Molecular-dynamics calculations of the velocity-autocorrelation function. Methods, hard-disk results

The velocity-autocorrelation function ${\ensuremath{\rho}}_{D}(t)$ for hard disks is computed for ten values of the reduced volume, ranging from 30 to 1.4 times the close-packed volume (${V}_{0}$) for systems of as few as 168 and as many as 5822 particles, by a Monte Carlo molecular-dynamics technique. For values of the time greater than roughly 20 mean-free times (${t}_{0}$), the results are compared with the predictions of a version of the mode-coupling theory for ${\ensuremath{\rho}}_{D}(t)$, modified to take into account the finite size of the system. Except at the highest densities, the data agree with the modified theory when one uses Knskog values for the transport coefficients in evaluating the theoretical ${\ensuremath{\rho}}_{D}(t)$, provided the comparison is limited to times beyond a value ${s}_{i}{t}_{0}$, with ${s}_{i}$ dependent on the density. The value of ${s}_{i}$ appears to increase from roughly 20 at the lowest densities to 40 at a volume of $2{V}_{0}$. At volumes of 1.6, 1.5, and $1.4{V}_{0}$, the theoretical ${\ensuremath{\rho}}_{D}(t)$'s are too large out to times as large as $320{t}_{0}$, unless we use values of the transport coefficients rather larger than the Enskog values in evaluating the theoretical ${\ensuremath{\rho}}_{D}(t)$. The velocity-autocorrelation-function results for $tl20{t}_{0}$ are presented as the difference relative to the Lorentz-Boltzmann-Enskog prediction, which has the exact slope at $t=0$.