Multidimensional phase space

The sin7o term reflects the fact that the intramolecular vector r is expressed in polar coordinates it is especially important for the dissociation of linear molecules and must not be forgotten The delta-function S(Hf — Ef) selects only those points (i.e., trajectories) in the multidimensional phase-space that have the correct energy Ef. It ensures that the quantum mechanical resonance condition Ef = Ei + Ephoton is fulfilled. 

Dynamical systems may be conveniently analyzed by means of a multidimensional phase space, in which to any state of the system corresponds a point. Therefore, to any motion of a system corresponds an orbit or trajectory. The trajectory represents the history of the dynamic system. For one-dimensional linear systems, as in the case of the harmonic series-resonance circuit, described by the differential equation... 

This result is easily transferred to the multidimensional phase space. The unit vector, normal to the differential surface element ds, is given by... 

The trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

Equation (369) indicates that to obtain the semiclassical reaction rate constant k T) one needs to carry out the multidimensional phase-space average for a sufficiently long time. This is far from trivial, since the integrand in Eq. (369) is highly oscillatory due to quantum interference effects between the sampling classical trajectories. The use of some filtering methods to dampen the oscillations in the integrand may improve the accuracy of the semiclassical calculation. 

GLOBAL ASPECTS OF CHEMICAL REACTIONS IN MULTIDIMENSIONAL PHASE SPACE... 

Connections among NHIMs in multidimensional phase space is impossible to visualize directly. Thus, we need methods to detect their connections indirectly based on, for example, time series of orbits. We discuss this problem briefly. 

T. Konishi, Slow Dynamics in Multidimensional Phase Space Arnold Model Revisited, Adv. Chem. Phys. Part B 130, 423 (2005). 

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