Cylindrical phase space

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. 

N. De Leon, M. A. Mehta, and R. Q. Topper, Cylindrical manifolds in phase space as mediators of chemical reaction dynamics and kinetics. I. Theory, J. Chem. Phys. 94, 8310 (1991). 

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. 

A beautiful classical theory of unimolecular isomerization called the reactive island theory (RIT) has been developed by DeLeon and Marston [23] and by DeLeon and co-workers [24,25]. In RIT the classical phase-space structures are analyzed in great detail. Indeed, the key observation in RIT is that different cylindrical manifolds in phase space can act as mediators of unimolecular conformational isomerization. Figure 23 illustrates homoclinic tangling of motion near an unstable periodic orbit in a system of two DOFs with a fixed point T, and it applies to a wide class of isomerization reaction with two stable isomer... 

Cylindrical Manifolds. There is one big advantage of looking at 2-DOF TS in phase space It puts emphasis on the existence of the tubes that determine the transport of classical probability in phase space. Existence of those tubes has been known for a long time [48]. These tubes are the set of trajectories that constitute the stable/unstable manifolds of PODS. Locally, in the vicinity of P, they immediatly generalize to higher dimensions. They are constructed as follows ... 

Oped a so-called reactive island theory the reactive islands are the phase-space areas surrounded by the periodic orbits in the transition state, and reactions are interpreted as occurring along cylindrical invariant manifolds through the islands. Fair et al. [29] also found in their two- and three-dof models of the dissociation reaction of hydrazoic acid that a similar cylinderlike structure emerges in the phase space as it leaves the transition state. However, these are crucially based on the findings and the existence of (pure) periodic orbits for all the dof, at least in the transition states. Hence, some questions remain unresolved, for example, How can one extract these periodic orbits from many-body dof phase space and How can the periodic orbits persist at high energies above the saddle point, where chaos may wipe out any of them ... 

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.

M. A. Mehta, Ph.D. Dissertation, Yale University, New Haven, CT, 1990. Classical and Quantum Dynamics of Phase Space Cylindrical Manifolds. 

Equations (1), (2), (4), and (5) are the equations of motion for a system of particles with cylindrical symmetry. In the following pages, we shall discuss a number of aspects of the numerical solution of these equations such as the choice of numerical integration algorithm, some shortcuts in the numerical integration, the difficult problem of choosing a potential function for polyatomic molecules, and the calculation of quantities of statistical-mechanical interest from the phase space trajectory. 

Figure 8.3 The eigenvalues of the operator [C ] C= [/ — C] for the cylindrical phase at j JV = 10.9, / = 0.428, showing the band structure of the eigenvalue spectrum. The insert illustrates the first Brillouin zone of the system. The smellest eigenvalue occurs at the zone boundary, as indicated by the arrow. The smallest eigenvalue is negative in this case, indicating that the cylindrical structure is unstable at this point of the phase space. (Reproduced from M. Laradji et al. Phys. Rev. Lett 78, 2577 (1997) Copyright (1997) with permission from the American Physical Society).

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

Calculate (a) the residence time, t, and (b) the space time, r, and (c) explain any difference between the two, for the gas-phase production of C2H4 from C2H6 in a cylindrical PFR of constant diameter, based on the following data and assumptions ... 

The size of reactor is chosen to accommodate this holdup. The diameter is determined from the gas flow rate, q, together with, for example, an allowable superficial linear gas velocity, u (in lieu of an allowable (- AP)) D = (4qhru)m for a cylindrical vessel. The volume could be determined from an appropriate bed density, together with an overhead space for disengagement of solid and gas phases (we assume no carryover of solid in the gas exit stream). 

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