Relaxation of One-dimensional Collisionless Gravitating Systems
  In an effort to better understand collisionless relaxation processes in
gravitational systems, we investigate one-dimensional models. Taking advantage
of a Hermite-Legendre expansion of relevant distribution functions, we present
analytical and numerical behaviors of Maxwell-Boltzmann entropy. In particular,
we modestly perturb systems about a separable-solution equilibrium and observe
their collisionless evolution to a steady state. We verify the
time-independence of fine-grained entropy in these systems before turning our
attention to the behavior of coarse-grained entropy. We also verify that there
is no analogue to the collisional H-theorem for these systems. Competing terms
in the second-order coarse-grained entropy make it impossible to guarantee
continuously increasing entropy. However, over dynamical time-scales the
coarse-grained entropy generally increases, with small oscillations occurring.
The lack of substantive differences between the entropies in test-particle and
self-gravitating cases suggests that phase mixing, rather than violent
relaxation associated with potential changes, more significantly drives the
coarse-grained entropy evolution. The effects of violent relaxation can be
better quantified through analysis of energy distributions rather than
phase-space distributions.
