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

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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.