Random Distance Distribution for Spherical Objects: General Theory and Applications to n-Dimensional Physics

TL;DR

This paper develops a general analytical framework for calculating the probability density function of distances between two random points in n-dimensional spherical objects, applicable to various scientific fields and including a new method for evaluating n-dimensional random number generators.

Contribution

It introduces a formalism for deriving the distance distribution in n-dimensional spheres with arbitrary density, extending known uniform cases and enabling diverse applications.

Findings

Derived a general formula for P_n(s) in n-dimensional spheres.

Validated the formalism for uniform density cases.

Proposed a new statistical method for assessing n-dimensional random number generators.

Abstract

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 stochastic geometry, 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 obtained from our formalism to study random number generators in (n)-dimensions used in Monte Carlo simulations.