Nearest-neighbor distribution functions and mean separation for impenetrable particles in one to three dimensions.

Abstract

Expressions for the hard-sphere nearest-neighbor probability distribution, ${\mathit{P}}_{\mathrm{NN}}$, and the associated mean nearest-neighbor distance, 〈${\mathit{r}}_{\mathrm{NN}}$〉, in D=1 to 3 dimensions, which were recently derived by Torquato, Lu, and Rubinstein [Phys. Rev. A 41, 2059 (1990)] are compared with earlier results of Reiss and Casberg [J. Chem. Phys. 61, 1107 (1974)] and Macdonald [Mol. Phys. 44, 1043 (1981)]. Full agreement is found for the D=1 results, where an exact solution can be found. But no such solution is possible for D=2 and 3. The equation of state, ${\mathit{P}}_{\mathrm{NN}}$, and 〈${\mathit{r}}_{\mathrm{NN}}$〉 can all be calculated from knowledge of the central function ${\mathit{G}}_{\mathit{D}}$, a conditional probability depending on both density and distance, r, from a particle center. The consequences of several different approximations for ${\mathit{G}}_{\mathit{D}}$ are explored for hard disks and spheres. In spite of the much more approximate ${\mathit{G}}_{\mathit{D}}$ functions used by Macdonald compared to those of Torquato, differences in the resulting ${\mathit{P}}_{\mathrm{NN}}$ responses are relatively small even at high densities, and the differences in 〈${\mathit{r}}_{\mathrm{NN}}$〉 predictions, the quantities of primary experimental value, are entirely negligible. Although the original D=2 Torquato expression for ${\mathit{P}}_{\mathrm{NN}}$ was not properly normalized, a sign correction restores normalization and leads to close agreement with the earlier work. For D=3, numerical results show that simple approximation of ${\mathit{G}}_{3}$ by the contact value of the ordinary radial distribution function, a quantity independent of r, yields results for ${\mathit{P}}_{\mathrm{NN}}$ very close to those of Torquato, and the corresponding 〈${\mathit{r}}_{\mathrm{NN}}$〉 results are completely indistinguishable.