  This paper summarises a numerical investigation of phase mixing in
time-independent Hamiltonian systems that admit a coexistence of regular and
chaotic phase space regions, allowing also for low amplitude perturbations
idealised as periodic driving, friction, and/or white and colored noise. The
evolution of initially localised ensembles of orbits was probed through lower
order moments and coarse-grained distribution functions. In the absence of
time-dependent perturbations, regular ensembles disperse initially as a power
law in time and only exhibit a coarse-grained approach towards an invariant
equilibrium over comparatively long times. Chaotic ensembles generally diverge
exponentially fast on a time scale related to a typical finite time Lyapunov
exponent, but can exhibit complex behaviour if they are impacted by the effects
of cantori or the Arnold web. Viewed over somewhat longer times, chaotic
ensembles typical converge exponentially towards an invariant or near-invariant
equilibrium. This, however, need not correspond to a true equilibrium, which
may only be approached over very long time scales. Time-dependent perturbations
can dramatically increase the efficiency of phase mixing, both by accelerating
the approach towards a near-equilibrium and by facilitating diffusion through
cantori or along the Arnold web so as to accelerate the approach towards a true
equilibrium. The efficacy of such perturbations typically scales
logarithmically in amplitude, but is comparatively insensitive to most other
details, a conclusion which reinforces the interpretation that the
perturbations act via a resonant coupling.
