Collisional relaxation of two-dimensional self-gravitating systems
  Systems with long range interactions present generically the formation of
quasi-stationary long-lived non-equilibrium states. These states relax to
Boltzmann equilibrium following a dynamics which is not well understood. In
this paper we study this process in two-dimensional inhomogeneous
self-gravitating systems. Using the Chandrasekhar -- or local -- approximation
we write a simple approximate kinetic equation for the relaxation process,
obtaining a Fokker -- Planck equation for the velocity distribution with
explicit analytical diffusion coefficients. Performing molecular dynamics
simulations and comparing them with the evolution predicted by the Fokker --
Planck equation, we observe a good agreement with the model for all the
duration of the relaxation, from the formation of the quasi-stationary state to
thermal equilibrium. We observe however an overestimate or underestimate of the
relaxation rate of the particles with the slower or larger velocities
respectively. It is due to systematic errors in estimating the velocities of
the particles at the moment of the collisions, inherent to the Chandrasekhar
approximation when applied to inhomogeneous systems. Theory and simulations
give a scaling of the relaxation time proportional to the number of particles
in the system.
