Statistical Mechanics of Single-Occupancy Systems of Spheres, Disks, and Rods

Abstract

An independent oscillator approximation proposed recently by Fixman has been numerically examined by using single-occupancy systems of hard spheres, disks, and rods over the entire density range. Numerical data on pressure, entropy and the singlet distribution function are compared with existing exact data to elucidate the theory's dependence on dimensionality. A considerable simplification which results after truncating the effective potential appearing in Fixman's theory by its first two nonvanishing terms in the Taylor expansion allows proof of the existence of high-density expansion in a parameter that characterizes available free volume for pressure and entropy, and the second moment of the singlet distribution function. Analytic expressions are given for the first one or two coefficients in the expansion for these quantities. However, complete calculations without the truncation procedure show significantly different high-density behaviors for pressure and entropy. In addition to this pressure and entropy, calculations for spheres from the second approach show the existence of two distinct branches: a “fluidlike” branch, which would be unstable without the cell wall, extending up to 75% of the close-packed density (ρ0), and a “solidlike” branch, which is stable even without the cell wall, starting from 0.73ρ0 and extending to the close-packed density. For the single-occupancy system of hard disks, a similar but continuous van der Waals-like transition is observed in the vicinity of the melting density of normal disks without the cell restriction. Finally, a Monte Carlo method is employed to obtain data for the singlet distribution and its mean-square width for single-occupancy systems of 72, 896, and 1400 hard disks. The resulting data are compared with those derived under the independent oscillator approximation.