Construction of Non-equilibrium and Non-extensive thermostatistics for Self-gravitating Systems

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It should be emphasized that gravity has an essential role in astrophysics. Since the gravitational energy is proportional to the square of mass of objects, it dominates other additive energies in cases of massive astrophysical objects. We investigate evolution of the stellar system, e.g. globular clusters, which can be recognized as the self-gravitating N-body system. We try to apply a framework of non-equilibrium and non-extensive thermostatistics proposed by Tsallis in order to analyze quasi-stationary evolution of self-gravitating N-body systems.We can summarize our research by three accomplishments. At first, we could generalize a famous notion of gravothermal catastrophe, which is crucially based on Boltzmann-Gibbs (B-G) entropy. By means of extending the framework for stability analysis to use Tsallis entropy, we were able to show that polytrope, which is maximal entropic state, can describe much wider class of states of self-gravitating systems. The polytrope has one index n and coincides with B-G equilibrium distribution in the limit n→infinity. In order to investigate time evolution of the polytrope, we performed N-body numerical simulation. We observed that polytrope evolves along a sequence of polytrope itself with increasing index n. This is our second interesting result that sequences of polytropes can represents quasi-stationary evolution of the stellar systems.Final our accomplishment is concerning to time evolution of the polytrope. Based on Fokker-Planck equation for stellar systems and a framework of generalized variational principle proposed by Prof.Prigodine, we succeeded to derive an integro-differential equation for index n, which nicely predicts quasi-stationary evolution of self-gravitating N-body systems.