Hard sphere gas state equation

Abstract

The dynamic evolution of a gas of N hard spheres determines the equation of state once the equilibrium is reached, after a short transient. The system is investigated in the thermodynamic limit. The algorithms based on a tricky management of the collisions list allow to simulate up to N = 10 6 spheres, rendering the statistical error sufficiently small in simulations involving 10 8 collisions. The effect of boundaries is discussed, and, to avoid any dependence, periodic boundary conditions are chosen in a box, whose edge is much larger than the spheres radius. The initial state is the symmetric close packing and, by reducing the hard spheres radius, we follow the evolution of the mean free path as a function of the density y (where y is the ratio between the volume of the spheres and the total volume). We observe the solid-fluid first-order phase transition and follow the fluid branch until the hard spheres gas is very dilute. The phase transition is well resolved due to the improved statistics and to the choice, as order parameter, of the mean free path rather than Z = PV / NkT , which has a singularity at zero mean free path. The equations of state in the fluid and solid branches are compared with the Taylor series for the mean free path, obtained from the virial expansions of Z. The second-order truncations P 2 ( y ) , appear to provide the best fit to the mean free path. This suggests an approximation to Z as Z = 1 + B 2 y / P 2 ( y ) , where B 2 is the second virial coefficient.