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

Abstract

Employing the well-known unitary equivalence between Fokker-Planck operators and Schrödinger 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 Schrödinger problem in one spatial dimension, and to a Coulomb Schrödinger 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.

Figures (3)

• Figure 1: Quasi-normal frequencies of nonrelativistic Fokker-Planck kinetic theory for imaginary $k$ (left) and real $k$ (right), in units of the spectral gap $\beta\nu/m$. The hydrodynamic mode is unique and nondegenerate, and exhibits an exact diffusive dispersion relation $\omega=-i\mathfrak{D}k^2$ at all $k$, with diffusion coefficient $\mathfrak{D}=(\beta^2\nu)^{-1}$. In addition, an infinite tower of nonhydrodynamic modes is present, equally spaced by $\nu\beta/m$ and degenerate for $D\ge2$. For real $k$, all modes remain purely damped, i.e. $\omega\in i\mathbb{R}$, in sharp contrast with RTA RomatschkeCutsandPoles:2015gicBajec:2024jezBrants:2024wrx, where the nonhydrodynamic modes form a continuous line $\omega\in\mathbb{R}-i\times\mathrm{gap}$. This qualitative difference originates from the microscopic dynamics encoded in the collision operator: in RTA, particles propagate ballistically between collisions, whereas the Fokker-Planck operator describes diffusion in momentum space induced by many small-angle scatterings, which in turn enforces diffusive motion in real space at all scales (at least in the Newtonian case).
• Figure 2: Quasi-normal frequencies of ultrarelativistic Fokker-Planck kinetic theory in one spatial dimension, for imaginary $k$ (left) and real $k$ (right), all in units of $\beta^2\nu/4$ (the spectral gap at $k{=}0$). Left panel: A single hydrodynamic mode exhibits diffusive dispersion, $\omega{=}-i\mathfrak{D}k^2$, with diffusion coefficient $\mathfrak{D}=(\beta^2\nu)^{-1}$. All remaining modes form a continuum (yellow region). The diffusive mode merges into the continuum before being able to cross the line $\mathfrak{Im}\omega\,{=}\,|\mathfrak{Im}k|$ (dashed), whose violation would signal acausality HellerBounds2022ejwGavassinoBounds2023myj. Right panel: For real $k$, the hydrodynamic mode is imaginary, while the nonhydrodynamic modes organize into two vertical continua emerging from $(\pm k,-i\,\text{gap})$. The appearance of propagating nonhydrodynamic modes in the relativistic theory, in contrast with the purely damped Newtonian spectrum, reflects the underlying microscopic dynamics: momentum diffusion in the Newtonian case leads to overdamped Brownian motion, whereas in the ultrarelativistic one-dimensional case the discrete velocity spectrum ($v{=}\pm1$) permits ballistic propagation despite stochastic momentum exchange.
• Figure 3: Quasi-normal frequencies of ultrarelativistic Fokker-Planck kinetic theory in three spatial dimensions, for imaginary $k$ (left) and real $k$ (right), all expressed in units of $\beta^2\nu/4$ (the spectral gap of the continuous band, "c.gap", at $k=0$). Left panel: A single hydrodynamic mode exhibits diffusive dispersion, $\omega=-i\mathfrak{D}k^2$, with diffusion coefficient $\mathfrak{D}=(\beta^2\nu)^{-1}$. Above it, we find an infinite tower of discrete modes that we could not determine analytically, but two of which have been calculated numerically for illustration (green). At $k=0$, the spectral degeneracy is the same as that of the hydrogen atom, and just like the hydrogen atom, these levels undergo Stark splitting at finite $k$. The remaining modes form a continuum (yellow region). Again, the diffusive mode merges into the continuum before it gets a chance of crossing the boundary of stability $\mathfrak{Im}\omega=|\mathfrak{Im}k|$ (dashed). Right panel: For real $k$, the hydrodynamic mode is imaginary, while the discrete non-hydrodynamic modes may acquire a real part, as shown by the two numerical examples (green). The continuous branch forms an infinitely long vertical rectangle with upper corners $(\pm k,-i\,\text{gap})$ (yellow), signalling the presence of ballistic propagation, whose existence is discussed in section \ref{['WhyBallistic']}. In both panels, the shade of red marks the forbidden region, where we are guaranteed not to find other modes.