Relativistic resistive magnetohydrodynamics for a two-component plasma

TL;DR

This work derives a covariant, kinetic-theory–based framework for relativistic resistive MHD in a two-component ultrarelativistic plasma by applying the 14-moment approximation to the Boltzmann–Vlasov equations in the Landau frame. It yields tightly coupled evolution equations for the net charge diffusion current $\mathcal{V}^\mu$, the total and relative diffusion currents, and the total and relative shear-stress tensors $\pi^{\mu\nu}$ and $\delta\pi^{\mu\nu}$, incorporating nonlinear feedback between electric/magnetic fields and dissipative quantities and inter-species collisions. The theory agrees with Israel–Stewart–type relaxation in the regime of small $\eta/s$, vanishing $B$, and moderate $E$, while revealing controlled deviations under strong electric fields due to back-reaction and a nonzero relative energy diffusion $\delta h^\mu$ that can generate anisotropy even without flow. Analyses of homogeneous and Bjorken-flow cases show that Ohmic relaxation is a good description at late times in neutral, homogeneous settings, whereas expansion and strong fields amplify nonlinear couplings, with diffusion and shear coupling persisting but damped in expanding geometries. This provides a kinetic-grounded, covariant description of two-component resistive MHD with potential applications to relativistic plasmas in heavy-ion collisions and astrophysical contexts.

Abstract

We derive relativistic resistive magnetohydrodynamics for a two-component ultrarelativistic plasma directly from kinetic theory. Starting with the Boltzmann--Vlasov equation and using the 14-moment approximation in the Landau frame, we obtain coupled evolution equations for the charge diffusion four-current and the shear-stress tensor. Benchmarking against the usual Israel-Stewart type relaxation form shows that this simplified description is accurate for small viscosity to entropy ($η/s$) ratio, vanishing magnetic field, and not so strong electric field. Outside this regime the dynamics depart in a controlled way, i.e., strong electric fields introduce nonlinear back-reaction that delays and reduces current peaks, and a sizable shear-stress is produced even without a flow profile.

Figures (5)

• Figure 1: Time evolution of the unnormalized diffusion current $\mathcal{V}_{q,x}(t)$ for different initial field strengths $E_0$, with $\eta/s = 1$, $|q| = 2/3$, and $\sigma_T = 0.1\,\sigma_T^{+-}$. Larger $E_0$ values lead to more pronounced transient peaks before relaxation. All trajectories converge to the same asymptotic limit, consistent with a steady Ohmic regime.
• Figure 2: Normalized time evolution of $\mathcal{V}_{q,x}/E_x$ for several initial electric field strengths $E_0$, with $\eta/s = 1$, $|q| = 2/3$, and $\sigma_T = 0.1\,\sigma_T^{+-}$. At late times, all curves converge to the same asymptotic value, consistent with the Ohmic limit $\mathcal{V}_{q,x} \simeq \sigma_{\rm E}\, E_x$. The early-time deviation reflects nonlinear relaxation effects under stronger applied fields.
• Figure 3: Time evolution of the diffusion current $\mathcal{V}_{q,x}(t)$ for different values of $\eta/s$, with fixed $|q|=1$, $\sigma_T = 0.1\,\sigma_T^{+-}$ and $E_0$ = $30\, fm^{-2}$. Increasing $\eta/s$ reduces collisional damping and gives rise to oscillatory relaxation, while smaller $\eta/s$ leads to a smooth exponential decay.
• Figure 4: Time evolution of the normalized shear-stress component $\pi_{xx}/\epsilon$ for various $E_0$ values, with $\eta/s = 1$, $\sigma_T/\sigma_T^{+-} = 0.1$. Larger $E_0$ enhances the transient peak due to stronger field-induced anisotropy, while all curves relax to zero at late times, restoring isotropy.
• Figure 5: Time evolution of the charge diffusion current $\mathcal{V}_{q,x}$ and normalized shear-stress $\pi_{xx}/\epsilon$ in Bjorken flow for different initial electric field strengths $E_0$. The left panel shows $\mathcal{V}_{q,x}$, exhibiting an early-time peak that increases with $E_0$ and decays slowly due to viscous and expansion damping. The right panel shows $\pi_{xx}/\epsilon$, which exhibits similar early peak trend but does not depend on $E_0$ values and thus is hugely governed by the hydrodynamical process.