A Balescu-Lenard type kinetic equation fot the collisional evolution of
  stable self-gravitating systems
  A kinetic equation for the collisional evolution of stable, bound, self
gravitating and slowly relaxing systems is established, which is valid when the
number of constituents is very large. It accounts for the detailed dynamics and
self consistent dressing by collective gravitational interaction of the
colliding particles, for the system's inhomogeneity and for different
constituent's masses. The evolution of the one-body distribution function is
described in action angle space. The collision operators are expressed in terms
of the collective response function allowed by the existing distribution
functions at any given time and involve particles in resonant motions. The set
of equations which describe the coupled evolution of the distribution functions
and of the potential is derived for spherical systems. In the homogeneous
limit, which sacrifices the description of the evolution of the spatial
structure of the system, but retains the effects of collective gravitational
dressing, the kinetic equation reduces to a form similar to the Balescu-Lenard
equation of plasma physics.
