Separatrixes defined

IS2N2I 1, S2N2 < 0. In this case there is only one singular point A (saddle), whose separatrix defines the collision cross section of drops (Fig. 13.9, a). 

The basic picture discussed above is quite general in Hamiltonian systems. Of particular importance is the concept of stable and unstable manifolds associated with unstable periodic orbits. Trajectories along the stable manifold will be mapped toward the periodic orbit, whereas trajectories along the unstable manifold will be mapped away from the periodic orbit. It turns out that the union of segments of the stable and unstable manifolds is very useful in defining the reaction separatrix and calculating the flux crossing the separatrix in few-dimensional systems. 

Gray, Rice, and Davis [12] developed an alternative RRKM (ARRKM) theory in an attempt to simplify the Davis-Gray theory for van der Waals predissociation reactions. Specifically, they replaced the exact separatrix with an approximate phase space dividing surface by dropping a number of small terms in the system Hamiltonian, and they replaced the exact mapping that defines the flux across the tme separatrix with an analytic treatment of the flux across the approximate separatrix. This simplification is schematically presented in Fig. 18. 

For a given diatom vibrational state with classical vibrational action v, Zhao and Rice defined a vibrational-state-dependent separatrix function by... 

One still has to carry out calculations for the rate constant ksep of crossing the separatrix S sep defined above. According to the ARRKM theory, for particular total angular momentum J one has... 

The separation of the phase-space inside the separatrix into a region inside and a region outside the intramolecular energy transfer dividing surface defines the initial conditions N 0) and A 2(0), and leads us to write... 

Since the flux Fgr E) in the Gray-Rice theory is defined by the flux crossing the separatrix from the trapped A state to state C = + C, we have... 

The saddle point between isomer B and isomer A is represented by yj = and the saddle point between isomer B and isomer A is represented by y = q g As in the case of double-well potentials, the MRRKM theory defines isomerization separatrix surfaces for the three isomer states B, A and B by... 

The MRRKM theory defines the separatrix with respect to the reaction coordinate y by the relation... 

There are a number of open issues associated with statistical descriptions of unimolecular reactions, particularly in many-dimensional systems. One fundamental issue is to find a qualitative criterion for predicting if a reaction in a many-dimensional system is statistical or nonstatistic al. In a recent review article, Toda [17] discussed different aspects of the Arnold web — that is, the network of nonlinear resonances in many-dimensional systems. Toda pointed out the importance of analyzing the qualitative features of the Arnold web— for example, how different resonance zones intersect and how the intersections further overlap with one another. However, as pointed out earlier, even in the case of fully developed global chaos it remains challenging to define a nonlocal reaction separatrix and to calculate the flux crossing the separatrix in a manydimensional phase-space. 

FIGURE 14 (PLATE 3). Localization domains of mono-X-substituted benzenes C6H5X (from left to right top X = H, OH, F, bottom X = Cl, Br, I). The ELF value defining the boundary isosurface, t (r) = 0.659 corresponds to the critical point of index 1 on the separatrix between adjacent V(C, C) basins of benzene. Colour code magenta = core, orange = monosynaptic, blue = protonated disynaptic, green = disynaptic. Adapted from Reference 220 with permission... 

Hence, the family of all residue curves that originate from one fixed composition point and terminate into another fixed composition point defines a distillation region. Two adjacent regions are separated by a separatrix. In this book we called it simply distillation boundary. 

Based on the separatrix, the reaction rate can be defined as follows. Each time the Poincare map is applied, new regions similar to the one denoted by A are created. Then, those orbits included in these regions will dissociate. Therefore, the reaction rate is proportional to the area of the region A multiplied by the number of regions created by a single application of the map. Thus, the concept of reaction rate is founded on the basis of dynamical behavior. 

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