Entropy of hard square lattice gas with k distinct species of particles: coloring problems and vertex models

Abstract

Coloring the faces of a two-dimensional square lattice with k distinct colors such that no two adjacent faces have the same color is considered by establishing connection between the kcoloring problem and a generalized vertex model. Associating the colors with k distinct species of particles with an infinite repulsive force between nearest neighbors of the same type and zero chemical potential µ associated with each species, the number of ways [W (k)] N for large N is related to the entropy of the hard square lattice gas at close packing of the lattice, where N is the number of lattice sites. We discuss the evaluation of W (k) using transfer matrix method with non-periodic boundary conditions imposed along at least one direction, and, show the characteristic Toeplitz block structure of the transfer matrix. Using this result, we present some analytical calculations for non-periodic models that remain finite in one dimension. The case k = 3 is found to approach the exact result obtained by Lieb for the residual entropy of ice with periodic boundary conditions. Finally, we show, by explicit calculation of the contribution of subgraphs and the series expansion of W (k), that the generalized Pauling type estimate (which is based on mean-field approximation) dominates at large values of k. We thus also provide an alternative series expansion for the chromatic polynomial of a regular square graph.

Highly confined mixtures of parallel hard squares: A density-functional-theory study

“…, N x , N x ∼ 100} allows to obtain a sufficiently accurate Gibbs free-energy per particle, g(x) [from Eq. (19)] to search for possible phase transitions and calculate the phase diagrams. This section is divided into two parts.…”

“…Classical work on the numerical calculation of virial coefficients [15,16] demonstrate the importance of hard squares as a simple model to elucidate important problems in statistical mechanics. The lattice-gas version of the model has attracted some attention [17][18][19] The parallel hard square model has also been investigated [20][21][22]. Simulations have shown that freely-oriented hard squares present nematic tetratic and crystal square phases [6].…”