Phase space cylinder

As an example take a gas in a cylindrical vessel. In addition to the energy there is one other constant of the motion the angular momentum around the cylinder axis. The 6A/-dimensional phase space is thereby reduced to subshells of 6N-2 dimensions. Consider a small sub volume in the vessel and let Y(t) be the number of molecules in it. According to III.2 Y(t) is a stochastic function, with range n = 0,1,2,. .., N. Each value Y = n delineates a phase cell ) one expects that Y(t) is a Markov process if the gas is sufficiently dilute and that pi is approximately a Poisson distribution if the subvolume is much smaller than the vessel. 

Figure 12. Caricatures of the cylinders as they wind about the phase space volume. All the surfaces and volumes are embedded in four-dimensional phase space IR. (A) The cylinders and overlap one another at resulting in direct back reactions. (B) These two cylinders do not overlap one another. [Reprinted with permission from N. De Leon, J. Chem. Phys. 96, 285 (1992). Copyright 1992, American Institute of Physics.]...

The robust existence of a skeleton composed of a NHIM and its spherical invariant cylinders in the phase space should play their essential roles not only to help us understand the physical origin of observed nonstatistical, dynamical behavior but also to provide us with a new scope to control chemical reaction dynamics in terms of geometrical feature of the phase space. Here, let us articulate some of the subjects we have to confront in the immediate future ... 

The phase portrait for the pendulum is more illuminating when wrapped onto the surface of a cylinder (Figure 6.7.4). In fact, a cylinder is the natural phase space for the pendulum, because it incor-... 

Since the system is 27r-periodic in 0, it may be considered as a vector field on a cylinder. (See Section 6,1 for another vector field on a cylinder.) The x-axis runs along the cylinder, and the 0-axis wraps around it. Note that the cylindrical phase space is finite, with edges given by the circles x = 0 and x = 1. 

As in Section 6.7 the phase space is a cylinder, since is an angular variable and y is a real number (best thought of as an angular velocity). 

Besides the plane and the cylinder, another important two-dimensional phase space is the torus. It is the natural phase space for systems of the form... 

The cylinder leading to the transition state is very obvious in this figure. There are several points to observe first, at the bottom of the figure, where the r2 arc length is 0, is the transition state. All trajectories at the transition state lie within the boundary of the perimeter family at this point. Second, as the trajectories leave the transition state, they remain clustered as they were at the transition state, sweeping out a cylinder in the phase space. Third, the cylinder moves as it evolves, following a somewhat twisted path in the phase space, predictable from the path of the central trajectory. Viewed in terms of (r, p,), the trajectories of the perimeter family oscillate around the outside of the cylinder. As r2 propagates, the trajectories then wind around the outside of the cylinder. 

Furthermore, this phase-space perspective reveals that there is even stronger correlation among the reactive mode and the nonreactive mode than is evident in the ensemble averages. Specifically, the variations from the ensemble average represented by the standard deviations are not random. Rather, they represent the organized distribution of phases about the cylinder as the trajectories wind around the cylinder. 

Figure 11 Phase-space surface px versus r, versus r2 arc length for the trajectories in the perimeter family (dotted lines) and for the central trajectory (solid line) of the two-mode hydrazoic acid model at a total energy of 0.0610 au. The reaction cylinder is conspicuous as these trajectories move through phase space.

Our question now becomes, why does the cylinder follow the specific path that it does We already have a preliminary answer to this question very large mode-mode couplings are required for activation of the reaction coordinate, so trajectories approaching the transition state must pass through a relatively restricted window in phase space to achieve these couplings. 

Figure 12 Phase-space surface of section p, versus r, at r2 = r" at a total energy of 0.0610 au. The dashed line is the total energy boundary. The asterisks indicate the points where the trajectories of the perimeter family pass through this surface after leaving the transition state. The solid line is the constraint determined by Eq. (24) for a fixed E2 = 0.0700 au. Equation 24 accurately predicts both the location and distribution of the trajectories as they pass through this surface and thus describes the location of the reactive cylinder in phase space.

There is now a division in phase space between trajectories that are instantaneously within the overlap between the two cylinders and those that are not. First of all, since the coupling is weak it is likely that most of the phase... 

Figure 20 Schematic drawing of two cylindrical manifolds within isomer A in the weak-coupling limit (refer to Figure 8 for an explanation of the symbols). The two-dimensional cylinders will intersect each other along one-dimensional lines. These lines are two homoclinic orbits. The small, thin tube spanning both isomers corresponds to a reactive KAM torus. Note that although we have stopped drawing the cylinders beyond a certain point for clarity, in reality the cylinders continue to wind about and explore the entire accessible region of chaotic phase space. Reprinted with permission from Ref. 108.

We employ the symbols x and p to denote the set of coordinates and conjugate momenta for the complete specification of the phase space of all the N water molecules contained in the cylinder. In this notation, the element x,- e x refers to the variables that are required to specify the position and orientation of the water molecule in the pore and is thus an abbreviation of the form ... 

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