Exact Finite Method of Lattice Statistics. I. Square and Triangular Lattice Gases of Hard Molecules

Abstract

A general, feasible approach is presented for the evaluation of the statistical thermodynamics of interacting lattice gases. Exact solutions are obtained for lattice systems of infinite length and increasing finite width, using the matrix method which treats all densities on an equivalent basis. Through the application of symmetry reduction and the use of an electronic computer to perform logical as well as arithmetical operations, widths of up to 24 sites for two‐dimensional lattices can be handled. For examples studied, rapid convergence is obtained away from transition regions and in the vicinity of phase transitions the behavior appears to be a sufficiently regular function of the width to allow meaningful extrapolation to systems of infinite width. Two problems of two‐dimensional lattice gases are solved as illustrations of the technique: the square and triangular lattice gases with infinite repulsive interactions preventing the simultaneous occupancy of adjacent lattice sites (excluded‐volume effect). Both systems exhibit phase transitions which are most likely second order at densities of 74.2% (square) and 83.7% (triangular) of the close‐packed density. For both lattices the compressibility is infinite at the transition point, becoming infinite linearly with the logarithm of the width of the lattice for the square lattice and perhaps slightly more strongly for the triangular lattice.