  By a natural nonextensive generalization of the conservation of energy in the
q-kinetic theory, we study the nonextensivity and the power-law distributions
for the many-body systems with the self-gravitating long-range interactions. It
is shown that the power-law distributions describe the long-range nature of the
interactions and the non-local correlations within the self-gravitating system
with the inhomogeneous velocity dispersion. A relation is established between
the nonextensive parameter q and the measurable quantities of the
self-gravitating system: the velocity dispersion and the mass density.
Correspondingly, the nonextensive parameter q can be uniquely determined from
the microscopic dynamical equation and thus the physical interpretation of q
different from unity can be clearly presented. We derive a nonlinear
differential equation for the radial density dependence of the self-gravitating
system with the inhomogeneous velocity dispersion, which can correctly describe
the density distribution for the dark matter in the above physical situation.
We also apply this q-kinetic approach to analyze the nonextensivity of
self-gravitating collisionless systems and self-gravitating gaseous dynamical
systems, giving the power-law distributions the clear physical meaning.
