Dynamical Evolution of Self-Gravitating Systems

轨道平均Fokker-Planck方程是研究碰撞恒星动力学的基本方程。然而，当自变量个数等于或大于2时，轨道平均的Fokker-Planck方程很难积分。因此，期望获得近似解。变分原理对于得到这样的解是有用的。由于轨道平均的Fokker-Planck方程是非线性和非自邻接的，普通的变分原理不再适用。我们利用Glansdorf和Prigogine提出的局域势，导出了一个变分原理，Itoh等（1988）研究了当星系初始分布为Poisson分布时星系的引力成团。在这一阶段的研究中，我们调查的情况下，星系的初始分布具有幂律。我们发现f（V）中出现的B与样品体积V有关，得到了B（V）与两粒子关联函数之间的解析关系，模拟结果与之一致。为了简单起见，我们采用两分量模型，其中质量较小的星系的质量为m_1，质量较大的星系的质量为m_2。我们发现当m_2/m_1 < 10时，Saslaw和汉密尔顿（1984）导出的f（N）是再生的.

Orbit-averaged Fokker-Planck equations are the fundamental evolution equation for collisional stellar dynamics. However, it is difficult to integrate the orbit-averaged Fokker-Planck equations when the number of the independent variables is equal or larger than two. Therefor, it is desirable to obtain approximate solutions. Variational Principles are useful to get such solutions. Since the orbit-averaged Fokker-Planck equations are nonlinear and non-self-adjoining, ordinary variational principles are out of use. We used the local potential proposed by Glansdorf and Prigogine and derived a variational principle.In Itoh et al. (1988) we studied the gravitational clustering of galaxies when the initial distribution of galaxies is Poisson distribution. In this period of the research, we investigated the case when the initial distribution of galaxies has power laws. We find that b appearing in f(V) depend on the volume of the sample, V. The relation between b(V) and the two-particle correlation function is obtained analytically and the results of the simulations are in agreement with it.We further investigated the effects of different mass components. For simplicity we adopted two component model where the mass of the less massive galaxy is m_1 and that of the massive galaxy is m_2. We find that f(N) derived by Saslaw and Hamilton (1984) is reproduced when m_2/m_1 < 10.