Stellar Dynamics around a Massive Black Hole II: Resonant Relaxation
  We present a first-principles theory of Resonant Relaxation (RR) of a low
mass stellar system orbiting a more massive black hole (MBH). We first extend
the kinetic theory of Gilbert (1968) to include the Keplerian field of a black
hole of mass $M_\bullet$. Specializing to a Keplerian stellar system of mass $M
\ll M_\bullet$, we use the orbit-averaging method of Sridhar & Touma (2015;
Paper I) to derive a kinetic equation for RR. This describes the collisional
evolution of a system of $N \gg 1$ Gaussian Rings in a reduced 5-dim space,
under the combined actions of self-gravity, 1 PN and 1.5 PN relativistic
effects of the MBH and an arbitrary external potential. In general geometries
RR is driven by both apsidal and nodal resonances, so the distinction between
scalar-RR and vector-RR disappears. The system passes through a sequence of
quasi-steady secular collisionless equilibria, driven by irreversible 2-Ring
correlations that accrue through gravitational interactions, both direct and
collective. This correlation function is related to a `wake function', which is
the linear response of the system to the perturbation of a chosen Ring. The
wake function is easier to appreciate, and satisfies a simpler equation, than
the correlation function. We discuss general implications for the interplay of
secular dynamics and non-equilibrium statistical mechanics in the evolution of
Keplerian stellar systems toward secular thermodynamic equilibria, and set the
stage for applications to the RR of axisymmetric discs in Paper III.
