Melting Transition and Communal Entropy for Hard Spheres

In order to confirm the existence of a first-order melting transition for a classical many-body system of hard spheres and to discover the densities of the coexisting phases, we have made a Monte Carlo determination of the pressure and absolute entropy of the hard-sphere solid. We use these solid-phase thermodynamic properties, coupled with known fluid-phase data, to show that the hard-sphere solid, at a density of 0.74 relative to close packing, and the hard-sphere fluid, at a density of 0.67 relative to close packing, satisfy the thermodynamic equilibrium conditions of equal pressure and chemical potential at constant temperature. To get the solid-phase entropy, we integrated the Monte Carlo pressure–volume equation of state for a “single-occupancy” system in which the center of each hard sphere was constrained to occupy its own private cell. Such a system is no different from the ordinary solid at high density, but at low density its entropy and pressure are both lower. The difference in entropy between an unconstrained system of particles and a constrained one, with one particle per cell, is the so-called “communal entropy,” the determination of which has been a fundamental problem in the theory of liquids. Our Monte Carlo measurements show that communal entropy is nearly a linear function of density.