Dynamical origin of hierachial time scale in many degrees of freedom systems Hamitonian systems with internal degreees of freedom

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1. Slow relaxation in dynamics of liquid water at room temperatures is studied and a consistent interpretation explaining why liquid water dynamics exhibits power-law relaxation behavior and can form the bottleneck in phase space is proposed even though it is a many dimensional and strongly chaotic system. Historically, it had long been believed that the liquid water dynamics is a typical expotentially fast relaxation. This was indeed observed experimentally, and theoretical arguments were made based on it. However, Sasai et al performed a long-time microcanonical MD simulations of liquid water and discovered that the total potential energy flunctuation shows 1/f-type spectrum in room temperatures, whereas simpler liquids such as argon exhibit nearly white spectra.Our idea is inspired by recent developments of perturbation theories of Hamiltonian systems, and is reminiscent of the so-called Boltzmann-Jeans conjecture. Within this scenario, it is natrual to expect that show relaxation i … More s not limited to liquid water dynamics. We found the our hypothesis works well in predicting the relaxation properties of other molecules. Relation to the potential landscape picture is also discussed.2. Two hard spheres in a rectangular box was investigated as a minimal model to see the slow dynamics in glassy or supercooled states. It was rigorously shown that two hard spheres in a box has mixing property and so ergodicity, and also numerically shown that two-step correlation profile is observed in a suitable condition. Here, a particular attention is paid on two types of periodic orbits in the model: (1) bouncing ball mode, which is a one parameter family of periodic orbits that cause slow relaxation in a single particle billiard model, and (2) periodic orbits that exist in the bottleneck between two hard balls. The latter can be regarded as the so-called normally hyperbolic invariant manifold that play the role of transition state in higher dimensional space. Extensive numerical simulations reveal that correlation functions show two-step profile, but both are exponential and thus no stretched type correlation does not occur in such a small system. Furthermore, it was found that the influence of bouncing ball mode is weaken as the degrees of freedom increases. Less