Pressure and Entropy for Hard Particles at High Density

Abstract

A hard-particle model is introduced. For this model it is possible to calculate the thermodynamic limit of both the pressure and the entropy at high (solid-phase) densities. The model embodies both the simplicity associated with a nearest-neighbor lattice-gas interaction, and the realism of a continuous configuration space. The volume of the system is divided into V cells, with particles in nearest-neighbor cells interacting as hard parallel squares (cubes, in the three-dimensional case). When written in terms of the cell occupation numbers the configurational integral for this system bears a superficial resemblance to a lattice-gas partition function. It is possible to calculate the maximum term in the lattice-gas partition function explicitly. This makes it possible to establish tight bounds on the continuum configurational integral; these bounds show that the free-volume form of the pressure and the entropy is exact at high density. At close packing, where the upper and lower bounds coincide, the absolute entropy is determined. It is shown that the communal entropy vanishes at close packing, and cannot be fully excited for densities greater than half the close-packed density.