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.

Monte Carlo simulations of two-dimensional hard core lattice gases

“…This system, which can also interpreted either as 45 o tilted hard-squares of linear size λ = √ 2 or as hard disks of radius √ 2/2, has been extensively studied and here we present some results for the sake of both completeness and comparison. Many different approaches have been used to describe its properties on a square lattice: series expansions [3,5,11,12,13], cluster variational and transfer matrix methods [5,14,15,16,17,18,19,20,21,22,23], renormalization group [24,25], Monte Carlo simulations [26,27,28,29,30,31,32,33], Bethe lattice [5,34,35,36,37,38], and more recently density functional theory [39]. Moreover, this model has also been considered because of its interesting mathematical [40,41,42] and dynamical [43,44,45,46,47,48,49,50,51,…”

Driven lattice gas with nearest-neighbor exclusion: shear-like drive

“…(This model also represents the zero-neighbor limit of the Biroli-Mézard lattice glass model [7].) The equilibrium version was studied in [8,9,10,11,12] both theoretically and numerically, for various lattice types.…”

Transfer matrix simulation of hard-core lattice gases on triangular lattice with up to third-neighbour exclusion

“…При k=1 (1-NN модель) существует один непрерывный фазовый переход, свойства которого хорошо изучены [17][18][19]. При k=2 (2-NN модель), так же как и при k=1, наблюдается единственный непрерывный фазовый переход [17, [20][21][22].…”

“…В качестве численного метода мы будем использовать метод трансфер-матрицы [18,26], некоторые детали которого будут описаны ниже.…”