Relaxation to universal non-Maxwellian equilibria in a collisionless plasma

The question of how the particle distribution of a turbulent collisionless plasma will evolve, and what it may eventually relax to, is of both fundamental theoretical interest and observational consequence. In quiescent collisional plasmas, where the chaos is provided by inter-particle collisions, the answer is provided by thermodynamics and the universal maximum-entropy state is the Maxwellian distribution. However, in collisionless systems, the chaos is typically provided by the turbulent mean-field dynamics which nominally conserve an infinite set of invariants (conservation of phase volume) giving the system some memory of its initial conditions and making relaxation to a Maxwellian impossible. We will argue theoretically, and show numerically, that such plasmas will still relax towards universal equilibria, which are non-Maxwellian and feature a power-law distribution of particle energies. These equilibria will still be obtained by a maximum-entropy principle, while taking into account the conservation of these collisionless invariants on short time scales. Of course, there is no such thing as a truly collisionless system. On longer timescales these invariants are subtly broken by a turbulent cascade giving the system a kind of 'turbulent amnesia': relaxation, driven by the turbulence, to equilibrium is controlled by the collisionless invariants (the system's memory) but this memory is made imprecise by that same turbulence. We demonstrate that the invariants are driven to a universal form which results in a distribution function with a universal power-law tail in energy, with exponent -2