The $p-$sphere and the geometric substratum of power law probability distributions

TL;DR

This paper explores the mathematical connection between power law distributions and uniform distributions on p-spheres, revealing how the Boltzmann-Gibbs distribution relates to power laws and linking parameters to non-extensive statistics.

Contribution

It provides a novel geometric framework connecting p-spheres to power law distributions and derives relationships between parameters and the non-extensivity parameter q.

Findings

Power law distributions can be derived from uniform laws on p-spheres.

Parameters p and n are linked to the non-extensivity parameter q.

The Boltzmann-Gibbs distribution is shown to pass through power law distributions.

Abstract

Links between power law probability distributions and marginal distributions of uniform laws on $p$-spheres in $\mathbb{R}^{n}$ show that a mathematical derivation of the Boltzmann-Gibbs distribution necessarily passes through power law ones. Results are also given that link parameters $p$ and $n$ to the value of the non-extensivity parameter $q$ that characterizes these power laws in the context of non-extensive statistics.