Exact interpolation between Fick and Cattaneo diffusion in relativistic kinetic theory

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.

💡 Research Summary

The authors investigate diffusion in a relativistic kinetic setting in one spatial dimension, focusing on how the macroscopic laws of Fick (parabolic diffusion) and Cattaneo (hyperbolic, causal diffusion) can be continuously connected by a single microscopic parameter. They start from the Boltzmann equation for mass‑less particles moving along a line and interacting linearly with an external thermal bath. Two well‑known collision operators are reviewed: (i) the Fokker‑Planck operator ν∂ₚ(∂ₚf+vf), which models infinitely soft but infinitely frequent momentum exchanges and reproduces exactly Fick’s law with diffusion constant D=ν⁻¹; (ii) the Anderson‑Witting relaxation‑time operator Γ(f_eq−f), which assumes a finite mean free time Γ⁻¹ and completely randomizing collisions, leading to the exact Cattaneo equation ∂ₜJ+J/Γ=−∂ₓn/Γ.

The central contribution is the construction of a one‑parameter family of collision terms that linearly combines these two limits. By introducing a weight a∈