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

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Wiggins这个研究项目关注的是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。 虽然对于两个自由度的系统，沿着这些路线已经有了很大的进展沿着，但是对于具有三个或更多自由度的系统，类似性质的进展很少。 动力系统理论的几何观点提供了一个框架，用于发现控制分子中反应速率和能量流的高维相空间结构的类型。 我们将这种方法应用于几个问题：三维里德堡原子在交叉电场和磁场，伦敦-Eyring-Polanti-Sato势，在甲醛中的旋转振动相互作用，并通过共振在硫化羰（OCS）的能量流的研究。 在每一种情况下，我们将表明，有更高维的相空间结构，作为一个“过渡态”。“事实上，甚至有可能将周期性轨道划分表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在开发计算方法，以实现这些特定问题中的这些几何结构。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。本研究将与化学家、物理学家以及计算科学家合作进行。本研究项目研究有关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中过渡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。 虽然对于简单的低维系统，这个理论已经有了相当完整的理解，但对于更物理现实的高维系统，还没有类似的理论框架。 这样的理论将涉及数学描述的表面尺寸大于四，并将需要开发计算工具，实现和可视化这样的表面。 这将导致对分子如何分裂以及原子如何联合收割机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.