Exact interpolation between Fick and Cattaneo diffusion in relativistic kinetic theory

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.

Figures (4)

• Figure 1: Dispersion relations $\omega=-i\mathfrak{D}k^2$ and $\omega=-i\mathfrak{D}(k^2-\omega^2)$ for Fick (blue) and Cattaneo (red) diffusion, respectively, as obtained from the corresponding kinetic theories. Left panel: Frequency plotted as a function of imaginary $k$. The Fick branch terminates at $|\mathfrak{Im}k|=(2\mathfrak{D})^{-1}$, beyond which the information current associated with the mode \ref{['ansatz']} diverges, rendering the mode unphysical (the particle density itself diverges at $|\mathfrak{Im}k|=\mathfrak{D}^{-1}$). By contrast, the Cattaneo branches extend to arbitrarily large $|k|$. As expected, none of the curves enter the region $\mathfrak{Im}\omega<|\mathfrak{Im}k|$ (shaded), where the modes would become unstable under Lorentz boosts GavassinoBounds2023myj. Right panel: Frequency plotted as a function of real $k$. Both Fick and Cattaneo branches extend over the full real axis. In the Cattaneo case, $i\mathfrak{D}\omega$ develops an imaginary part (red dashed) for $|k|>(2\mathfrak{D})^{-1}$, indicating the onset of propagating behavior Pu2010BAGGIOLI20201GavassinoGENERIC:2022isg. In both cases, one finds $\mathfrak{Im}\omega\leq 0$, consistent with linear stability.
• Figure 2: Hydrodynamic dispersion relation $\omega(k)$ of the Fokker-Planck-Anderson-Witting kinetic theory \ref{['interpuls']} for imaginary wave number and various values of the parameter $a$ (see equation \ref{['parameterization']}). Left panel: As $a$ increases (orange: $a=1/5$, brown: $a=2/3$, red: $a=0.98$), the dispersion relation interpolates continuously between Fick’s law (upper dashed curve), with $\mathfrak{D}=\tau/4$, and Cattaneo’s law (lower dashed curve), with $\mathfrak{D}=\tau$. Throughout this deformation, the continuous spectrum remains fixed and occupies the region $i\omega\geq -|ik|+1/\tau$ (yellow shading). Consistently with causality HellerBounds2022ejwGavassinoDistrubingMoving:2026klp, none of the branches enters the unstable region $\mathfrak{Im}\omega<|\mathfrak{Im}k|$ (red shading). Right panel: For $a<2/3$, the hydrodynamic branch merges with the continuous spectrum at a finite value $(ik\tau)_{\text{max}}$, beyond which it ceases to exist. In particular, at $a=0$ one recovers Fick’s law, which is defined only for $|\mathfrak{Im}(k\tau)|<2$. As $a$ increases, $(ik\tau)_{\text{max}}$ grows monotonically and diverges as $a\to 2/3$, signaling the transition to the Cattaneo regime, where the dispersion relation extends over the entire imaginary axis.
• Figure 3: Left panel: Non-propagating branch of the discrete dispersion relation $\omega(k)$ of the interpolating kinetic theory \ref{['interpuls']} for real wavenumber and various values of the parameter $a$ (blue: $a=0.5$, brown: $a=0.9$, orange: $a=0.99$, red: $a=0.999$). As $a$ increases, the curve interpolates continuously between the parabolic Fick geometry and the circular Cattaneo geometry. The non-propagating modes, characterized by $i\omega\in\mathbb{R}$, are obtained by setting $\lambda_1=r-is$ and $\lambda_2=r+is$ with $r>0$ and exploring the corresponding parametric curve $(k(r),\omega(r))$ over the interval $1/2<r\le (2\sqrt{1-a})^{-1}$. For all $a<1$, $\chi$ spans the entire imaginary axis, and the curves extend to arbitrarily large $|k|$. In the limit $a\to 1$, the branch approaches a circle of radius $1$, together with a vertical half-line emerging from $(k\tau,i\omega\tau)=(0,1)$. This additional line is absent in the Cattaneo theory ($a\equiv 1$) and reflects an exchange-of-limits issue. If one fixes $r\in[1/2+\varepsilon,(2\sqrt{1-a})^{-1}]$ with $\varepsilon>0$ and then sends $a\to1$, only the Cattaneo circle survives. If instead $a$ is fixed and $r\to1/2$, then $s$ diverges and the wavefunction $\psi(p)$ becomes highly oscillatory (see right panel for an explicit example). In this ultraviolet regime, the diffusive Fokker-Planck term \ref{['FokkerPLanck']} dominates, even when the momentum diffusivity $\nu$ is small, causing the mode to decay on a timescale much shorter than $\tau$. For the same reason, the continuous spectrum is given by \ref{['continuousspectrumaless1']} for all $a<1$, while it collapses to $\pm ik+1/\tau$ at $a=1$: arbitrarily oscillatory states are always damped by the Fokker-Planck term at finite $1-a$. Right panel: Real part of the effective wavefunction $\psi(p)$ corresponding to $a=0.9$ and $r=1/2+0.0001$, illustrating the highly oscillatory ultraviolet behavior near the endpoint $r=1/2$.
• Figure 4: Real (blue) and imaginary (red) parts of the dispersion relations $i\omega(k)$ for real $k$, including both the non-propagating branch (cf. figure \ref{['fig:NonPropagating']}) and the propagating branches, shown for representative values of $a$. A pair of propagating modes bifurcates from the continuous spectrum, where the latter occupies the region $\mathfrak{Re}\,(i\omega)\ge 1/\tau$ (yellow shading) and $\mathfrak{Im}\,(i\omega)=\pm k$ (black dashed), at a finite value of $k$. For $a<a_{\text{critical}}\approx 0.87$, this pair persists up to $k\to\infty$. For $a\ge a_{\text{critical}}$, the propagating modes are absorbed by the non-propagating branch at the forward tilt point and subsequently reemerge at the backward tilt point (note that $a_{\text{critical}}$ is the value of $a$ at which the non-propagating dispersion relation ceases to be single-valued). In the Cattaneo limit, only the branch emanating from the backward tilt point remains.