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

我们在通常双曲不变流形（NHIMs）概念的基础上建立了反应的动力学理论。这使我们能够获得反应过程中“过渡态”概念的坚实数学基础。此外，它还提供了一种复习传统反应观念的可能性。首先是NHIM上的混沌导致正常双曲度崩溃的可能性。注意，法向高双性意味着沿法向的不稳定性大于沿切向的不稳定性。然而，当NHIM上的混沌与沿法向的不稳定性相当时，法向双曲性处于崩溃的边缘。我们建立了一个模型系统，在这个模型系统中，这种现象可以用李变换和帕德求和来分析。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新的视野。第二个成果是表明，当势阱中的动力学是非遍历的时，可以观察到幂律行为。我们建立了一个模型，其中井中的阿诺德网是稀疏且不均匀的。停留时间的分布表现为幂律性质，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，反应动力学理论为认识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.