Plasma oscillations within Israel-Stewart theory

TL;DR

The work shows that plasma oscillations predicted by a Drude-like kinetic model persist within Israel-Stewart magnetohydrodynamics when coupled to Maxwell equations, revealing that the nonhydrodynamic sector can describe genuine plasmons. Through the Onsager-Casimir principle, it is shown that uncharged kinetic theories at $k=0$ are purely relaxational due to PT symmetry, while coupling to the PT-odd electric field introduces an antisymmetric interaction that yields at most one pair of oscillatory nonhydrodynamic modes per spatial direction. A concrete bound is established: in a kinetic theory with $N$ non-equilibrium moments, at most two plasmons can exist, with all other modes remaining on the imaginary axis. The results broaden the predictive scope of Israel-Stewart MHD to timescales around the plasma period $\omega_p^{-1}$, clarifying its relevance for electromagnetic processes in metals, astrophysical plasmas, and possibly heavy-ion collision contexts, by linking the macroscopic relaxation dynamics to underlying plasmons via a thermodynamic symmetry framework.

Abstract

It is well known that, at zero wavenumber, the non-hydrodynamic frequencies of uncharged kinetic theory are purely imaginary. On the other hand, it was recently shown that, in resistive magnetohydrodynamics, the interplay between the Israel-Stewart relaxation equation and the Ampère-Maxwell law can give rise to a pair of oscillating non-hydrodynamic modes. In this work, we analyze this phenomenon in detail. We first demonstrate that these oscillatory modes are exact solutions of the Drude model, corresponding to ordinary plasma oscillations. We then invoke the Onsager-Casimir principle to explain that their oscillatory nature reflects the distinct PT-transformation properties of the degrees of freedom: the distribution function is even, while the electric field is odd. Finally, we establish that, in a kinetic theory of charged particles, there can be at most one such pair of oscillatory modes per spatial dimension, while all other modes still must sit on the imaginary axis.

Figures (5)

• Figure 1: Hydrodynamic (red) and non-hydrodynamic (blue) modes of a microscopic theory at $k=0$ according to the conventional wisdom. Left panel: Typical excitation spectrum of a kinetic theory. All the non-hydrodynamic frequencies are scattered along the imaginary axis (some may form a continuum Moore:2018mma, thin blue line), and their magnitude is of the order of the particle mean free time $\tau$. Right panel: Typical excitation spectrum of a holographic theory. The non-hydrodynamic relaxation rates $-i\omega$ come in (often regularly spaced) complex conjugate pairs, and their magnitude is of the order of the temperature $T$.
• Figure 2: A minimal model of plasma oscillations can be constructed by rigidly displacing the electron gas (yellow) in a finite conductor (blue), which produces two oppositely charged layers at the boundaries. The resulting electric field (red arrow) exerts a restoring force on the electrons. Including both the electron inertia and the electron-proton friction leads directly to the equation of a damped harmonic oscillator.
• Figure 3: Non-hydrodynamic sector of two Boltzmann-Vlasov-Maxwell models with 10 non-equilibrium degrees of freedom, governed by equation \ref{['dopo']}, with $\sigma_n$ and $\mathcal{J}^n$ given by \ref{['randomiamo']}. For a given realization of the random variables, we let $g$ run from 0 (green poles) to 1 (blue poles), and we mark the trajectory drawn by the moving poles in red.
• Figure 4: Dispersion relations $\omega(k)$ of the transversal modes of a conductor with $\omega_p\tau=0.3$ (left panel) or $1$ (right panel), modeled according to Israel-Stewart theory, see equation \ref{['transversalIS']}. The blue lines are the real parts, the red lines are the imaginary parts, and the orange band marks the qualitative regime of validity. Note that there are two relevant scales in \ref{['transversalIS']}, namely $\tau$ and $\sigma^{-1}$, but applicability of the theory only requires $k\tau \ll 1$, which is why the regime of validity of the left panel encloses more structure (in the left panel, $\tau \ll \sigma^{-1}$, so the modes with $k\sim \sigma^{-1}$ are physical).
• Figure 5: Dispersion relations $\omega(k)$ of the longitudinal modes of a conductor with $\omega_p\tau=0.3$ (left panel) or $1$ (right panel), modeled according to Israel-Stewart theory, see equation \ref{['islongitud']}. The blue lines are the real parts and the red lines are the imaginary parts. The regime of applicability is defined by the condition $k\tau\ll 1$, which in the plot corresponds to $kb\ll b/\tau$.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof