Random distance distribution for spherical objects: General theory and applications to physics

A formalism is presented for analytically obtaining the probability density function, P_n(s), for the random distance s between two random points in an n-dimensional spherical object of radius R. Our formalism allows P_n(s) to be calculated for a spherical n-ball having an arbitrary volume density, and reproduces the well-known results for the case of uniform density. The results find applications in geometric probability, computational science, molecular biological systems, statistical physics, astrophysics, condensed matter physics, nuclear physics and elementary particle physics. As one application of these results, we propose a new statistical method derived from our formalism to study random number generators used in Monte Carlo simulations.