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

1. 研究了液态水在室温下的慢弛豫动力学，并提出了一个一致的解释，解释了为什么液态水动力学表现出幂律弛豫行为，并在相空间中形成瓶颈，即使它是一个多维和强混沌系统。历史上，人们一直认为液态水动力学是典型的非潜在快速弛豫。这确实是在实验中观察到的，并在此基础上提出了理论论点。然而，Sasai等人对液态水进行了长时间的微规范MD模拟，发现在室温下总势能波动表现为1/f型光谱，而更简单的液体如氩气则表现出接近白色的光谱。我们的想法是受到哈密顿系统微扰理论最近发展的启发，并使人想起所谓的玻尔兹曼-金斯猜想。在这种情况下，很自然地期望显示松弛…更多的不限于液态水动力学。我们发现我们的假设在预测其他分子的弛豫特性方面很有效。与潜在景观画面的关系也进行了讨论。研究了矩形盒子中的两个硬球作为最小模型，以观察玻璃态或过冷态的慢动力学。严格地证明了盒子内的两个硬球具有混合性和遍历性，并通过数值计算证明了在适当的条件下可以观察到两步相关曲线。在这里，我们特别关注了模型中的两种类型的周期轨道：(1)弹跳球模式，这是一种单参数周期轨道族，在单粒子台球模型中引起缓慢弛豫；(2)存在于两个硬球之间的瓶颈处的周期轨道。后者可以看作是所谓的通常双曲不变流形，在高维空间中起过渡态的作用。大量的数值模拟表明，相关函数呈两步分布，但都是指数型的，因此在这样一个小系统中不会发生拉伸型相关。此外，随着自由度的增加，弹跳球模式的影响逐渐减弱。少

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