Double-Adiabatic Equations of State for Relativistic Plasmas

Abstract

The adiabatic equation of state $P \propto n^Γ$ describes the pressure evolution of highly collisional, isotropic plasmas in terms of their density, providing a possible closure of the fluid moment hierarchy in the absence of heat fluxes and dissipation. An analogous closure exists for collisionless, magnetised plasmas, whose pressure tensor is anisotropic with respect to the magnetic field, and the closure is therefore double-adiabatic, prescribing the evolution of the parallel and perpendicular pressures in terms of the magnetic-field strength and density. Here, we present a general first-principle formalism to derive adiabatic laws using the symmetries of the system. With this theory we recover the adiabatic equation of state $P \propto n^Γ$ for isotropic plasmas and the double-adiabatic equations of state for collisionless, magnetised plasmas. We extend the latter to the relativistic regime, finding that their exact functional form depends on the pressure anisotropy and is not a simple power law. Our double-adiabatic equations of state describe simple geometries, like magnetic mirrors or compressed homogeneous plasmas, as well as complex high-energy astrophysical processes, such as the evolution of plasmoid structures formed during magnetic reconnection.

Figures (4)

• Figure 1: Ratio $P_\perp/P_\parallel$ as a function of $A \equiv B'^3/n'^2$ (see \ref{['eq: A']}), shown by the green line. The solid light- and dark-grey lines indicate $P_\perp/P_\parallel = A$ and $P_\perp/P_\parallel = A^{4/5}$, respectively. In the limits $P_\perp/P_\parallel \ll 1$ and $P_\perp/P_\parallel \gg 1$, the ratio scales as $P_\perp/P_\parallel \sim A$, whereas close to isotropy it scales as $P_\perp/P_\parallel \sim A^{4/5}$.
• Figure 2: Generalised adiabatic indices $\Gamma_\perp$ (left) and $\Gamma_\parallel$ (right), defined in Section \ref{['sec: Adiabatic Indices']}, as functions of the pressure anisotropy parameter $A = B'^3/n'^2$. The pressure-evolution equations written in terms of these generalised adiabatic indices are \ref{['eq: adiabatic indices']}. The dashed vertical lines correspond to the point $A= 1$, i.e., isotropic plasma.
• Figure 3: The true \ref{['eq: p pressure definitions']} and modified \ref{['eq: p hat pressure definitions']} pressures, calculated from PIC simulations of plasma undergoing compression perpendicular (left panel) and parallel (right panel) to the magnetic field, plotted as functions of the normalised density $n'$. In the perpendicular compression case, $B' = n'$, while in the parallel case $B' = 1$. The theoretical predictions given by \ref{['eq: pressure perp evolution compressing']}, \ref{['eq: pressure parallel evolution compressing']}, and \ref{['eq: double adiabatic evolution']} are shown as solid grey lines. In the left panel, the grey dashed lines correspond to $P_\perp' = n'^{8/5}$ and $P_\parallel' = n'^{4/5}$, while in the right panel to $P_\perp' = n'^{4/5}$ and $P_\parallel' = n'^{12/5}$. At larger compressions, a clear departure from double-adiabatic theory is observed due to the excitation of the relevant pressure-anisotropy-driven kinetic instabilities.
• Figure 4: Our simulation set up. The background magnetic field is $\mathbf{B}_0 = B_0 \hat{\mathbf{x}}$, the box is compressed in the $z$ and $y$ directions. Only allow variations in the $(x,y)$ plane are allowed in the simulation.