Monte Carlo Evaluation of the Partition Function for a Hard-Disk System

A Monte Carlo technique has been developed for evaluating the partition function of a hardisk system which should be generalizable to other many-body integrals. Evaluating the partition function directly, and hence the free energy per particle, instead of simulating the system allows much smaller systems to be used as realistic samples of an infinite system since a canonical ensemble of such systems is actually sampled. This eliminates the metastable-state problem that tends to lock simulated systems in one phase or another. The free energy has been evaluated for a 16-particle system with periodic boundary conditions for $0\le \rho \le 0.9$ with $\rho =\frac{\mathrm{density}}{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}-\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{e}\mathrm{d}}$ density. The free energy is below that of a free-volume calculation which assumes crystallization for $\rho \le 0.79$ and equal to it for higher densities. At $\rho =0.8\mathrm{}$ the free-volume equation can be extrapolated to $N=\infty$ and gives $\frac{A\left(0.8\right)\mathrm{}}{\mathrm{NkT}}=4.01\mathrm{}\pm 0.04$, which compares well with the 3.96 ± 0.01 which Ree and Hoover obtained by integrating the pressure curve. The pressure in the phase-change region can also be extrapolated to give $\frac{P{V}_0}{\mathrm{NkT}}=7.6\mathrm{}$, which compares well with the Alder-Wainwright result of 7.72. The phase-change region, the only part which cannot be easily extrapolated, is $1.25\le \frac{V}{V_0}\le 1.28$ for 16 particles, which is about the correct density but only about 60% of the width of theirs ($1.266\le \frac{V}{V_0}\le 1.312$) for 870 particles because of a combination of $\frac{1}{N}$ effects and smoothing in fitting to the $\frac{A}{\mathrm{NkT}}$ data.