Lorentz-boosted diffusion: initial value formulation and exact solutions

TL;DR

This paper reformulates the Lorentz-boosted diffusion problem using kinetic theory, resulting in a well-posed initial-value problem with an explicit Green function solution.

Contribution

It introduces a kinetic-theory-based formulation for Lorentz-boosted diffusion, providing exact solutions and addressing the ill-posedness of the classical approach.

Findings

The dynamics are well-posed in the kinetic framework.

The Green function is derived in closed form.

Evolution can be expressed as a superposition of sampled initial data.

Abstract

It is well known that the diffusion equation, when treated as a stand-alone partial differential equation, exhibits exponential instabilities in boosted frames, which render the corresponding initial-value problem ill-posed. Recently, however, it was shown that Fick-type diffusion arises as the exact hydrodynamic sector of relativistic Fokker-Planck kinetic theory. In this work, we exploit this kinetic embedding to formulate a modified initial-value problem for one-dimensional Lorentz-boosted diffusion. We show that the resulting dynamics are well posed both forward and backward in time, provided the boosted density profiles admit a kinetic-theory realization. Such profiles form a space of band-limited functions, within which the evolution can be expressed as a discrete superposition of spatially sampled initial data, weighted by a Shannon-Whittaker-type Green function defined on the full Minkowski plane. The Green function is obtained in closed analytic form.