  A massive black hole (MBH) consumes stars whose orbits evolve into the small
phase-space volume of unstable orbits, the "loss-cone", which take them
directly into the MBH, or close enough to interact strongly with it. The
resulting phenomena: tidal heating and tidal disruption, binary capture and
hyper-velocity star ejection, gravitational wave (GW) emission by inspiraling
compact remnants, or hydrodynamical interactions with an accretion disk, are of
interest as they can produce observable signatures and thereby reveal the
existence of the MBH, affect its mass and spin evolution, probe strong gravity,
and provide information on stars and gas near the MBH. The continuous loss of
stars and the processes that resupply them shape the central stellar
distribution. We investigate relativistic stellar dynamics near the loss-cone
of a non-spinning MBH in steady-state analytically and by Monte Carlo
simulations of the diffusion of the orbital parameters. These take into account
Newtonian mass precession due to enclosed stellar mass, in-plane precession due
to general relativity, dissipation by GW, uncorrelated two-body relaxation,
correlated resonant relaxation (RR) and adiabatic invariance due to secular
precession, using a rigorously derived description of correlated post-Newtonian
dynamics in the diffusion limit. We argue that general maximal entropy
considerations strongly constrain orbital diffusion in steady-state,
irrespective of the relaxation mechanism. We identify the exact phase-space
separatrix between plunges and inspirals, predict their steady-state rates, and
verify they are robust under a wide range of assumptions. We derive the
dependence of the rates on the mass of the MBH, show that the contribution of
RR is small, and discuss special cases where unquenched RR in restricted
volumes of phase-space may affect the steady-state substantially.
