Universal equilibria in relaxing electrostatic plasma

In the absence of collisions, or in the presence of weak collisions, turbulent plasmas have little impetus to relax to a Maxwellian equilibrium. Nevertheless, distributions in turbulent plasmas often exhibit universal properties with only weak dependence on the nature of the turbulence. It is therefore an open question as to whether there are any universal classes of equilibria to which nearly collisionless systems are naturally predisposed. We identify such a class based on the statistical-mechanical arguments of Lynden-Bell (1967), which utilised the conservation of phase volume as a set of additional invariants (Casimir invariants), like total energy, in the relaxation of the system—essentially endowing the system with memory of its initial conditions. We show however, in the presence of turbulent fluctuations, that the Casimir invariants are broken by the generation of small-scale velocity-space structure. Just as energy is still a relevant quantity in systems where energy is not an invariant (e.g., in the presence of heating), these former Casimir invariants still control the equilibrium to which the system relaxes—the only difference from the Lynden-Bell paradigm being that the system's `memory' is now specified by the turbulence rather than the initial conditions.

We numerically confirm this class of equilibria using nearly-collisionless 1D particle-in-cell simulations of a prototypical turbulent system: the two-stream instability. The evolution of the Casimir invariants in fact makes the equilibria of this system more universal, causing the distribution function of particle energies to achieve a power-law tail with exponent -2.