  Everything you ever wanted to know about what has come to be known as
``chaotic mixing:'' This paper describes the evolution of localised ensembles
of initial conditions in 2- and 3-D time-independent potentials which admit
both regular and chaotic orbits. The coarse-grained approach towards an
invariant, or near-invariant, distribution was probed by tracking (1) phase
space moments through order 4 and (2) binned reduced distributions f(Z_a,Z_b,t)
for a,b=x,y,z,p_x,p_y,p_z, computed at fixed time intervals. For ``unconfined''
chaotic orbits in 2-D systems not stuck near islands by cantori, the moments
evolve exponentially: Quantities like the dispersion in p_x, which start small
and eventually asymptote towards a larger value, initially grow exponentially
in time at a rate comparable to the largest short time Lyapunov exponent.
Quantities like |<p_x>|, that can start large but eventually asymptote towards
zero, decrease exponentially. With respect to a discrete L^p norm, reduced
distributions f(t) generated from successive decay exponentially towards a
near-invariant f_{niv}, although a plot of Df(t)=||f(t)-f_{niv}|| can exhibit
considerable structure. Regular ensembles behave very differently, both moments
and Df evolving in a fashion better represented by a power law time dependence.
``Confined'' chaotic orbits, initially stuck near regular islands because of
cantori, exhibit an intermediate behaviour. The behaviour of ensembles evolved
in 3-D potentials is qualitatively similar, except that, in this case, it is
relatively likely to find one direction in configuration space which is ``less
chaotic'' than the other two, so that quantities like L_{ab} depend more
sensitively on which phase space variables one tracks.
