Saddle-point entropy states of equilibrated self-gravitating systems
  In this Letter, we investigate the stability of the statistical equilibrium
of spherically symmetric collisionless self-gravitating systems. By calculating
the second variation of the entropy, we find that perturbations of the relevant
physical quantities should be classified as long- and short-range
perturbations, which correspond to the long- and short-range relaxation
mechanisms, respectively. We show that the statistical equilibrium states of
self-gravitating systems are neither maximum nor minimum, but complex
saddle-point entropy states, and hence differ greatly from the case of ideal
gas. Violent relaxation should be divided into two phases. The first phase is
the entropy-production phase, while the second phase is the entropy-decreasing
phase. We speculate that the second-phase violent relaxation may just be the
long-wave Landau damping, which would work together with short-range
relaxations to keep the system equilibrated around the saddle-point entropy
states.
