Isomerization dynamics and the transition to chaos

Microcanonical rates, gap times, and phase space dividing surfaces

“…Phase points in the crossing region M A C do, however, eventually cross the dividing surface, and so lie on trajectories that contribute to the reactive flux. In general, however, as a consequence of the existence of trapped trajectories ͑e.g., trajectories on invariant trapped n-tori 97,98 or trajectories asymptotic to other invariant objects of zero measure͒, we have the inequality 32,42,98…”

“…Apart from a set of measure zero, all phase points x M A can be classified as either trapped ͑T͒ or crossing ͑C͒. 98 ͑Further discussion of this division of the reactant phase space in terms of the Poincaré recurrence theorem is given in Appendix A.͒ A phase point in the trapped region M A T never crosses the DS so that the associated trajectory does not contribute to the reactive flux. Phase points in the crossing region M A C do, however, eventually cross the dividing surface, and so lie on trajectories that contribute to the reactive flux.…”

“…Result ͑2.7͒ is essentially the content of the socalled classical spectral theorem. 55,56,156,157,184 If all points in the reactant region of phase space eventually react ͑that is, all points lie on crossing trajectories 97,98 ͒ then A C ͑E͒ = A ͑E͒, the full reactant phase space density of states. Apart from a set of measure zero, all phase points x M A can be classified as either trapped ͑T͒ or crossing ͑C͒.…”

“…Provided that the flow is everywhere transverse to DS in,out ͑E͒, those phase points in the reactant region M A that lie on crossing trajectories 97,98 ͑i.e., that will react, and so are "interesting" in Slater's terminology 10 ͒ can be specified uniquely by coordinates ͑q , p , ͒, where ͑q , p͒ DS in ͑E͒ is a point on DS in ͑E͒, the incoming half of the DS, specified by 2͑n −1͒ coordinates ͑q , p͒, and is a time variable. ͑Divid-ing surfaces constructed by the NF algorithm are guaranteed to be transverse to the vector field, except at the NHIM, where the vector field is tangent.…”

“…94 ͒ Correlations between features of the classical phase space structure and the behavior of the computed reactive flux 95,96 were explored relatively early on for two DoF systems by DeLeon and co-workers. 97,98 Important theoretical advances were subsequently made relating nonstatistical behavior to molecular phase space structure, 99 in particular, the existence of intramolecular bottlenecks to energy transfer, [100][101][102] "vague tori," 103,104 broken separatrices, [105][106][107] and reactive islands and cylinders. [108][109][110][111][112][113][114] The multimode case remains less thoroughly explored and understood.…”