Stellar Dynamics around a Massive Black Hole III: Resonant Relaxation of
  Axisymmetric Discs
  We study the Resonant Relaxation (RR) of an axisymmetric low mass (or
Keplerian) stellar disc orbiting a more massive black hole (MBH). Our recent
work on the general kinetic theory of RR is simplified in the standard manner
by ignoring the effects of `gravitational polarization', and applied to a
zero-thickness, flat, axisymmetric disc. The wake of a stellar orbit is
expressed in terms of the angular momenta exchanged with other orbits, and used
to derive a kinetic equation for RR under the combined actions of self-gravity,
1 PN and 1.5 PN relativistic effects of the MBH and an arbitrary external
axisymmetric potential. This is a Fokker-Planck equation for the stellar
distribution function (DF), wherein the diffusion coefficients are given
self-consistently in terms of contributions from apsidal resonances between
pairs of stellar orbits. The physical kinetics is studied for the two main
cases of interest. (1) `Lossless' discs in which the MBH is not a sink of
stars, and disc mass, angular momentum and energy are conserved: we prove that
general H-functions can increase or decrease during RR, but the Boltzmann
entropy is (essentially) unique in being a non-decreasing function of time.
Therefore secular thermal equilibria are maximum entropy states, with DFs of
the Boltzmann form; the two-Ring correlation function at equilibrium is
computed. (2) Discs that lose stars to the MBH through an `empty loss-cone': we
derive expressions for the MBH feeding rates of mass, angular momentum and
energy in terms of the diffusive flux at the loss-cone boundary.
