Bottlenecks phase-space structure

The Davis-Gray theory teaches us that by retaining the most important elements of the nonhnear reaction dynamics it is possible to accurately locate the intramolecular bottlenecks and to have an exact phase space separatrix as the transition state. Unfortunately, even for systems with only two DOFs, there may be considerable technical difficulties associated with locating the exact bottlenecks and the separatrix. Exact calculations of the fluxes across these phase space structures present more problems. For these reasons, further development of unimolecular reaction rate theory requires useful approximations. 

The fact that classical unstable periodic trajectories can manifest themselves in the Wigner function implies that nonstatistical behavior in the quanmm dynamics can be intimately related to the phase-space structure of the classical molecular dynamics. Consider, for example, the bottlenecks to intramolecular energy flow. Since the intramolecular bottlenecks are caused by remnants of the most robust tori, they are presumably related to the least unstable periodic trajectories. Hence quantum scars, being most significant in the case of the least unstable periodic trajectories, are expected to be more or less connected with intramolecular bottlenecks. Indeed, this observation motivated a recent proposal [75] to semiclassically locate quantum intramolecular bottlenecks. Specifically, the most robust intramolecular bottlenecks are associated with the least unstable periodic trajectories for which Eq. (332) holds, that is,... 

A complete model for the non-ergodic classical dynamics of a polyatomic molecule will need to represent the complete Arnold web structure of the phase space. There may be multiple bottlenecks for IVR and vague tori may exist in the vicinity of invariant tori. These complex phase space structures, leading to non-ergodic dynamics, are the origins of the... 

In research similar to that described in section 4.3.1, phase space structures and phase space bottlenecks have been used to analyze unimolecular reaction dynamics (Davis and Gray, 1986 Gray et al., 1986b Gray and Rice, 1987 Zhao and Rice, 1992 Jain et al., 1993 DeLeon, 1992a,b Davis and Skodje, 1992). Important phase space structural properties are illustrated in figure 8.11, for the one-dimensional pendulum Hamiltonian (Lichtenberg and Lieberman, 1991) ... 

It is ensured that the NHIMs, if they exist, survive under arbitrary perturbation to maintain the property that the stretching and contraction rates under the linearized dynamics transverse to dominate those tangent to In practice, we could compute the only approximately with a finite-order perturbative calculation. Therefore, the robustness of the NHIM against perturbation (referred as to structurally stable [21,53]) is expected to provide us with one of the most appropriate descriptions of a phase-space bottleneck of reactions, if such an approximation of the Ji due to a finite order of the perturbative calculation can be regarded as a perturbation. One of the questions arising is, How can the NHIMs composed of a reacting system in solutions survive under the influence of solvent molecules (This is closely relevant to the subject of how the system and bath should be identified in many-body systems.)... 

Hierarchical structures in the timescales of IVR imply existence of intramolecular bottlenecks, which result in breakdown of ergodicity. In other words, the phase space consists of several regions that are connected by narrow pathways. Moreover, these regions themselves would consist of hierarchical structures of their own. As we will discuss in Section V.C, the network of nonlinear resonances provides one possibility for such hierarchical structures of phase space. Then, the existence of hierarchy in timescales of IVR would not be specific to acetylene. It can be quite generic under the condition that ergodicity of IVR does not hold. 

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