core square lattice gas

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Abstract
A five parameter symmetric distribution of atom patterns on the unit cell of the square lattice is statistically continued to construct a five parameter distribution of atom patterns on the 2×∞ lattice, i.e., on two adjacent rows of sites. The probability of a pattern on a single row, and the probability of a pattern on one row given the pattern on the other, define a row by row (Markovian) probability generating process on the double infinite square lattice. The probability of any finite pattern on this lattice is determined by a finite number of steps in this process. The five parameter distribution of atom pattern defined by the row by row process contains a three parameter family of isotropic distributions. A subfamily of these isotropic distributions is the exact solution of the Ising model for the hard core lattice gas, with exclusion of nearest neighbor sites and second neighbor interaction for cases where the interaction parameter is suitably related to the coverage. The free energy and the partition function for this gas are derived mathematically and are presented explicitly in terms of the concentration.