Universal non-Maxwellian equilibria in near-collisionless plasmas

A fundamental tenet of thermodynamics is that chaotic systems will relax to maximum-entropy states. In plasmas, the chaos is conventionally provided by

interparticle collisions and the universal maximum-entropy equilibrium is a Maxwellian distribution. However, in collisionless plasmas, the chaotic state is due to collective

turbulent dynamics. We will argue theoretically, and show numerically, that such plasmas still relax towards universal equilibria that are non-Maxwellian, featuring a

power-law distribution of particle energies. This is achieved via an entropy-maximization procedure that accounts for the short-time conservation of certain collisionless

invariants. The conservation of these collisionless invariants endows the system with a partial ‘memory' of its prior conditions, but is imperfect on long time scales due to the

development of a turbulent cascade to small scales, which breaks the precise conservation of phase volume, making this memory imprecise. The equilibria are still

determined by the short-time collisionless invariants, but the invariants themselves are driven to a universal form by the nature of the turbulence. This is numerically confirmed

for the case of beam instabilities in one-dimensional electrostatic plasmas (see Ewart et al. 2024, E-print arXiv:2409.01742), where sufficiently strong turbulence appears to

cause the distribution function of particle energies to develop a universal power-law tail, with exponent -2 (as predicted in Ewart et al. 2023, J. Plasma Phys. 89. 905890516).