Timescale Analysis

As explained in Sect. 2.1, a full description of the time-dependent progress of a chemical reaction system requires a mechanism containing not just reactants and products but also important intermediate species. The rate of consumption of the species within the mechanism can vary over many orders of magnitude depending on the species type. Radical intermediates, for example, usually react on quicker timescales than stable molecular species. This can lead to numerical issues when attempting to solve initial value problems such as that expressed in Eq. (5.1), since the variation in timescales can lead to a stiff differential equation system which may become numerically unstable unless a small time step is used or special numerical 

Tnranyi, A.S. T(nnlin, Analysis of Kinetic Reaction Mechanisms, 

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E. Foumouo, P. Antoine, H. Bachau, B. Piraux, Attosecond timescale analysis of the dynamics of two-photon double ionization of helium, New J. Phys. 10 (2008) 025017. 

In the related work of Kim and Hynes [50], Equations (3.107) and (3.112) have been designated, respectively, by the labels SC (self-consistent or mean field) and BO (where Born-Oppenheimer here refers to timescale separation of solvent and solute electrons). More general timescale analysis has also been reported [50,51], Equation (3.112) is similar in spirit to the so-called direct RF method (DRF) [54-56], The difference between the BO and SC results has been related to electronic fluctuations associated with dispersion interactions [55], Approximate means of separating the full solute electronic densities into an ET-active subspace and the remainder, treated, respectively, at the BO and SC levels, have also been explored [52],... 

N and 2H relaxation can be employed to detect fluctuations of backbone dynamics of protein kinases on the nano- to picosecond and milli- to microsecond timescales. Analysis of the relaxation data allows for a semiquantitative estimation of the conformational entropy change for the main chain of protein kinases dependent on ligand binding or point mutation. 

Eggels and Somers [125] obtained an expression for the collision operator in Eq. (142) by using an asymptotic expression for iV, that is valid provided that the lattice gas is close to equilibrium and assuming that the magnitude of the fluid velocity, u, is small compared to unity. The asymptotic result was obtained by Frisch et al. [111] by using a multiple timescale analysis together... 

Let us determine the matrices J and F belonging to the kinetic system of ODEs above. These two types of matrices will be used several dozen times in the following chapters. For example, the Jacobian is used within the solution of stiff differential equations (Sect. 6.7), the calculation of local sensitivities (Sect. 5.2) and in timescale analysis (Sect. 6.2), whilst matrix F is used for the calculation of local sensitivities (Sect. 5.2). Carrying out the appropriate derivations, the following matrices are obtained ... 

The implication of distinguishing between fast and slow variables is that a short time after the perturbation, the values of the fast variables are determined by the values of the slow ones. Appropriate algebraic expressions to determine the values of the fast variables as functions of the values of the slow ones can therefore be developed. This is the starting point of model reduction methods based on timescale analysis. One such method was introduced in Sect. 2.3 where the quasi-steady-state approximation (QSSA) was demonstrated for the reduction in the number of variables of a simple example. In this case, the system timescales were directly associated with chemical species. We shah see in the later discussion that this need not always be the case. 

We hinted in Sect. 2.3.6 that the timescale separation present in most kinetic systems can be exploited in terms of model reduction. The next sections will therefore cover the use of timescale analysis for the reduction of the number of... 

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