Second-order solutions of the equilibrium statistical mechanics for
  self-gravitating systems
  In a previous study, we formulated a framework of the entropy-based
equilibrium statistical mechanics for self-gravitating systems. This theory is
based on the Boltzmann-Gibbs entropy and includes the generalized virial
equations as additional constraints. With the truncated distribution function
to the lowest order, we derived a set of second-order equations for the
equilibrium states of the system. In this work, the numerical solutions of
these equations are investigated. It is found that there are three types of
solutions for these equations. Both the isothermal and divergent solutions are
thermally unstable and have unconfined density profiles with infinite mass,
energy and spatial extent. The convergent solutions, however, seem to be
reasonable. Although the results cannot match the simulation data well, because
of the truncations of the distribution function and its moment equations, these
lowest-order convergent solutions show that the density profiles of the system
are confined, the velocity dispersions are variable functions of the radius,
and the velocity distributions are also anisotropic in different directions.
The convergent solutions also indicate that the statistical equilibrium of
self-gravitating systems is by no means the thermodynamic equilibrium. These
solutions are just the lowest-order approximation, but they have already
manifested the qualitative success of our theory. We expect that higher-order
solutions of our statistical-mechanical theory will give much better agreement
with the simulation results concerning dark matter haloes.
