  The orbits of stars in galaxies are generically chaotic: the chaotic behavior
arises in part from the intrinsically grainy nature of a potential that is
composed of point masses. Even if the potential is assumed to be smooth,
however, orbits in non-axisymmetric galaxies can be chaotic due to the presence
of central density cusps or black holes. The chaotic nature of orbits implies
that perturbations will grow exponentially and this in turn is expected to
result in a diffusion in phase space. We show that the degree of orbital
evolution is not well predicted by the growth rate of infinitesimal
perturbations, i.e. by the Liapunov exponent. A more useful criterion is
whether perturbations continue to grow exponentially until their scale is of
order the size of the system. We illustrate these ideas in a potential
consisting of N fixed point masses. Liapunov exponents are large for all values
of N, but orbits become increasingly regular in their behavior as N increases;
the reason is that the exponential divergence saturates at smaller and smaller
distances as N is increased. The objects which impede diffusion are the
invariant tori. When there are few stable tori, an ensemble of chaotic orbits
evolves rapidly toward a nearly stationary state. This mixing process occurs on
timescales of a few crossing times in triaxial potentials containing massive
central singularities, consistent with the rapid evolution observed in N-body
simulations of galaxies with central black holes.
