  This article is devoted to the study of the dynamical behavior of a
collisionless kinetic gas in d=1,2,3 space dimensions 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 d 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 article, we provide a mathematically precise
formulation for this relaxation process which 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. Thus, the final state of the gas is described by an
effective distribution function depending only on integrals of motion, which
considerably reduces the degrees of freedom of the gas configuration. We
discuss some applications to gravitational physics, including the propagation
of a collisionless gas in typical potentials arising in stellar dynamics and
the modeling of dark matter halos, and we also generalize our results to a
relativistic gas whose individual particles follow bound timelike trajectories
in the exterior region of a static, spherically symmetric black hole spacetime.
