Phase-space region

The scaling prescription (59) embodies the assumption that the external force is so weak that it does not drive the TS trajectory out of the phase-space region in which the normal form expansion is valid. In the autonomous version of geometric TST, one generally assumes that this region is sufficiently large to make the normal form expansion a useful tool for the computation of the geometric objects. Once this assumption has been made, the additional condition imposed by Eq. (59) is only a weak constraint. 

Fig. 6.1. A cartoon depiction of the phase space T and important phase space regions and their relationship. The important phase space regions of systems 0 (To ) and 1 (T) ) are abstractly represented by shaded and open oval shapes, respectively. These regions can be related in four ways (a) subset, (b) coincidence, (c) partial overlap, and (d) no overlap. Also sketched is the important phase space region of the intermediate M in a two-stage calculation (see section on Multiple-Stage Design ). The appropriate staging strategy differs according to the different overlap relationships between F0 and 77...

Fig. 6.3. To ensure the accuracy of a nonequilibrium work free energy calculation, the switching paths should go down the funnel. The important phase space regions for the intermediate states along the ideal funnel paths are illustrated in this plot, for the case where r0 and / are partially overlapped. Two funnel paths need to be constructed to transfer the systems from both 0 and 1 to a common intermediate M where rm is inside the r0 and J overlap region. The construction of such paths is discussed in Sect. 6.6...

It would be valuable if one could proceed with a reliable free energy calculation without having to be too concerned about the important phase space and entropy of the systems of interest, and to analyze the perturbation distribution functions. The OS technique [35, 43, 44, 54] has been developed for this purpose. Since this is developed from Bennett s acceptance ratio method, this will also be reviewed in this section. That is, we focus on the situation in which the two systems of interest (or intermediates in between) have partial overlap in their important phase space regions. The partial overlap relationship should represent the situation found in a wide range of real problems. 

The phase-space region Af/ corresponding to the lattice vector / is partitioned into cells A. The probability that the system is found in the cell A at time t is given by... 

The diffusive random walk of the Helfand moment is mled by a diffusion equation. If the phase-space region is defined by requiring Ga(t) < x/2, the escape rate can be computed as the leading eigenvalue of the diffusion equation with these absorbing boundary conditions for the Helfand moment [37, 39] ... 

As shown by TalkneP there is a direct connection between the Rayleigh quotient method and the reactive flux method. Two conditions must be met. The first is that phase space regions of products must be absorbing. In different terms, the trial function must decay to zero in the products region. The second condition is that the reduced barrier height pyl" 1. As already mentioned above, differences between the two methods will be of the order e P. ... 

Similar to unstable periodic orbits, an NHIM has stable and unstable manifolds that are of dimension 2 — 2 and are also structurally stable. Note that a union of the segments of the stable and unstable manifolds is also of dimension 2n — 2, which is only of dimension one less than the energy surface. Hence, as far as dimensionality is concerned, it is possible for a combination of the stable and unstable manifolds of an NHIM to divide the many-dimensional energy surface so that reaction flux can be dehned. However, unlike the fewdimensional case in which a union of the stable and unstable manifolds necessarily encloses a phase space region, a combination of the stable and unstable manifolds of an NHIM may not do so in a many-dimensional system. This phenomenon is called homoclinic tangency, and it is extensively discussed in a recent review article by Toda [17]. 

Figure 18. (a) A schematic composite surface of section for nomotating T-shaped Hel2. (b) Idealization of surface of section indicating flow out of various phase-space regions. The hatched areas represent regions of quasi-periodic motion. [From S. K. Gray, S. A. Rice, and M. J. Davis, J. Phys. Chem. 90, 3470 (1986).]... 

Clearly, the A and B isomer states should be inside the separatrix, and the state C should be in the phase-space region outside of the separatrix but inside the energy boundary. A schematic diagram of this three-state isomerization model is presented in Fig. 20. From the results of previous analyses of predissociation we expect that within the A and B domains there are, in general, intramolecular bottlenecks to energy transfer. However, these bottlenecks are... 

A simple kinetics model of the three-state mechanism that takes into account bottlenecks to intramolecular energy transfer can be developed by splitting the phase space region A into the quasi-periodic motion region A and the highly chaotic region A2, with A = A + A2. Such a kinetics model is presented in Fig. 22. Let be the rate constant for flow of phase space points from Ai into... 

These results opened the way to neutrino astronomy and also to the use of cosmic v to study fundamental particle physics in phase-space regions not accessible to accelerators based experiments. 

It is first necessary to impose conditions on the three phase space regions in Fig. 10. Specifically, we require that the transition states S be defined so that recrossing from R+ to R or R to R- does not occur. In addition, it proves necessary to impose additional conditions on R to ensure statistical decay to R+. Details are provided elsewhere,46 but, qualitatively, they are designed, in the first instance, to exclude direct trajectories from consideration. 

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