The diffusion equation is compatible with special relativity

TL;DR

This paper demonstrates that the diffusion equation can be compatible with special relativity by linking it to relativistic kinetic theory, resolving apparent conflicts related to causality and stability.

Contribution

It shows that diffusion can be derived from a relativistic kinetic framework, challenging the notion of fundamental incompatibility.

Findings

Diffusion solutions correspond to solutions of the relativistic Vlasov-Fokker-Planck equation.

Instability arguments are based on non-kinetic solutions that lack microscopic counterparts.

Causality violations disappear when signals are defined through microscopic data.

Abstract

Due to its parabolic character, the diffusion equation exhibits instantaneous spatial spreading, and becomes unstable when Lorentz-boosted. According to the conventional interpretation, these features reflect a fundamental incompatibility with special relativity. In this Letter, we show that this interpretation is incorrect by demonstrating that any smooth and sufficiently localized solution of the diffusion equation is the particle density of an exact solution of the relativistic Vlasov-Fokker-Planck equation. This establishes the existence of a causal, stable, and thermodynamically consistent relativistic kinetic theory whose hydrodynamic sector is governed exactly by diffusion at all wavelengths. We further demonstrate that the standard arguments for instability arise from considering solutions that admit no counterpart in kinetic theory, and that apparent violations of causality disappear once signals are defined in terms of the underlying microscopic data.