  The long-standing puzzle surrounding the statistical mechanics of
self-gravitating systems has not yet been solved successfully. We formulate a
systematic theoretical framework of entropy-based statistical mechanics for
spherically symmetric collisionless self-gravitating systems. We use an
approach that is very different from that of the conventional statistical
mechanics of short-range interaction systems. We demonstrate that the
equilibrium states of self-gravitating systems consist of both mechanical and
statistical equilibria, with the former characterized by a series of
velocity-moment equations and the latter by statistical equilibrium equations,
which should be derived from the entropy principle. The velocity-moment
equations of all orders are derived from the steady-state collisionless
Boltzmann equation. We point out that the ergodicity is invalid for the whole
self-gravitating systems, but it can be re-established locally. Based on the
local ergodicity, using Fermi-Dirac-like statistics, with the nondegenerate
condition and the spatial independence of the local microstates, we rederive
the Boltzmann-Gibbs entropy. This is consistent with the validity of the
collisionless Boltzmann equation, and should be the correct entropy form for
collisionless self-gravitating systems. Apart from the usual constraints of
mass and energy conservation, we demonstrate that the series of moment or
virialization equations must be included as additional constraints on the
entropy functional when performing the variational calculus; this is an
extension to the original prescription by White & Narayan. Any possible
velocity distribution can be produced by the statistical-mechanical approach
that we have developed with the extended Boltzmann-Gibbs/White-Narayan
statistics. Finally, we discuss the questions of negative specific heat and
ensemble inequivalence for self-gravitating systems.
