Invariant manifolds, chemical reaction

Up to moderately high energy ( 179%) of the activation barrier for reactant product in the Are isomerization reaction, the fates of most trajectories can be predicted more accurately by Eq. (11) as the order of perturbation calculation increases, except just in the vicinity of the (approximate) stable invariant manifolds (e.g., see Eig. 5), and that the transmission coefficient K observed in the configurational space can also be reproduced by the dynamical propensity rule without any elaborate trajectory calculation (see Eig. 6). Our findings indicate that almost all observed deviations from unity of the conventional transmission coefficient k may be due to the choice of the reaction coordinate whenever the k arises from the recrossings, and most transitions in chemical... 

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. 

The special form of the invariant submanifold li(k) in Eq. (2.3) is due to the explicit existence of fast-reacting radical species. Well known (Maas and Pope [1], Lam [2]) is the existence of invariant manifolds also in reactions systems without simple separation of radical (fast) and nonradical (slow) chemical species, but with a complex hierarchy of reactions at very different time scales. In this situation we can at least suppose the existence of an invariant submanifold U C in concentration space representing the slow... 

Gorban, A.N., Karlin, I.V., 2005. Invariant Manifolds for Physical and Chemical Kinetics. Springer, New York. Gorban, A.N., Radulescu, O., 2008. Dynamic and static limitation in multiscale reaction networks, revisited. 

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