Phase space structure dynamics

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

The KAM theorem demonstrates the existence of KAM tori when the perturbations to the motion are small. What happens when a nearly integrable Hamiltonian is strongly perturbed For example, with increasing perturbation strength, what is the last KAM torus to be destroyed and how should we characterize the phase space structures when all KAM tori are destroyed Using simple dynamical mapping systems, which can be regarded as Poincare maps in Hamiltonian systems with two DOFs, MacKay, Meiss, and Percival [8,9] and Bensimon and Kadanoff [10] showed that the most robust KAM curve... 

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 phase space structure of classical molecular dynamics is extensively used in developing classical reaction rate theory. If the quanmm reaction dynamics can also be viewed from a phase-space perspective, then a quantum reaction rate theory can use a significant amount of the classical language and the quantum-classical correspondence in reaction rate theory can be closely examined. This is indeed possible by use of, for example, the Wigner function approach. For simplicity let us consider a Hamiltonian system with only one DOF. Generalization to many-dimensional systems is straightforward. The Wigner function associated with a density operator /)( / is defined by... 

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,... 

In this section we will develop the phase-space structure for a broad class of n-DOF Hamiltonian systems that are appropriate for the study of reaction dynamics through a rank-one saddle. For this class of systems we will show that on the energy surface there is always a higher-dimensional version of a saddle (an NHIM [22]) with codimension one (i.e., with dimensionality one less than the energy surface) stable and unstable manifolds. Within a region bounded by the stable and unstable manifolds of the NHIM, we can construct the TS, which is a dynamical surface of no return for the trajectories. Our approach is algorithmic in nature in the sense that we provide a series of steps that can be carried out to locate the NHIM, its stable and unstable manifolds, and the TS, as well as describe all possible trajectories near it. 

The most illuminating consequence of multi-dimensional vibrational dynamics in polyatomic molecules are fluctuations of resonance widths. In particular, narrow resonances can often be found far above the first dissociation threshold. We have seen that in systems with one degree of freedom the sequence of resonance states is rather short. Since the excitation energy is deposited directly into the reaction coordinate, the complex breaks apart very quickly and the resonances become broad even close to the dissociation threshold. In polyatomic molecules, energy can be temporarily stored in additional degrees of freedom. The lifetime is then determined not only by the total energy, but also by the rate with which the excitation can be redistributed and transferred to the dissociation bond (see the discussion of the classical phase space structure in Sect. 8). 

It is not immediately obvious, by simply looking at a molecule s Hamiltonian and/or its PES, whether the unimolecular dynamics will be intrinsic RRKM or not and computer simulations as outlined here are required. Intrinsic non-RRKM dynamics is indicative of mode-specific decomposition, since different regions of phase space are not strongly coupled and a micro-canonical ensemble is not maintained during the fragmentation. The phase space structures, which give rise to intrinsic RRKM or non-RRKM behavior, are discussed in the next section. 

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... 

Thus the phase space structure of the classical dynamics depends not on field strength e and frequency co independently, but only on the scaled field strength defined by Eq. (7). Scaled energies E and scaled actions T derived from the equations of motion in the scaled variables are related to the energies E and actions / in atomic units by... 

It is natural to conceive that this short-time behavior should be due to some time interval for a trajectory to spend to look for exit ways to the next basins in the complicated structure of phase space. In the next section, we will propose a geometrical view that shows what this complexity is. Hence we consider that the hole of Na- b(t) in the short-time region should be a reflection of chaos, which is just opposite to the behavior arising from nonchaotic direct paths as observed in Hj" dynamics. The present effect is therefore expected to be more significant as the molecular size increases or the potential surface and corresponding phase-space structure become more complicated. Another important aspect of the hole in Na-,b t) is an induction time for a transport of the flow of trajectories in phase space. It is of no doubt that the RRKM theory does not take account of a finite speed for the transport of nonequilibrium phase flow from the mid-area of a basin to the transition states. Berblinger and Schlier [28] removed the contribution from the direct paths and equate the statistical part only to the RRKM rate. One should be able to do the same procedure to factor out the effect of the induction time due to transport. We believe that the transport in phase space is essentially important in a nonequilibrium rate theory and have reported a diffusion model to treat them [29]. 

In contrast to hyperbolic systems, the phase space structure in the mixed system is quite intricate and inhomogeneous, which brings about transport phenomena and relaxation processes essentially different from uniformly hyperbolic cases [3]. A remarkable fact is that qualitatively different classes of motions such as quasi-periodic motions on invariant tori and stochastic motions in chaotic seas coexist in a single phase space. The ordered motions associated with invariant tori are embedded in disordered motions in a self-similar way. The geometry of phase space then reflects the dynamics. 

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