Mean Nearest-Neighbor Distance in Random Packings of HardD-Dimensional Spheres

We derive the first nontrivial rigorous bounds on the mean distance between nearest neighbors $\lambda$ in ergodic, isotropic packings of hard $D$-dimensional spheres that depend on the packing fraction and nearest-neighbor distribution function. Several interesting implications of these bounds for equilibrium as well as nonequilibrium ensembles are explored. For an equilibrium ensemble, we find accurate analytical approximations for $\lambda$ for $D=2\mathrm{}$ and 3 that apply up to random close packing. Our theoretical results are in excellent agreement with available computer-simulation data.