Relativistic Heat Equation via L\'{e}vy stable distributions: Exact Solutions

TL;DR

This paper extends the classical heat equation to a relativistic context using Lévy stable distributions, providing exact solutions and detailed statistical analysis of the resulting relativistic diffusion process.

Contribution

It introduces a relativistic heat equation with exact solutions derived via Lévy stable distributions, highlighting the role of Bessel polynomials and comparing with non-relativistic diffusion.

Findings

Relativistic diffusion is slower than non-relativistic diffusion.

Exact solutions are obtained for various initial conditions.

The statistical properties of the relativistic process are fully characterized.

Abstract

We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one-sided L\'{e}vy stable distributions. We note a natural appearance of Bessel polynomials which allow one the obtention of closed form solutions for a number of initial conditions. The resulting relativistic diffusion is slower than the non-relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and is compared with the non-relativistic case.