Exact interpolation between Fick and Cattaneo diffusion in relativistic kinetic theory

TL;DR

This paper develops a family of exactly solvable relativistic kinetic models that interpolate between diffusive and hyperbolic transport laws, providing analytical insight into the transition from diffusion to wave-like propagation.

Contribution

It introduces a parameter-controlled family of models that smoothly connect Fick and Cattaneo diffusion, with fully analytical quasinormal mode spectra.

Findings

Analytical spectrum for all interpolation parameters

Continuous deformation of diffusive modes into propagating modes

Microscopic realization of mixed diffusive-telegraphic behavior

Abstract

We construct a family of exactly solvable relativistic kinetic theories in $1+1$ dimensions whose hydrodynamic sector continuously interpolates between Fick's and Cattaneo's laws of diffusion. The interpolation is controlled by a single parameter $a\in[0,1]$, which tunes the microscopic scattering dynamics from infinitely soft but infinitely frequent scatterings ($a=0$), reproducing standard diffusion, to maximally hard but finite-rate scatterings ($a=1$), yielding hyperbolic Cattaneo-type transport. For intermediate values of $a$, the dynamics combines frequent weak scatterings with rare strong randomizing events, providing a concrete microscopic realization of mixed diffusive-telegraphic behavior. Remarkably, the full quasinormal mode spectrum can be obtained analytically for all $a$. This allows us to track explicitly how purely diffusive modes continuously deform into damped propagating modes as the collision structure is varied.