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

1.研究了室温下液态水动力学中的慢弛豫现象，对液态水动力学中的幂律弛豫行为和相空间瓶颈现象给出了一致的解释，尽管液态水动力学是一个多维的强混沌系统.历史上，人们一直认为液态水动力学是一种典型的外势快弛豫。然而，Sasai等人对液态水进行了长时间的微正则分子动力学模拟，发现在室温下总势能涨落呈现1/f型谱，而较简单的液体如氩则呈现近白色谱。我们的想法受到了最近发展的哈密顿系统微扰理论的启发，这让人想起了所谓的玻尔兹曼-金斯猜想。在这种情况下，自然会期望显示放松， 关于我们 不仅限于液态水动力学。我们发现我们的假设在预测其他分子的弛豫性质方面效果很好。并讨论了与潜在景观图的关系.以矩形盒中的两个硬球为最小模型，研究了玻璃态和过冷态下的慢动力学。严格地证明了盒中两个硬球具有混合性和遍历性，并在适当的条件下观察到两步关联分布。在这里，特别注意两种类型的周期轨道的模型：（1）弹跳球模式，这是一个参数家庭的周期轨道，导致慢松弛的单粒子台球模型，和（2）周期轨道存在于两个硬球之间的瓶颈。后者可以看作是高维空间中起过渡态作用的所谓的常双曲不变流形。大量的数值模拟表明，相关函数显示两个步骤的轮廓，但都是指数，因此没有拉伸型相关不会发生在这样一个小的系统。此外，还发现随着自由度的增加，弹跳模态的影响逐渐减弱。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