The secular evolution of discrete quasi-Keplerian systems. I. Kinetic
  theory of stellar clusters near black holes
  We derive the kinetic equation that describes the secular evolution of a
large set of particles orbiting a dominant massive object, such as stars bound
to a supermassive black hole or a proto-planetary debris disc encircling a
star. Because the particles move in a quasi-Keplerian potential, their orbits
can be approximated by ellipses whose orientations remain fixed over many
dynamical times. The kinetic equation is obtained by simply averaging the BBGKY
equations over the fast angle that describes motion along these ellipses. This
so-called Balescu-Lenard equation describes self-consistently the long-term
evolution of the distribution of quasi-Keplerian orbits around the central
object: it models the diffusion and drift of their actions, induced through
their mutual resonant interaction. Hence, it is the master equation that
describes the secular effects of resonant relaxation. We show how it captures
the phenonema of mass segregation and of the relativistic Schwarzschild barrier
recently discovered in $N$-body simulations.
