Phase transitions in hard rods on a lattice

Abstract

Systems of particles with hard core interactions are used to study phase transitions in equilibrium statistical mechanics. Lattice models are analytically more tractable. The problem of hard rods of length k on lattices is well studied. The system undergoes two transitions. First from a disordered phase to an orientationally aligned nematic phase, then from the nematic phase to a disordered phase again, as the density is increased. The first transition is well studied. However, the second transition remains a mystery - there are inconclusive results, or sometimes even contradictory results. We focused on numerically studying the problem of phase transitions in hard rigid rods of length k on a two dimensional square lattice. First using simple arguments we predicted the behaviour of quantities like the chemical potential, the density and the entropy near the transition as a function of k for a d−dimensional hypercubic lattice. We tested these predictions by studying the k×∞ lattice. We introduced a toy model for the second transition of hard rods on a 2-d square lattice, and argued that the asymptotic behaviour of the chemical potential, and density of holes is the same as that found for the k × ∞ lattice for large k. We gave evidence that the transition from the nematic phase to the high-density disordered phase is a first order transition.