'The Nonlinear Dynamical Foundations of Transition State Theory in Systems with Three or More Degrees-of-Freedom'

数学科学：具有三个或更多自由度的系统中过渡状态理论的非线性动力学基础wiggins这个研究项目是关于理解具有三个或更多自由度的系统中过渡状态理论的非线性动力学基础。虽然对于两个自由度的系统在这些方面已经取得了很大的进展，但是对于具有三个或更多自由度的系统，几乎没有类似性质的进展。动力系统理论的几何观点为发现控制反应速率和分子能量流的高维相空间结构类型提供了一个框架。我们将把这种方法应用于几个问题：交叉电场和磁场中的三维里德伯原子，伦敦-埃灵-波兰反佐藤势，甲醛中的旋转-振动相互作用，以及羰基硫化物（OCS）中共振的能量流研究。在每种情况下，我们都将展示高维相空间结构作为“过渡态”。事实上，我们甚至可以将周期轨道分面这个非常有用的概念推广到具有三个或更多自由度的系统。我们的研究也将集中在开发计算方法来实现这些几何结构在这些具体问题。我们也将开发一个基于小波的频率图分析工具来研究共振中的能量流动。这项工作将与化学家和物理学家以及计算科学家合作进行。这个项目研究有关化学反应的基本问题。潜在的数学问题涉及具有三个或更多自由度的系统中过渡状态理论的非线性动力学基础。这是预测化学反应动力学的基本理论。对于简单的低维系统，这个理论已经有了相当完整的理解，但对于更现实的高维系统，还没有类似的理论框架。这样的理论将涉及对大于4维的曲面的数学描述，并将需要开发计算工具来实现和可视化这样的曲面。这将导致对分子如何分解和原子如何结合形成分子的动力学的详细理解。这项研究是跨学科的，涉及化学家、物理学家和计算科学家的合作。

NSF Award Abstract - DMS-0071338Mathematical Sciences: The Nonlinear Dynamical Foundations of Transition State Theory in Systems with Three or More Degrees of FreedomAbstract0071338 WigginsThis research project is concerned with understanding the nonlinear dynamical foundations of transition state theory in systems with three or more degrees of freedom. While there has been much progress along these lines for two degree-of-freedom systems, there has been little progress of a similar nature for systems with three or more degrees-of-freedom. The geometrical point of view of dynamical systems theory offers a framework for discovering the types of higher dimensional phase space structures that govern reaction rates and energy flow in molecules. We will apply this approach to several problems: three dimensional Rydberg atoms in crossed electric and magnetic fields, the London-Eyring-Polanti-Sato potential, rotation-vibration interaction in formaldehyde, and a study of energy flow through resonances in carbonyl sulphide (OCS). In each case we will show that there are higher dimensional phase space structures that act as a "transition state." Indeed, it may even be possible to generalize the very useful idea of a periodic orbit dividing surface to systems with three or more degrees of freedom. Our research will also focus on developing computational approaches to realize these geometric structures in these specific problems. We will also develop a wavelet-based frequency map analysis tool for studying energy flow in resonances. This work will be carried out in collaboration with chemists and physicists, as well as computational scientists.This project investigates fundamental questions concerning chemical reactions. The underlying mathematical questions involve the nonlinear dynamical foundations of transition state theory in systems with three or more degrees of freedom. This is the fundamental theory that allows predictions concerning chemical reaction dynamics. While there is a fairly complete understanding of this theory for simple, low dimensional, systems, there is as yet no analogous theoretical framework for more physically realistic high dimensional systems. Such a theory will involve mathematical description of surfaces in dimensions larger than four and will require development of computational tools for realizing and visualizing such surfaces. This will result in a detailed understanding of the dynamics of how molecules break up and of how atoms combine to form molecules. The research is interdisciplinary, involving the collaboration of chemists, physicists, and computational scientists.