New Aspects of non-thermal chaos and ergodic problem

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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