  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.
