By Schaaf W.L.
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Extra resources for A bibliography of recreational mathematics (5 vols.)
It shows three major quasi-periodic regions, the resonance islands, the location of the dividing surface for intramolecular energy transfer, and a typical turnstile. [From M. J. Davis, J. Chem. Phys. ] The ﬁrst application of transport bottlenecks to a molecular system was in the study of vibrational relaxation of a model OCS molecule with two DOFs. Davis  found that the correct location of the bottleneck requires ﬁnding a set of unstable orbits that have a set of resonance conditions oCO 3 5 8 13 21 34 ; ; ;...
1. If the frequency ratio, or the winding number o1 =o2, is a rational number, the two DOFs are in resonance and an individual trajectory will close on itself on the torus. By contrast, if o1 =o2 is an irrational number, then as time evolves a single trajectory will eventually cover the torus. The motion in the latter case is called conditionally periodic. Kolmogoroff, Arnold, and Moser (KAM) established a theorem regarding the qualitative features of a perturbed Hamiltonian system. The KAM theorem states that, under small perturbation and for an analytical H1 ðJ1 ; J2 ; f1 ; f2 Þ in a Figure 1.
The separatrix deﬁnes the boundary in phase space between products and reactants—for example, the boundary between bounded motion and unbounded motion in a fragmentation reaction. Davis and Gray also incorporated the bottlenecks to intramolecular energy transfer into their reaction rate theory. This modiﬁed statistical theory accurately predicts, for 6 meishan zhao et al. example, the decay rate of a simpliﬁed classical model of He-I2 ; that decay rate is overestimated by conventional RRKM theory by at least an order of magnitude.