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

应当强调的是，引力在天体物理学中具有重要作用。由于引力能与物体质量的平方成正比，在大质量天体的情况下，它支配着其他加性能量。我们研究了恒星系统的演化，例如球状星团，它可以被认为是自引力N体系统。我们尝试应用Tsallis提出的非平衡非广延热统计框架来分析自引力N体系统的准定态演化。首先，我们可以推广一个著名的重力热突变概念，它的关键是基于玻尔兹曼-吉布斯（B-G）熵。通过扩展Tsallis熵的稳定性分析框架，我们能够证明，最大熵的polytrope状态，可以描述更广泛的自引力系统的状态。该多面体有一个指数n，在极限n→无穷大时符合B-G平衡分布。为了研究多面体的时间演化，我们进行了N体数值模拟。我们观察到，随着指数n的增加，多变性沿着一系列多变性本身演化。这是我们的第二个有趣的结果，即多面体序列可以代表恒星系统的准定常演化，最后我们的成果是关于多面体的时间演化。基于恒星系统的Fokker-Planck方程和Prigodine教授提出的广义变分原理框架，我们成功地导出了一个指数为n的积分-微分方程，它很好地预言了自引力N体系统的准定态演化。

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.