Rigid Disks at High Density. II

Abstract

As shown in a previous publication, the Helmholtz free energy per particle (divided by kBT) for a rigid disk system has the following asymptotic form in the limit of the reduced density θ=A0/A→1:F/NkBT∼2ln(λ/a)−2ln(θ−1−1)+C+D(θ−1−1)+···, in which C and D are appropriate numerical constants, λ is the mean thermal de Broglie wavelength, a is the disk diameter, and A0 is the close-packed value of the area, A, of a system of N particles. Moreover, a technique for estimating the value of C was developed. This technique was based upon a product representation for the partition function which considered a sequence of correlated motion of larger and larger sets of contiguous particles. An approximate value of C was previously obtained from all contributions for four or fewer correlated disks. This article extends these calculations to fifth order (contributions of all sets of five particles) with the summary result: C=0.14384–0.013857+0.014322–0.0073211–0.004222+···=0.14384–0.011078+···, in which the first number is the value of C obtained from the one-particle free-area model, and the remaining terms are the corrections from two-, three-, etc., particle correlations. These results are in slight disagreement with the estimate of C obtained by Hoover and Alder from molecular dynamics calculations, namely C≅0.14384–0.06±0.02. The sequential approximation scheme for estimating C is applied to the tunnel model (where the exact value is known) and shows a rapid convergence. An analysis of the most compact clusters or sets also supports the fifth-order estimate of C.