We study the foundation and limitations of the statistical reaction theory. In particular, we focus our attention to the question of whether the rate constant can be defined for nonergodic systems. Based on the analysis of the Arnold web in the reactant well, we show that the survival probability exhibits two types of behavior: one where it depends on the residential time as the power-law decay and the other where it decays exponentially. The power-law decay casts a doubt on definability of the rate constant for nonergodic systems. We indicate that existence of the two types of behavior comes from sub-diffusive motions in remote regions from resonance overlap. Moreover, based on analysis of nonstationary features of trajectories, we can understand how the normally hyperbolic invariant manifold (NHIM) is connected with the Arnold web. We propose that the following two features play a key role in understanding the reactions where ergodicity is broken, i.e., whether the Arnold web is nonuniform and how the NHIM is connected with the Arnold web.
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