A New Geometric Probability Technique for an N-dimensional Sphere and Its Applications to Physics

TL;DR

This paper introduces a novel geometric probability method to analytically compute the distribution of distances between random points in an n-dimensional sphere with arbitrary density, extending known results and enabling diverse scientific applications.

Contribution

The paper develops a new formalism for calculating the probability density function of distances in n-dimensional spheres with arbitrary densities, generalizing existing uniform-density results.

Findings

Derived a general expression for P_n(s) for arbitrary density distributions.

Validated the formalism by reproducing known uniform-density results.

Highlighted applications in physics and stochastic geometry.

Abstract

A new formalism is presented for analytically obtaining the probability density function, \( P_{n}(s) \), for the distance between two random points in an \( n \)-dimensional sphere of radius \( R \). Our formalism allows \( P_{n}(s) \) to be calculated for a sphere having an arbitrary density distribution, and reproduces the well-known results for the case of a sphere with uniform density. The results find applications in stochastic geometry, probability distribution theory, astrophysics, nuclear physics, and elementary particle physics.