Slow and Fast Variables

In Eqs. (11.3.26) and (11.3.30) the dot denotes matrix multiplication. If K(/) and F(/) are known, these equations represent a set of closed equations from which the time evolution of the properties Ai,, Am can be computed. Equations (11.3.26) and (11.3.30) are an exact consequence of the equations of motion. 

In Section 10.3 it was shown that the Fourier component SA(q, t) of the fluctuation of a conserved density has a lifetime x(q) such that x(q) - oo as q - 0 that is, SA(q, t) varies slowly for small q. Thus we expect that the small (q — 0) wave number Fourier components of the densities of all the conserved properties form a good set of variables. For example, in an isotropic monatomic fluid we surmise that a good set consists of the low q Fourier components of the mass, linear momentum, and energy densities. 

Another good example of a separation of time scales is Brownian motion. Because the Brownian particle is much more massive than the solvent particles, it moves much more slowly. Thus the position and velocity of the Brownian particle should constitute a good set of variables. 

Highly anisotropie molecules reorient slowly in dense fluids and liquid crystals. In these fluids the conserved densities do not by themselves constitute a good set. It is necessary to include densities of orientational properties. This is made more specific later. 

Let us assume that we can list the independent variables whose decay is slow that is, those whose relaxation times rr satisfy 

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Note that in this case, the variables = (x) and t) = k(x) represent the true slow and fast variables, respectively, since the fast transients are observed only in the r] variables. 

Note that this method ultimately leads to a set of state-space realizations for the reduced-order models for each time scale of a multiple-time-scale system, but does not identify the slow and fast variables associated with the individual... 

The plan of this chapter is as follows. In Section II and the Appendix we review some basic topics in statistical mechanics that underlie questions concerning the regimes of validity of slow and fast variable descriptions of irreversible motion. 

In Section II we compare particle mechanics in the slow and fast variable timescale regimes. We start the discussion by showing the following. For damped macroscopic particles, the potential energy function whose minima locate the particle s points of static equilibrium also produces the forces which drive its dynamics. For damped microscopic particles, in contrast, the potential that determines the particle s statics may or may not produce the forces that drive its dynamics. 

This concludes our comparison of particle motion in the slow and fast variable timescale regimes. 

Within a chemical system, the long lifetime variables are called slow variables. For such variables, the distance between the original and the perturbed trajectories remains almost constant in time, whilst for the short lifetime, the so-called fast variables, the perturbed trajectory quickly approaches the original trajectory (see Fig. 6.1) (Klonowski 1983 Lee and Othmer 2010). It is important to note that there is no relationship between the magnitude of the production rate and the separation of slow and fast variables. This partition is based only on the rate of response to a perturbation. A high production rate (quickly changing concentration) may belong... 

Methods based on the partitioning of a reaction system into fast and slow components have been proposed by several authors [158-160], A key assumption made in this context is the separation of the space of concentration variables into two orthogonal subspaces and Qf spanned by the slow and fast reactions. With this assumption the time variation of the species concentrations is given as... 

Importantly, these clinical studies were executed at a time when the molecular basis of the slow and fast acetylator phenotypes were not well understood. We now know that there are two isoforms of the /V-acetyltrans-ferase enzyme arising from two different genes, NAT1 and NAT2. Constitutive expression of NAT1 accounts for basal enzyme function, and functionally important polymorphisms in NAT2, are thought to contribute to variability in overall enzymatic activity and thus to define the slow- and fast-acetylators phenotypes (23,24). 

Within the framework proposed in Kumar et al. (1998), a time-scale decomposition is initially used to derive separate representations of the slow and fast dynamics of (2.36) in the appropriate time scales and to provide some insight into the variables that should be used as part of the desired coordinate change. Specifically, by multiplying Equation (2.36) by e and considering the limit e —> 0,... 

One interesting research issue currently receiving much attention is placement of grid points for the discretization. For a single distributed variable such as temperature, one can see how to place the points, i.e., put more points where the variable is changing more rapidly and fewer where it is not. How is this to be done for many distributed but coupled variables that are changing at quite different rates in different parts of the space This problem is similar to that of slow- and fast-moving units in a dynamic simulation, only here no natural modularity occurs within a flowsheet of interconnected units. 

It is well known that the O2 reduction site of bovine heart cytochrome c oxidase in the fuUy oxidized state exhibits variable reactivity to cyanide and ferrocytochrome c, which is dependent on the method of purihcation (Moody, 1996). Some preparations react with cyanide extremely slowly at an almost immeasurable rate and are known as the slow form. Other preparations, which react at a half-Ufe of about 30 s, are known as the fast form (Brandt et al., 1989). Electronic absorption spectra of the slow-and fast-form preparations exhibit Soret bands at 418 and 424 nm, respectively. The two forms often coexist in a single preparation (Baker et al., 1987). Both forms exhibit an identical visible-Soret spectrum in the fully reduced state. The slow-form preparation can be converted to the fast form by dithionite reduction followed by reoxidation with O2. The fast form thus obtained returns to the slow form spontaneously at a rate much slower than the enzymatic turnover rate. Thus, the slow form is unlikely to be involved in the enzymatic turnover (Antoniniei a/., 1977). It should be noted that no clear experimental evidence has been reported for direct involvement of the fast form in the enzyme turnover, although its direct involvement has been widely accepted. The third species of the fully oxidized O2 reduction site, which appears in the partially reduced enzyme, reacts with cyanide 10 —10 times more rapidly than the fast form (Jones et al., 1984). In the absence of a reducing system, no interconversion is detectable between the slow and the fast forms (Brandt et al., 1989). Thus, the heterogeneity is expected to inhibit the crystallization of this enzyme. In fact, the enzyme preparations providing crystals showing X-ray diffraction at atomic resolution are the fast form preparation. 

The reduction of system (5.6) turns out to be much more delicate because the variables cannot be readily separated into slow and fast . Each of the kinetic equations (5.6) indeed contains both slow and fast terms. Thus, in the equation for the fraction of free, active receptor p, the first term ki(-p -i- 8), which relates to the reversible modification... 

The use of labeled cholesterol or its precursor, mevalonate, has the appeal that a limited number of products are presumably formed and that the lipid is believed to turn over very slowly within the nervous system. It should be noted, however, that the observed slow cholesterol turnover reflects primarily the major brain pool of this lipid, myelin. Axonal flow studies are however directed at neurons, not at the glial cells that synthesize myelin. MacGregor et al., (1973) noted that following injection of cholesterol into the lumbar region of the chick, aproximodistal gradient of cholesterol was found in the sciatic nerve. The rate was thought to be about that observed for protein. Both cholesterol and cholesterol ester were detected, but the relative proportions were variable. A slow and fast rate of axonal flow were... 

Continuing with the more general case where both slow and fast dfs arc present, we consider the case where the fast variables are localized, as happens for vibrational dfs. We approximate the internal Hamiltonian by... 

Now that we have a good picture of how SN2 reactions occur, we need to see how they can be used and what variables affect them. Some SN-2 reactions are fast, and some are slow some take place in high yield and others, in low yield. Understanding the factors involved can be of tremendous value. Let s begin by recalling a few things about reaction rates In general. 

The method developer should identify critical points in the method. Frequently, the Youden test may be used to determine if temperature, time, flow rate for solid-phase extraction, weight, volume, and other variables in the method are critical. The developer needs to identify if it is acceptable to take a break during a procedure, length of the break, and steps that need to be completed quickly. Because of differences in background and training between analysts, method developers should not assume that other analysts will perform a technique in the same way as in the developer s laboratory. Often analysts will have different interpretations of simple terms such as shake , slow , complete , and fast . 

The breathing rate data used to define the BR variable were adapted from the reported distribution generated from Shamoo et al.3 In the Shamoo study, a different distribution was identified for several activity patterns, and for this simulation the slow, medium, and fast rate classifications were combined. The distribution is shown in Figure 3. 

Another variable in CMP recipes is the back pressure. Usually, if the nonuniformity problem is identified to be due to a center-slow-edge-fast process, back pressure can be used to push the back of a wafer and accelerate the center polish rate. Thus, the uniformity can be improved accordingly. 

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