Shiino Masatoshi
Department of Applied Physics, Faculty of Science, Tokyo Institute of Technology, 2-12-1 Ohokayama, Meguro-ku, Tokyo, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 May;67(5 Pt 2):056118. doi: 10.1103/PhysRevE.67.056118. Epub 2003 May 23.
Multidimensional nonlinear Fokker-Planck equations of mean-field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions. Two types of eigenvalue equations are studied. The nonlinear Fokker-Planck equations are shown to exhibit an H theorem with a Liapunov functional that takes the form of a free energy involving generalized entropies of Tsallis. The second-order variation of the Liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean-field coupling potential. Assuming quasiequilibrium for the velocity distribution, the reduced eigenvalue equation with space coordinates alone is also obtained. The alternative type of eigenvalue equation based on the linearization of the nonlinear Fokker-Planck equations is presented. Taking the mean-field coupling potential to be the gravitational one, the nonlinear Fokker-Planck equation in terms of three-dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean-field model of self-gravitating system. By solving the eigenvalue equation for the eigenfunction with 0 eigenvalue, the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed.
在广义统计力学框架内，提出了平均场类型的多维非线性福克 - 普朗克方程，以建立其解的稳定性分析的一般公式。研究了两种类型的特征值方程。结果表明，非线性福克 - 普朗克方程表现出一个具有李雅普诺夫泛函的H定理，该泛函具有涉及Tsallis广义熵的自由能形式。计算李雅普诺夫泛函的二阶变分以进行局部稳定性分析，并针对任意形式的平均场耦合势导出相关的特征值方程。假设速度分布为准平衡态，还得到了仅含空间坐标的简化特征值方程。给出了基于非线性福克 - 普朗克方程线性化的另一种类型的特征值方程。将平均场耦合势取为引力势，表明包含三维速度和空间坐标的非线性福克 - 普朗克方程以及稳定性分析框架适用于自引力系统的平均场模型。通过求解具有零特征值的特征函数的特征值方程，讨论了具有球对称性的平衡概率密度稳定性变化的发生情况。
