The statistical mechanics of relativistic orbits around a massive black
  hole
  Stars around a massive black hole (MBH) move on nearly fixed Keplerian
orbits, in a centrally-dominated potential. The random fluctuations of the
discrete stellar background cause small potential perturbations, which
accelerate the evolution of orbital angular momentum by resonant relaxation.
This drives many phenomena near MBHs, such as extreme mass-ratio gravitational
wave inspirals, the warping of accretion disks, and the formation of exotic
stellar populations. We present here a formal statistical mechanics framework
to analyze such systems, where the background potential is described as a
correlated Gaussian noise. We derive the leading order, phase-averaged 3D
stochastic Hamiltonian equations of motion, for evolving the orbital elements
of a test star, and obtain the effective Fokker-Planck equation for a general
correlated Gaussian noise, for evolving the stellar distribution function. We
show that the evolution of angular momentum depends critically on the temporal
smoothness of the background potential fluctuations. Smooth noise has a maximal
variability frequency $\nu_{\max}$. We show that in the presence of such noise,
the evolution of the normalized angular momentum $j=\sqrt{1-e^{2}}$ of a
relativistic test star, undergoing Schwarzschild (in-plane) General
Relativistic precession with frequency $\nu_{GR}/j^{2}$, is exponentially
suppressed for $j<j_{b}$, where $\nu_{GR}/j_{b}^{2}\sim\nu_{\max}$, due to the
adiabatic invariance of the precession against the slowly varying random
background torques. This results in an effective Schwarzschild
precession-induced barrier in angular momentum. When $j_{b}$ is large enough,
this barrier can have significant dynamical implications for processes near the
MBH.
