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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.

Saddle‐point entropy states of equilibrated self‐gravitating systems