Entropy principle and complementary second law of thermodynamics for
  self-gravitating systems
  (abbreviated) The statistical mechanics of self-gravitating systems is a
long-held puzzle. In this work, we employ a phenomenological entropy form of
ideal gas, first proposed by White & Narayan, to revisit this issue. By
calculating the first-order variation of the entropy, subject to the mass- and
energy-conservation constraints, we obtain an entropy stationary equation.
Incorporated with the Jeans equation, and by specifying some functional form
for the anisotropy parameter beta, we numerically solve the two equations, and
demonstrate that the velocity anisotropy parameter plays an important role to
attain a density profile that is finite in mass, energy, and spatial extent. If
incorporated again with some empirical density profile from simulations, our
theoretical predictions of the anisotropy parameter, and the radial
pseudo-phase-space density in the outer non-gravitationally degenerate region
of the dark matter halo, agree rather well with the simulation data, and the
predictions are also acceptable in the middle weak-degenerate region of the
dark halo. The second-order variational calculus reveals the seemingly
paradoxical but actually complementary consequence that the equilibrium state
of self-gravitating systems is the global minimum entropy state for the whole
system under long-range violent relaxation, but simultaneously the local
maximum entropy state for every and any small part of the system under
short-range two-body relaxation and Landau damping. This minimum-maximum
entropy duality means that the standard second law of thermodynamics needs to
be re-expressed or generalized for self-gravitating systems. We believe that
our findings, especially the complementary second law of thermodynamics, may
provide crucial clues to the development of the statistical physics of
self-gravitating systems as well as other long-range interaction systems.
