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.