The quasi-normal modes of relativistic Fokker-Planck kinetic theory

TL;DR

This paper analyzes the quasi-normal modes of relativistic Fokker-Planck kinetic theories by mapping the collision operator to quantum Hamiltonians, revealing unique spectral features and diffusion behaviors.

Contribution

It establishes a novel connection between relativistic kinetic operators and quantum Schrödinger problems, providing explicit spectral characterizations and insights into relativistic diffusion.

Findings

Collision operator maps to Dirac-delta Schrödinger problem in 1D

Hydrodynamic mode exhibits exact Fick diffusion at all wavenumbers

Relativistic effects produce a continuous ballistic band absent in Newtonian physics

Abstract

Employing the well-known unitary equivalence between Fokker-Planck operators and Schr\"odinger Hamiltonians, we compute the quasi-normal-mode spectrum of ultrarelativistic kinetic theories with momentum-space diffusion. We show that the collision operator reduces to a Dirac-delta Schr\"odinger problem in one spatial dimension, and to a Coulomb Schr\"odinger operator with hydrogenic spectrum in three dimensions. Finite spatial wavenumber appears as a perturbation of the associated quantum potential. The hydrodynamic mode is found to obey exact Fick-type diffusion at all real wavenumbers, whereas relativistic kinematics generically produces a continuous ballistic band in the non-hydrodynamic sector, a feature absent in the Newtonian regime.