Phase space asymptotic

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

So far we have defined the local stability ("there exists such <3 as. . . ). Now let us define the global stability for rest points. The rest point c0 is called globally asymptotically stable (as a whole) within the phase space D if it is stable according to Lyapunov, and for any initial conditions d0e D the solution c(t, k, cLa) tends to approach c 0 at t - oo. 

Since every achiral substrate is eventually consumed a(t = oo) = 0 and all the reactions stop asymptotically, Eq. 39 tells us that the product rs should vanish. If there is more R than S initially, S monomer disappears ultimately, for instance. But S molecules do not disappear nor decompose back into achiral substrate. They are only incorporated into the heterodimer RS. The system is not determined solely by monomer concentrations r and s, (or (p and q ) but also depends on heterodimer concentration [.RS]. The flow takes place in a three-dimensional phase space of r, s, [RS], as shown in Fig. 5a. 

The asymptotics discussed above force both the concentrations of the substrate a = c- r - s - 2[RS] and the product rs to vanish ultimately. These two conditions define a line of fixed points in the three-dimensional r - s - [RS] phase space. If the initial state has a prejudice to R enantiomer such as ro > s(l, then the system ends up on a fixed line r + 2[RS] = c on a s = 0 plane, as shown in Fig. 5a. Otherwise with ro < so> the system flows to another fixed... 

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,... 

Even the states of systems with infinite dimension, like systems described by partial differential equations, may lie on attractors of low dimension. The phase space of a system may also have more than one attractor. In this case the asymptotic behavior, i.e., the attractor at which a trajectory ends up, depends on the initial conditions. Thus, each attractor is surrounded by an attraction basin, which is the part of the phase space in which the trajectories from all initial conditions end up. 

Fig. 8. Phase space for the dynamics generated by the Hamiltonian (12) for = 2. The initial conditions of scattering trajectories are defined by the asymptotic incoming energy E, and a phase 27iT relative to the oscillating potential. Initial conditions leading to trajectories x t) with a total of two zeros are marked black. The white regions in between correspond to trajectories with three or more zeros (From [62])...

Asymptotic stability never appears, because it is not possible for the eigenvalues A3, A4 to be both inside the unit circle. This is also a consequence of the fact that the volume in phase space is conserved. 

There are many types that represent a set of measure zero in phase space orbits with an original and/or final parabolic escape, orbits with an asymptotic motion to some periodic or quasi-periodic solution, orbits with an oscillating motion of the first type that we have already met in Section 9, orbits open in the past and bounded in the future (complete capture) or inversely, motions leading to a collision of the two point-masses of the binary, etc. and there are also three main types, three types that represent sets of positive measure in phase space ... 

The regions of phase space, filled with trajectories of similar asymptotic behaviour, are separated by the sets (for n = 2 these are straight lines) called separatrices. The directions along which separatrices reach a stationary point are called the main directions (axes). The method of determination of separatrices is given in Appendix A2.6. 

Another related method to reduce the grid requirement is based on the superposition principle. The wave function and the propagation are split between two overlapping grids (79). This method allows the separation of the asymptotic dynamics from the interaction part. Care must be taken that the transmission of amplitude from one grid to the other is gradual in space to avoid numerical problems of overflowing phase space by a sharp transmission function. 

Asymptotically, hyperspherical coordinates become inadequate since the energetically allowed space contains fewer and fewer grid points. It is therefore necessary to map the wave function onto other coordinates (e.g., Jacobi coordinates). However, in the semiclassical treatment of the problem this is not possible since the wave function is known only in a restricted phase space, i.e., in either (6, ) or (p, 0, < >) space. It is therefore necessary either to carry out the projection in these coordinates by using variable grid methodology or to introduce a mixed Jacobi-hyperspherical coordinate treatment. This latter procedure is possible since we can express the Hamiltonian as... 

Figure 11. Kinetic energy release distribution for metastable loss of CH4 from nascent Co(C3Hg)+ collision complexes. The "unrestricted" phase space theory curve assumes the entrance channel contains only an orbiting transition state, the exit channel has only an orbiting transition state (no reverse activation barrier), and there are no intermediate tight transition states that affect the dynamics. The "restricted" phase space theory calculation includes a tight transition state for insertion into a C-H bond located 0.08 eV below the asymptotic energy of the reactants.

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