A rigorous solution to the superluminal issue in the diffusion equation

TL;DR

This paper presents a rigorous solution to the superluminal propagation issue in the diffusion equation by modeling particles as undergoing a random flight process, improving accuracy in scenarios like cosmic-ray propagation.

Contribution

The work introduces a new rigorous solution based on random flight processes, addressing superluminal issues in diffusion equations more accurately than previous phenomenological models.

Findings

The solution aligns well with simulations of the random flight process.

It significantly deviates from the Jüttner propagator-based solutions.

Applicable to cosmic-ray and burst-like particle injection scenarios.

Abstract

Superluminal propagation is an intrinsic problem in the diffusion equation and has not been effectively addressed for a long time. In this work, a rigorous solution to this issue is obtained under the assumption that particles undergo a random flight process, where they move isotropically at a constant speed while experiencing random scatterings. We validate this solution by comparing it with comprehensive simulations of the random flight process and find that it significantly deviates from the solution derived from the J\"{u}ttner propagator. This solution is broadly applicable to various diffusion phenomena, such as cosmic-ray propagation. We emphasize that our rigorous solution is particularly crucial in scenarios involving burst-like particle injection, where previous phenomenological approaches to the superluminal diffusion problem may not yield accurate results.