Phase space mixing in the equatorial plane of a Kerr black hole

It is shown that a collisionless, relativistic kinetic gas configuration propagating in the equatorial plane of a Kerr black hole undergoes a relaxation process and eventually settles down to a stationary, axisymmetric configuration surrounding the black hole. The underlying mechanism for this relaxation process is due to phase space mixing, which implies that although the one-particle distribution function f satisfying the collisionless Boltzmann equation is quasiperiodic in time, the associated macroscopic observables computed from averages over f possess well-defined limits as time goes to infinity. The final state of the gas is described by an effective distribution function depending only upon the constants of motion, and it can be determined by an appropriate average of the initial distribution function.