Multi-dimensional system

I. Generalized Langevin equation. Zwanzig s Hamiltonian, n. Evaluation of quantum rates for multi-dimensional systems, ni. Beyond the Langevin equation/quantum Kramers paradigm ... 

This result can be generalized for multi-dimensional systems in which a limit set for every motion is a fixed point or a limit cycle, linear approximation matrices at fixed points have no eigenvalues in the imaginary axis and limit cycles have no multiplicators on the unit circle. In this case, k should be treated for fixed points as the sums of those eigenvalues that have positive real parts (they are "unstable ), and for limit cycles as the sums of unstable characteristics indices. 

One may obtain now (see [7]) the general expression of the density matrix of multi-dimensional system that is invariant to relation of the unitary transformation S ... 

Laskar, J. (1993). Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Physica D, 67 257-281. 

D. S. Selengut, Variational Analysis of Multi-Dimensional Systems, HW-59126, p. 89. General Electric Hanford Lab. (1959). 

Let, for example, Ri denote the reaction coordinate of interest in a multi-dimensional system. The corresponding reduced probability density is defined as... 

Thus the perfect diabatic representation does not in fact generally exist unless the basis set is complete [198]. Practical theories of diabatic representation, including those for multi-dimensional systems, have been studied extensively by Smith [376] and Baer [27, 28]. Another scheme was proposed using a different perspective [23, 292]. In this approximate but practical treatment, a diabatization is pursued by requiring the basis states to retain their individual characters smoothly, rather than minimizing the magnitude of derivative coupling. 

With the above simple background, we consider ET in complex systems. In this case, eqn (12.4) cannot be used straightforwardly because the nonadiabatic transition probability is explicitly dependent on the reaction coordinate and one does not know how to select this one-dimensional reaction coordinate from the multi-dimensional systems. Therefore, our strategy is to start from a generalized quantum rate expression. Miller et al. have shown that the... 

In eqn (12.19), the effects of nonadiabatic transition including the nuclear tunneling are properly taken into account by rj and naturally the main task is to evaluate the thermally averaged transition probability P fl, < ), which has to be evaluated using the Monte Carlo technique for multi-dimensional systems. It is easily shown that the Marcus-Hush formula in adiabatic and nonadiabatic limits can be recovered from eqn (12.19) and eqn (12.20) within the high-temperature approximation. 

ET rate via electronic coupling for a multi-dimensional system in the Marcus inverted regime, (a) pEa—6.7, (b) pSa = 10.0, and (c) pEa = 20.0. Ea represents the minimum energy on the seam surface. Solid line present result dashed line the results predicted from the LZ formula dotted line results from perturbation theory. 

Because this model system is one dimensional and dissociative, making a uniform grid appropriate, it is likely that the FFT split exponential prqpagator 2 would be advantageous. In multi-dimensional systems the DVR should be conq)etitive. 

We have not touched upon the special problems arising from multi-dimensional systems and steady-state problems, which are dealt with separately in Chapt. 8. 

Levanon, Y., Gefen, A., Lerman, Y, Given, U. Ratzon, N.Z. (2012) Multi dimensional system for evaluating preventive program for upper extremity disorders among computer operators. Work, 41 Suppl 1, 669-75. 

Next, a separable multi-dimensional system will be considered. The Hamiltonian of this system can be written as... 

A very basic microscopic measure of a dynamical system is its Lyapunov exponent or, for multi-dimensional systems, set of exponents. The latter is often referred to as the Lyapunov spectrum of the dynamical system. The equations of motion of a simulated MD system can formally be written as... 

Gonchenko, S. V., Turaev, D. V. and Shilnikov, L. P. [1993b] Dynamical phenomena in multi-dimensional systems with a structurally unstable homoclinic Poincare curve, Russian Acad. Sci. Dokl. Math. 47(3), 410-415. 

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