Nearest-neighbor statistics for packings of hard spheres and disks

The probability of finding a nearest neighbor at some radial distance from a given particle in a system of interacting particles is of fundamental importance in a host of fields in the physical as well as biological sciences. A procedure is developed to obtain analytical expressions for nearest-neighbor probability functions for random isotropic packings of hard D-dimensional spheres that are accurate for all densities, i.e., up to the random close-packing fraction. Using these results, the mean nearest-neighbor distance λ as a function of the packing fraction is computed for such many-body systems and compared to rigorous bounds on λ derived here. Our theoretical results are found to be in excellent agreement with available computer-simulation data.