Timescale separation

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],... 

Below we will use the timescale separation between the (fast) thermal relaxation within the L and R subsystems and the (slow) transition between them in one additional way We will assume that relative equilibrium within each subsystem is maintained, that is. 

Equation (10.155) was obtained under three approximations. The first two are the neglect of initial correlations and the assumption of weak coupling that was used to approximate Eq. (10.110) by Eq. (10.112). The third is the assumption of timescale separation between the (fast) bath and the (slow) system used to get the final Markovian form. 

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

Let us consider the last point. The reader is already familiar with two important implications of the timescale separation between electronic and nuclear motions in molecular systems One is the Bom-Oppenheimer principle which provides the foundation for the concept of potential energy surfaces for the nuclear motion. The other is the prominent role played by the Franck-Condon principle and Franck-Condon factors (overlap of nuclear wavefunctions) in the vibrational structure of molecular electronic spectra. Indeed this principle, stating that electronic transitions occur at fixed nuclear positions, is a direct consequence of the observation that electronic motion takes place on a timescale short relative to that of the nuclei. 

Obviously any process in a closed system must conserve the overall energy. The transition described here has to conserve the energy of the fast mode since, because of the timescale separation, eneigy cannot be exchanged between the two modes. 

The plan of Section IV is as follows In section IV.A, we qualitatively outline the general picture of reaction dynamics that emerges from fast variable physics. Next, in section IV.B, we examine liquid phase-activated barrier crossing in the short time regime of Section II.C. In Section IV.C we note that the fast variable/slow bath timescale separation also applies to liquid phase vibrational energy relaxation and then discuss that process from the fast variable standpoint. Finally, in Section IV.D, we discuss some related work of others. 

The fast variable/slow bath timescale separation arises in liquid phase VER because of a frequency mismatch between the solute and solvent molecules, rather than because of a speed difference, as in typical reactions. Despite this, the fast variable timescale separation yields a picture of solute VER very similar to the picture of short time-activated barrier crossing reflected in Eq. (3.53). We expect similar pictures to emerge for other processes. 

We next develop the Machlup-Onsager equation from Eq. (A.19) by making the familiar assumption that the macroscopic parameters A(f) are slow variables. This assumption is usually justified by the idea that a timescale separation exists between the parameters A(t) and their bath variables, due to the macroscopic nature of the former and microscopic nature of the latter. 

Here e, a, and b are real, positive parameters, e is chosen small in order to guarantee a clear timescale separation between the the fast x-variable (activator) and the slow y-variable (inhibitor). The variables a and b determine the position of the so-called nullclines, the two functions y x) that are determined by setting time derivatives dx/dt = 0 and dy/dt = 0. Depending on the parameters the FHN system has different dynamical regimes. Fig. 1.2 shows phase space portraits together with the nullclines and timeseries for three qualitatively different cases. 

The free-energy difference between these two intermediate states should not strongly depend on concentrations of the solutes. On the state level, internal energy, intrinsic entropy, and free-energy relations are based on an assumption of a timescale, separation between transitions within each state, and the slower and observable transitions between these states. The second law for any time-dependent ensemble leads to both stochastic entropy and the local detailed balance condition for the rates. 

Consider what happens as the molecule rotates in the terms of the i.r example. If the neighbours were to relax very rapidly to their new equilibrium positions upon a change in the molecular orientation then the mean induced moment will follow the molecular orientation and behave as an extra contribution to the molecular dipole, the fluctuating part of the induced moment (AM) will rapidly forget the earlier molecular orientation. Under this condition of timescale separation the cross-term will be small at all times (it is zero at t=0, by 5.11) and the first term in eqn 5.10 will contain all the reorientational time dependence. 

Cl2 and CO2 and used the DID model for the induced polarizability. They showed that for CO2 the collision induced contribution to the depolarised Rayleigh intensity was 25% of the total intensity and that the second moment was increased by about 50% by induced effects. The timescale separation was examined in N2 and C02 In their terminology is the collision induced contribution it was found to relax in a very similar way to the orientational function <°M(t).°M> and the spectra of the two terms were indistinguishable for practical purposes. Furthermore the cross-term was quite large. The net effect of the non-orientational terms was to reduce the amplitude of the spectrum at low frequencies and to increase it in the wings. Frenkel and McTague s results on nitrogen have been carefully compared with experiment by Sampoli de Santis and co-workers(54). 

Recently Ladanyi has extended the work on N2 and CO2 by examining the induced terms given by a site-site DID model. This model allows for the distribution of polarizable matter within the molecule (c.f the discussion of section 2) by representing the molecular polarizability by an isotropic point polarizability on each interaction site and taking all orders of intra-molecular DID interactions into account. For COo her results are appreciably different from Frenkel and McTague s. In particular there is an enhanced projection of the induced terms along and a better timescale separation between the allowed and collision-induced processes so that the total spectrum resembles the reorientational spectrum more closely. 

Madden and Tildesley have studied the spectra of a model of CS2 they used the normal centre-centre DID model. In CS2 the molecular reorientation is slow and diffusional in contrast to the rapid inertial tumbling found in N2 and CO2 in this regard CS2 may be regarded as a representative normal organic fluid. In figure 5 the simulated depolarised Raman spectrum (of the mode is shown for a density and temperature appropriate to STP. In this example clear evidence of a timescale separation is found, the total spectrum is dominated by the reorientational spectrum at low frequencies. This separation of timescales improves even further as the fluid becomes denser and cooler. By comparing figures 4 and 5 it can be seen thatthe intuitive separation of the experimental spectrum is not far from the truth. 

As shown in the last section a spectral intensity (which for convenience will be taken to cover e(0) - e( ))may always be formally divided into a reorientational part, Ir, and a collision-induced part. Under the favourable circumstance that a timescale separation exists (as in the CS2 and CH3CN examples of 5) it may be possible to separately determine the two parts and obtain a measured value for I-... 

While the detailed analysis of the Kerr profiles in pure liquids leads to the conclusion that four distinct responses underlie the overall rise and decay of the field-induced polarization anisotropy A n(t), naturally in a real liquid these responses are coupled together. For clarity of discussion we will continue to describe them separately, since the timescale separation assumption in Eq. (3) proved quite valid. Summarized as follows in terms of r (t), which are associated with the instantaneous electronic and noninstantaneous, field-driven nuclear responses linked to intra- and intermolecular motions, the responses are labeled r (electronic) and r2> r, r (nuclear). The underlying assumption of a harmonic potential v j, and a continuum fluid [g(r) = 1] might clearly lead to some over simplifications but if molecules reside at or near the bottom of the potential well, the situation is not unreasonable. The results are very instructive for modeling short-time behavior for molecular solvents at room temperature, which is the next, vastly more complex step (see references on simulations and the next chapter). 

We have already emphasized several times that fast but correlated fluctuations give rise to dissipation on the coarse-grained level of description, which is described here by the friction matrix M, Eq. (7.10) or (7.28). The notion fast is defined here by times t smaller than the timescale Xs, which separates the evolution of the relevant variables X from rapid dynamics of the remaining degrees of freedom. The existence of such a timescale (which is equivalent to the crucial assumption of timescale separation discussed in Section 7.3) is not obvious. Here, we observe that the correlation... 

For larger systems such as those typically fotmd in complex chemical problems, non-dimensionalisation may be impractical, and hence, numerical perturbation methods are generally used to investigate system dynamics and to explore timescale separation. By studying the evolution of a small disturbance or perturbation to the nonlinear system, it is possible to reduce the problem to a locally linear one. The resulting set of linear equations is easier to solve, and information can be obtained about the local timescales and stability of the nonlinear system. Several books on mathematics and physics (see e.g. Pontryagin 1962) discuss the linear stability analysis of the stationary states of a dynamical system. In this case, the dynamical system, described by an ODE, is in stationary state, i.e. the values of its variables are constant in time. If the stationary concentrations are perturbed, one of the possible results is that the stationary state is asymptotically stable, which means that the perturbed system always returns to the stationary state. Another possible outcome is that the stationary point is unstable. In this case, it is possible that the system returns to the stationary state after perturbation towards some special directions but may permanently deviate after a perturbation to other directions. A full discussion of stationary state analysis in chemical systems is given in Scott (1990). 

The errors induced within methods based on timescale separations will be discussed in more detail in Sect 7.8 below. On the other hand, since equilibrations will exist within the groups, the introduction of such families is likely to lead to the elimination of fast timescales, thus reducing the stiffness of the reduced system of differential equations with resultant increases in simulation speed. 0 D, for example, has an atmospheric lifetime of the order of 10 s (see Sect. 6.3), and therefore, its presence within a scheme can lead to large stiffness ratios when treated as an individual species. Within reactive flow models, further computational gains may also be made by advecting these families within the transport step rather than individual species, thereby reducing the number of transported variables. 

Linear Lumping in Systems with Timescale Separation... 

We now briefly provide a formalised framework for linear lumping in systems with timescale separation which is based on a similar approach to that presented in Sect. 6.3. We start with the initial value problem... 

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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