Collisionless relaxation of a Lynden-Bell plasma

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.

46 pages, 4 figures, submitted to JPP

References in corpus (4)

• Available energy and ground states of collisionless plasmas
• A solvable model of Vlasov-kinetic plasma turbulence in Fourier-Hermite phase space
• Generalized entropy production in collisionless plasma flows and turbulence
• Kinetic theory of collisionless relaxation for systems with long-range interactions

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