Lorentz-boosted diffusion: initial value formulation and exact solutions

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.

13 pages, 5 figures, published in PRD (see https://journals.aps.org/prd/abstract/10.1103/k646-mgz5)

References in corpus (7)

• Relativistic Brownian Motion
• Rigorous bounds on transport from causality
• Bounds on transport from hydrodynamic stability
• Unified Extended Irreversible Thermodynamics and the stability of relativistic theories for dissipation
• The stochastic relativistic advection diffusion equation from the Metropolis algorithm
• Heat propagation in rotating relativistic bodies
• Noncovariant parabolic theories of relativistic diffusion