Geometric Structure of Multi-dimensional Chaos and Its Application to Reactions in Non-equilibrium

我们建立了基于正常双曲不变流形(NHIM)概念的反应动力学理论。这使我们能够在数学上为反应过程中的“过渡态”概念奠定坚实的基础。此外，它还提供了一种对反应的传统观点进行回顾的可能性。第一个问题是NHIM上的混沌导致正常双曲度破裂的可能性。请注意，正常的超高涡度意味着沿法向方向的不稳定性大于沿切向方向的不稳定性。然而，当NHIM上的混沌变得与法线方向的不稳定性相当时，正常双曲性处于崩溃的边缘。我们已经建立了一个模型系统，可以用Lie变换和Pade求和来分析这一现象。分解意味着，在反应的背景下，一组新的自由度开始贡献描述反应的集体自由度。这为研究反应过程开辟了一个新的视野。第二个成果是，当势垒中的动力学是非遍历的时，观察到了幂定律行为。我们建立了一个模型，其中井中的阿诺德腹板是稀疏的和非均匀的。驻留时间分布呈幂函数分布，井内动力学表现为反常扩散。这些现象表明，传统的反应速率常数概念已不复存在。反应过程的上述两个方面表明，反应动力学理论为理解反应提供了一种新的可能性。

We have formulated the dynamical theory of reactions based on the concept of normally hyperbolic invariant manifolds (NHIMs). This enables us to obtain a mathematically firm foundation of the concept of "transition states" in reaction processes. Moreover, it offers a possibility of going over the conventional ideas of reactions. The first is the possibility of breakdown of the normal hyperbolicity caused by chaos on the NHIM. Note that normal hyperbilicity means that instability along the normal directions is larger than that along the tangential directions. However, when chaos on the NHIM becomes comparable to the instability along the normal directions, the normal hyperbolicity is at the edge of breaking down. We have constructed a model system where this phenomenon can be analyzed using the Lie transformation and Pade summation. The breakdown means, in the context of reactions, that a new set of degrees of freedom start to contribute collective degrees of freedom describing reactions. This opens a new horizon in studying the reaction processes. The second achievement is to indicate that a power-law behavior is observed when the dynamics in the potential well is non-ergodic. We have constructed a model where the Arnold web in the well is sparse and non-uniform. The distribution of residence times exhibit a power-law properties and the dynamics in the well show anomalous diffusion. These phenomena indicate that the traditional concept of the reaction rate constat does not exists any more. The above two aspects of the reaction processes indicate that the dynamical theory of reactions offer a new possibility of understanding reactions.