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

NSF奖摘要-DMS-0071338数学科学：三个或更多自由度系统中过渡态理论的非线性动力学基础这个研究项目致力于理解三个或更多自由度系统中过渡态理论的非线性动力学基础。虽然两自由度系统在这些方面取得了很大进展，但对于三自由度或更多自由度系统，几乎没有类似性质的进展。动力学系统理论的几何观点为发现控制分子中反应速率和能量流动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Rydberg原子，London-Eyring-Polanti-Sato势，甲醛中的旋转-振动相互作用，以及在羰基硫化物(OCS)中通过共振的能量流动的研究。在每种情况下，我们都将证明存在起“过渡态”作用的高维相空间结构。事实上，甚至有可能将周期轨道分割面的非常有用的想法推广到具有三个或更多自由度的系统。我们的研究还将集中于开发计算方法来实现这些特定问题中的几何结构。我们还将开发一个基于小波的频率图分析工具，用于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。这个项目研究关于化学反应的基本问题。基本的数学问题涉及三个或更多自由度系统的过渡态理论的非线性动力学基础。这是关于化学反应动力学预测的基本理论。虽然对于简单、低维系统的这一理论已经有了相当完整的理解，但对于更接近物理现实的高维系统，目前还没有类似的理论框架。这种理论将涉及对维度大于四的表面进行数学描述，并需要开发实现这种表面并使其可视化的计算工具。这将导致对分子如何分解的动力学以及原子如何结合形成分子的详细了解。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.