Collisionless relaxation of a Lynden-Bell plasma

TL;DR

This paper investigates the collisionless relaxation process in plasmas with minimal Coulomb collisions, deriving effective collision integrals that explain how Lynden-Bell equilibria are dynamically reached and their implications for plasma behavior.

Contribution

It introduces a method to derive collisionless collision integrals with H-theorems, explaining the dynamic approach to Lynden-Bell equilibria in plasmas.

Findings

Lynden-Bell equilibria are fixed points of derived collision integrals.

Collisionless relaxation can be rapid with sufficient electric fluctuations.

The formalism recovers classical and quantum collision integrals efficiently.

Abstract

Plasmas whose Coulomb-collision rates are very small may relax on shorter time scales to non-Maxwellian quasi-equilibria, which, nevertheless, have a universal form, with dependence on initial conditions retained only via an infinite set of Casimir invariants enforcing phase-volume conservation. These are distributions derived by Lynden-Bell (1967) via a statistical-mechanical entropy-maximisation procedure, assuming perfect mixing of phase-space elements. To show that these equilibria are reached dynamically, one must derive an effective `collisionless collision integral' for which they are fixed points -- unique and inevitable provided the integral has an appropriate H-theorem. We describe how such collision integrals are derived and what assumptions are required for them to have a closed form, how to prove the H-theorems for them, and why, for a system carrying sufficiently large electric-fluctuation energy, collisionless relaxation should be fast. It is suggested that collisionless dynamics may favour maximising entropy locally in phase space before converging to global maximum-entropy states. Relaxation due to interspecies interaction is examined, leading, inter alia, to spontaneous transient generation of electron currents. The formalism also allows efficient recovery of `true' collision integrals for both classical and quantum plasmas.