Hamiltonian dynamical systems correction

It is important to notice that the solution of the GLE depends only on y(t) and not on the particular set of parameters Ck, mk, cok that generate it through Eq. (8). In order to make this result more intelligible we should emphasize that the modes k in the Zwanzig Hamiltonian Eq. (7) do not (except in the crystalline case) refer to actual modes of the system rather, they represent a hypothetical environment " that generates the correct dynamical friction y(t) through Eq. (8), such that when entered in the GLE Eq. (2) it provides an accmate description of the dynamics. 

All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist Second, which features of the overall control process are most efficiently subject to feedback control ... 

The outstanding feature of the set (4.1.57) is that it is completely self-contained. Its structure does not depend on the specific form of the Hamiltonian (4.1.47). In fact, it describes the dynamics of energy levels for all Hamiltonian systems that can be split into two parts according to (4.1.47). The characteristics of a particular Hamiltonian enter only in the initial conditions. Thus we obtain the important result that if the system forgets the initial conditions after a while e, we can use the methods of equilibrium statistical mechanics to compute the statistical properties of the energy levels of the system. Pechukas (1983) used this approach to predict Wignerian statistics for the level fluctuations of chaotic systems. As discussed in the previous section, there is now ample evidence for the correctness of this prediction. 

Warshel and co-workers > ° >i studied the dynamics of this Sn2 reaction using an interesting and different approach the empirical valence bond (EVB) method, which has been described in detail in a recent book by Warshel. The fundamental idea behind the application of the EVB method to this Sn2 reaction is that the reaction can be treated as a two state system, where the reactants and products are each taken to be separate quantum mechanical states with Hamiltonians and Hi- These states can be coupled together by an empirical coupling Hamiltonian so that when the two state Hamiltonian is diagonalized, the correct features of the ground state surface on which the reaction occurs are obtained. H, and H2 are taken by Warshel and coworkers to have analytic forms based on the gas phase parameters. 

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