New Aspects of non-thermal chaos and ergodic problem

具有多个自由度的保守系统(哈密顿系统)的动力学性质是非常有趣和重要的，不仅从非线性动力学的角度来看，而且作为统计物理的一个基本问题，在诸如化学反应的系统中，组分(分子)的动力学行为是重要的。在动力系统理论中，一个称为“Arnold扩散”的过程被认为是具有多个自由度的哈密顿系统的典型行为，是一个基本的松弛过程。关于阿诺德扩散，我们澄清了--阿诺德扩散确实存在--以前通过线性扰动分析得到的关于阿诺德扩散大小的估计是正确的--尽管它被称为“扩散”，但阿诺德扩散与一般的布朗运动驱动的扩散相反，具有很强的时间相关性。有了这些结果，可以说对阿诺德扩散的认识有了很大的进步。此外，我们的…进一步发现Arnold扩散具有很强的时间相关性，这意味着，对于化学反应系统和生物系统中的分子，从运动方程而不是从热涨落来重新研究动力学行为是很重要的。对于动力学结构的形成，我们引入了一个新的模型来进一步研究我们在质片模型中发现的分形结构的机制。在该模型中，两体相互作用势被推广到|r|^k的幂函数形式，其中r是两粒子之间的距离，k是正参数。参数“k”的不同取值导致了两体关联的不同行为，并且我们发现两体势是幂函数型的事实并不总是产生幂函数型关联的事实。2005年9月在早稻田大学举行的“第三届21COE研讨会：作为交叉学科的天体物理学”会议上，对动力学结构的形成和哈密顿动力学系统进行了综述。较少

Dynamical properties of conservative systems (Hamiltonian systems) with many degrees of freedom are quite interesting and important, not only from the viewpoint of nonlinear dynamics but also as a fundamental problem of statistical physics, and in systems such as in chemical reaction where dynamical behavior of components (molecules) is important. In dynamical systems theory a process called "Arnold diffusion" is expected as a typical behavior of Hamiltonian systems with many degrees of freedom and as a fundamental process of relaxation. About Arnold diffusion we have clarified-- the Arnold diffusion really exists-- the estimation about the magnitude of Arnold diffusion previously obtained from linear perturbation analysis is correct-- although it is called "diffusion", Arnold diffusion has strong temporal correlation, contrary to ordinary diffusion driven by Brownian motion.With these results it can be said that the knowledge about Arnold diffusion has progessed much.In addition, our … More finding that Arnold diffusion has strong temporal correlation implies that, in chemical reaction systems and molecules in bio systems, few degrees of systems, it is important to reinvestigate the dynamical behavior not from thermal fluctuation but from equation of motion.As for dynamical structure formation, we introduced a new model to investigate further the mechanism of fractal structure we had discovered in mass-sheet model. In the model two-body interaction potential is extended to power-law type of |r|^k, where r is the distance between two particles and "k" is a positive parameter. Different values of the parameter "k" give rise to different behavior in two-body correlation, and we found that the fact that two-body potential is power-law type does not always produce power-law type correlation.A general review talk about dynamical structure formation and Hamiltonian dynamical systems is presented as invited talk in a conference "The Third 21COE symposium : Astrophysics as Interdisciplinary Science" held at Waseda University in Sep.2005. Less