Quasiperiodic trapping

In this spirit, we will also briefly describe the basis for some of the microscopic kinetic theories of unimolecular reaction rates that have arisen from nonlinear dynamics. Unlike the classical versions of Rice-Ramsperger-Kassel-Marcus (RRKM) theory and transition state theory, these theories explicitly take into account nonstatistical dynamical effects such as barrier recrossing, quasiperiodic trapping (both of which generally slow down the reaction rate), and other interesting effects. The implications for quantum dynamics are currently an active area of investigation. 

The second EOF that describes 14.2% of the total dispersion represents an alongshore quasiperiodical structure with a wavelength of 300-400 km and coastal trapping of amplitudes, which decreases with the distance from the coast. The annual cycle of the variability of the second EOF coefficient (curve 2 in Fig. 9d) is shifted by a quarter of the period with respect to the first EOF. This mode of the main pycnocline variability should be most clearly manifested in the summer in the salinity fields at a depth of 100 m. 

When classical trajectories are calculated for the H2O model with two O—H stretches discussed in section 4.3.2 (p. 76), local-mode type motion is found for which energy is trapped in individual O—H bonds (Lawton and Child, 1979, 1981). The trajectories are quasiperiodic and application of the EBK semiclassical quantization condition [Eq. (2.72)] results in pairs of local-mode states in which there are n quanta in one bond and m in the other or vice versa. The pair of local-mode states (n,m) and (m,n) have symmetry-related trajectories which have the same energy. The local-mode trajectory for the (5,0) state is depicted in Figure 4.6d. 

As discussed above (section 4.3.2) these local-mode states are not the eigenstates for the system, but superposition states. However, since the classical motion is quasiperiodic for these local mode states (i.e., the state is a torus in the phase space), the system is trapped in the initially excited local-mode state and the quantum periodic oscillation between the n,m) and (m,n) local mode states is not observed classically. Thus, classical mechanics severely underestimates the rate of energy transfer. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

Figure 12 (Top) two trapped quasiperiodic orbits at fixed energy superimposed on potential energy contours for the De Leon-Berne Hamiltonian at = 0.65 (see Figure 23 for the potential energy surface). (Below) The Poincare map for this system at this energy, with the surface of section defined at fixed <]2 with p2 > 0. The trajectories above are connected with their corresponding map locations below. Note that whereas these relatively regular motions lie on tori, most of the Poincare map is broken up, indicating that most motions at this energy are chaotic. Reprinted with permission from Ref. 119.

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