Perturbation theory time scales

The last assumption is very fundamental. It results in time-independent transition probabilities and makes a clean theory possible. It requires that the product of the time scale of the decay time for the tcf (called the correlation time and denoted x ) and the strength of the perturbation (in angular frequency units) has to be much smaller than unity (17-20). This range is sometimes denoted as the Redfield limit or the perturbation regime. 

It is probably the complexity of these theories that prohibited this particular aspect of electrode kinetics from being attractive for application in the study of homogeneous reaction kinetics per se. Yet it must be clear that the electrochemical techniques, together providing an extremely wide range of time scales, should be preeminently suited for investigations of both slow and (very) fast homogeneous reactions. This is the more true since, nowadays, the problem of the non-availability of a closed-form expression for the response—perturbation or response—time relation has been overcome by numerical analysis procedures conducted with the aid of computers. 

Thus to calculate the cross section, we need only calculate the asymptotic part of the scattered electron wave function. This is straightforward, at least for first-order perturbation theory. Provided that the time scale during which the perturbation of the molecule by the impact electron occurs is small compared to the time scale for electronic motion, we find26... 

The present chapter introduces the reader to singular perturbation theory as the framework for modeling and analyzing systems with multiple-time-scale dynamics, which we will make extensive use of throughout the text. 

Such nested applications of single-parameter singular perturbation theory (i.e., the extension of the analysis of two-time-scale systems presented in Chapter 2 to multiple-time-scale systems) have been used for stability analysis of linear (Ladde and Siljak 1983) and nonlinear (Desoer and Shahruz 1986) systems in the standard form. However, as emphasized above (Section 2.3), the ODE models of chemical processes are most often in the nonstandard singularly perturbed form, with the general multiple-perturbation representation... 

The basic ideas that are necessary for the first program stage are explained in Sections II, III, and IV. In Section II, we formulate the problem of how to analyze a system that has a gap in characteristic time scales. Our method is to use perturbation theory with respect to a parameter that is the ratio between a long time scale and a short time scale, which is a version of singular perturbation theory. The reason will be explained in Section II. In Section III, the concept of NHIMs is introduced in the context of singular perturbation theory. We will give an intuitive description of NHIMs and explain how the description is implemented, leaving the precise formulation of the NHIM concept to the literature in mathematics. In Section IV, we will show how Lie perturbation theory can be used to transform the system into the Fenichel normal form locally near a NHIM with a saddle with index 1. Our explanation is brief, since a detailed exposition has already been published [2]. 

This Hamiltonian must be put in the form of equation (Al) (See Appendix A ) for the Redfield theory to be applicable, and depending on the origin, different treatments of the perturbation is necessary. How the direct product is handled is determined by the correlation between the different parts. For simple liquids, the dipole-dipole tensor fluctuates on the picosecond to nanosecond time scale and it is thus not correlated with the nuclear spins. 

Strictly, the application of the algebraic method in non-linear perturbation theory requires the existence of a small parameter e in the equations, and this can be revealed by a non-dimensionalization procedure. However, even for such a simple set of equations, this is a complicated process which can be avoided by carrying out a numerical investigation of the time-scales present in the problem. By examining the eigenvalues of a linear approximation to the system as described in Section 4.7, it becomes clear that there are two negative fast modes in the above equations over all conditions tested. These are indicated by the presence of two large... 

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