Stable/unstable manifolds phase-space transition states

In statistical reaction rate theory, the concept of transition state plays a key role. Transition states are supposed to be the boundaries between reactants and products. However, the precise formulation of the transition state as a dividing surface is only possible when we consider transition states in phase space. This is the place where the concepts of normally hyperbolic invariant manifolds (NHIMs) and their stable and unstable manifolds come into play. 

This enables us to extract and visualize the stable and unstable invariant manifolds along the reaction coordinate in the phase space, to and from the hyperbolic point of the transition state of a many-body nonlinear system. PJ AJI", Pj, Qi, t) and PJ AJ , Pi, q, t) shown in Figure 2.13 can tell us how the system distributes in the two-dimensional (Pi(p,q), qi(p,q)) and PuQi) spaces while it retains its local, approximate invariant of action Jj (p, q) for a certain locality, AJ = 0.05 and z > 0.5, in the vicinity of... 

In chemical terms, normally hyperbolic invariant manifolds play the role of an extension of the concept of transition states. The reason why it is an extension is as follows. As already explained, transition states in the traditional sense are regarded as normally hyperbolic invariant manifolds in phase space. In addition to them, those saddle points with more than two unstable directions can be considered as normally hyperbolic invariant manifolds. Such saddle points are shown to play an important role in the dynamical phase transition of clusters [14]. Furthermore, as is already mentioned, a normally hyperbolic invariant manifold with unstable degrees of freedom along its tangential directions can be constructed as far as instability of its normal directions is stronger than its tangential ones. For either of the above cases, the reaction paths in the phase space correspond to the normal directions of these manifolds and constitute their stable or unstable manifolds. 

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