A probability theory for non-equilibrium gravitational systems
  This paper uses dynamical invariants to describe the evolution of
collisionless systems subject to time-dependent gravitational forces without
resorting to maximum-entropy probabilities. We show that collisionless
relaxation can be viewed as a special type of diffusion process in the
integral-of-motion space. In time-varying potentials with a fixed spatial
symmetry the diffusion coefficients are closely related to virial quantities,
such as the specific moment of inertia, the virial factor and the mean kinetic
and potential energy of microcanonical particle ensembles. The non-equilibrium
distribution function (DF) is found by convolving the initial DF with the Green
function that solves Einstein's equation for freely diffusing particles. Such a
convolution also yields a natural solution to the Fokker-Planck equations in
the energy space. Our mathematical formalism can be generalized to potentials
with a time-varying symmetry, where diffusion extends over multiple dimensions
of the integral-of-motion space. The new probability theory is in many ways
analogous to stochastic calculus, with two significant differences: (i) the
equations of motion that govern the trajectories of particles are fully
deterministic, and (ii) the diffusion coefficients can be derived
self-consistently from microcanonical phase-space averages without relying on
ergodicity assumptions. For illustration we follow the cold collapse of
$N$-body models in a time-dependent logarithmic potential. Comparison between
the analytical and numerical results shows excellent agreement in regions where
the potential evolution does not depart too strongly from the adiabatic regime.
