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

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.