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By a natural nonextensive generalization of the conservation of energy in theq-kinetic theory, we study the nonextensivity and the power-law distributionsfor the many-body systems with the self-gravitating long-range interactions. Itis shown that the power-law distributions describe the long-range nature of theinteractions and the non-local correlations within the self-gravitating systemwith the inhomogeneous velocity dispersion. A relation is established betweenthe nonextensive parameter q and the measurable quantities of theself-gravitating system: the velocity dispersion and the mass density.Correspondingly, the nonextensive parameter q can be uniquely determined fromthe microscopic dynamical equation and thus the physical interpretation of qdifferent from unity can be clearly presented. We derive a nonlineardifferential equation for the radial density dependence of the self-gravitatingsystem with the inhomogeneous velocity dispersion, which can correctly describethe density distribution for the dark matter in the above physical situation.We also apply this q-kinetic approach to analyze the nonextensivity ofself-gravitating collisionless systems and self-gravitating gaseous dynamicalsystems, giving the power-law distributions the clear physical meaning.Comment: 21 pages,45 reference

Nonextensivity and the power-law distributions for the systems with self-gravitating long-range interactions