We discuss some recent results regarding the dynamics of a collisionless kinetic gas which is trapped in a rotationally symmetric potential well. Although at the microscopic level the trajectories of individual gas particles are quasi-periodic and characterized by their fundamental frequencies, at the macroscopic level the gas relaxes in time to a stationary state, provided the potential satisfies a certain non-degeneracy condition. In this talk, we will provide a mathematically precise formulation for this relaxation process and will show that it is due to phase space mixing. In particular, we prove that a physically relevant class of macroscopic observables computed from the one-particle distribution function, such as particle and energy densities, pressure and stress tensors, converge in time to the corresponding observables associated with an averaged distribution function. The latter can be determined from the initial datum and depends only on integrals of motion. Some applications to gravitational physics will be presented, including: (i) the propagation of a collisionless gas in typical potentials arising in stellar dynamics and the modeling of dark matter halos, (ii) the propagation of a relativistic kinetic gas whose individual particles follow bound timelike trajectories in the exterior region of a black hole spacetime.
Based on joint work with Paola Rioseco (arXiv:2005.05988, arXiv:1807.10794).