Fluid-like entropy and equilibrium statistical mechanics of
  self-gravitating systems
  The statistical mechanics of self-gravitating systems has not been well
understood, and still remains an open question so far. In a previous study by
Kang & He, we showed that the fluid approximation may give a clue to further
investigate this problem. In fact, there are indeed many dynamical similarities
between self-gravitating and fluid systems. Based on a fluid-like entropy, that
work explained successfully the outer density profiles of dark matter halos,
but there left some drawbacks with the calculation concerning extremizing
process of the entropy. In the current paper, with the improved extremizing
calculation -- including an additional differential constraint of dynamical
equilibrium and without any other assumptions, we confirm that
statistical-mechanical methods can give a density profile with finite mass and
finite energy. Moreover, this density profile is also consistent with the
observational surface brightness of the elliptical galaxy NGC 3379. In our
methods, the density profile is derived from the equation of state, which is
obtained from entropy principle but does not correspond to the maximum entropy
of the system. Finally, we suggest an alternative entropy form, a hybrid of
Boltzmann-Gibbs and Tsallis entropy, whose global maximum may give rise to this
equation of state.
