Saddle regions phase-space transition states

In contrast, conventional reaction rate theory replaces the dynamics within the potential well by fluctuations at equilibrium. This replacement is made possible by the assumption of local equilibrium, in which the characteristic time scale of vibrational relaxation is supposed to be much shorter than that of reaction. Furthermore, it is supposed that the phase space within the potential well is uniformly covered by chaotic motions. Thus, only information concerning the saddle regions of the potential is taken into account in considering the reaction dynamics. This approach is called the transition state theory. 

The validity of the transition state concept that separates the phase space into the two distinct regions has not been clarified in cases in which the system passes through higher-rank saddles at least as frequently as through the lowest, first-rank, saddle. 

The reaction bottleneck or the transition state is located somewhere in the vicinity of the first-rank saddle, and is not delocalized through the whole region of configurational (or phase) space. [Note, however, that a delocalized, rigorous transition state certainly exists for some situations (e.g., Refs. 10-12).]... 

In addition to the transition state, the theory on the dynamics in the saddle region has elucidated other important objects in the phase space. An invariant manifold is a set of points in the phase space such that any point in that set will remain in that set perpetually during the time evolution. As a consequence, if an invariant manifold divides the whole phase space into two disjoint regions, the system can never cross the invariant manifold from one region to the other. 

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