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Edited by means of Nobel Prize winner Ilya Prigogine and well known authority Stuart A. Rice, the Advances in Chemical Physics sequence offers a discussion board for serious, authoritative reviews in each region of the self-discipline. In a layout that encourages the expression of person issues of view, specialists within the box current accomplished analyses of matters of curiosity. Advances in Chemical Physics continues to be the most suitable venue for shows of recent findings in its box.
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Extra info for Geometric Structures of Phase Space in Multidimensional Chaos Applications to Chemical Reaction Dynamics in Complex Systems. (Advances in Chemical Physics, Volume 130, Part A)
Sample text
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 [36] 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 ; ; ;... ¼ ; ; ; oCS 1 2 3 5 8 13 ð72Þ This series of resonance conditions converges to 2:618 . , implying that the most important bottleneck to intramolecular energy transfer is determined by a golden mean cantorus, that is, oCO ¼ ð2 þ gÞoCS ¼ 2:618 .
With stronger perturbations, more KAM tori are expected to be destroyed and therefore more trajectories become chaotic. E. Poincare´ Surface of Section The classical dynamics of a system can also be analyzed on the so-called Poincare´ surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare´ map on a surface of section. One important property of the Poincare´ map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare´ showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS.
An invariant measure, classical, semiclassical, and quantum mechanical rate theory 11 denoted by mi, has the property that it gives the same measure for VðtÞ and V 0 ðt0 Þ. That is, mi ½VðtÞ ¼ mi ½V0 ðt0 Þ ð11Þ Because the transformation from ½qðtÞ; pðtÞ to ½qðt0 Þ; pðt0 Þ is canonical, an almost trivial example of invariant measure in its differential form is given by dmi ¼ dq dp ð12Þ The energy of a time-independent Hamiltonian system is a conserved quantity. In this case an invariant measure can be constructed on the energy shell Hðq; pÞ ¼ E, that is, dmi ¼ NdðH À EÞdp dq ð13Þ where N is a normalization constant.