Time-scale separation

Suppose that the process time scale (or the time window of interest) is bounded between and tmax (tmm t W) and is a small subset of the entire reaction time scale spectrum. Then the species reacting on time scales longer than /niax remain dormant their concentrations are hardly different from their initial values. The state of these species may be treated as system parameters. On the other hand, species reacting with time scales shorter than tnim niay be considered relaxed. The relaxed state of these fast-reacting species may be treated as system initial conditions. These considerations naturally help reduce the system dimensionality. 

Briefly, a formal treatment is as follows. Let X be the matrix of the eigenvectors x, of K and A be the diagonal matrix of the corresponding eigenvalues Kx, = /l,x,). The linear transformation c = X c provides 

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From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

Finding that the scattering functions at low temperature are amenable to an MCT description, we are faced with a dilemma. On the one hand, the high-temperature mean-square displacement curves lead us to conclude that dihedral barriers constitute a second mechanism for time scale separation in super-cooled polymer melts besides packing effects. On the other hand, the... 

The answer to our question at the beginning of this summary therefore has to be as follows. When you want to locate the glass transition of a polymer melt, find the temperature at which a change in dynamics occurs. You will be able to observe a developing time-scale separation between short-time, vibrational dynamics and structural relaxation in the vicinity of this temperature. Below this crossover temperature, one will find that the temperature dependence of relaxation times assumes an Arrhenius law. Whether MCT is the final answer to describe this process in complex liquids like polymers may be a point of debate, but this crossover temperature is the temperature at which the glass transition occurs. 

Time-Scale Separation and Metabolic Pathways Control... 

TIME-SCALE SEPARATION AND METABOLIC PATHWAYS CONTROL... 

Delgado et aL recently demonstrated that time-scale separation is an effective way to localize metabolic control to only a few enzymes. They considered model pathways in which the eigenvalues of the Jacobian of the system are widely separated (i.e., systems with time-scale separation). Their treatment assumes the system possesses a unique, asymptotically stable steady-state and that the reaction steps of the system under analysis are... 

To summarize, the reactive flux method is a great help but it is predicated on a time scale separation, which results from the fact that the reaction time (1/T) is very long compared to all other times. This time scale separation is valid, only if the reduced barrier height is large. In this limit, the reactive flux method, the population decay method and the lowest nonzero eigenvalue of the Fokker-Planck equation all give the same result up to exponentially small corrections of the order of For small reduced barriers, there may be noticeable differences between the different definitions and as aheady mentioned each case must be handled with care. 

Such a time scale separation between system and bath may often be appropriate when dealing with intramolecular vibrational motions of molecules but is likely never appropriate for electronic transitions in solution near room temperature. In the past 10 years much effort has been devoted to dynamical aspects of the solvation process in polar liquids utilizing experiments [2-4], theory [5, 6], and computer simulations of molecular dynamics [7-10]. The... 

The observations of vibrational coherence in optically initiated reactions described above clearly show that the standard assumption of condensed-phase rate theories—that there is a clear time scale separation between vibrational dephasing and the nonadiabatic transition—is clearly violated in these cases. The observation of vibrational beats has generally been taken to imply that vibrational energy relaxation is slow. This viewpoint is based on the optical Bloch equations applied to two-level systems. In this model, the total dephasing rate is given by... 

At the heart of the mode coupling theory of liquids is the assumption that a separation of time scale exists between different dynamical events. While the time scale separation between the fast collisional events and the slower collective relaxation is explicitly exploited in the formulation of the theory, there is also an underlying assumption of the separation of length scales between different relaxation modes. Much of the success of MCT depends on the validity of this separation of length and time scales. 

In the Kramers approach the friction models collisions between the particle and the surrounding medium, and it is assumed that the collisions occur instantaneously. There is a time-scale separation between the reactive mode and its thermal bath. The dynamics are described by the Langevin equation (4.141). The situation where the collisions do not occur instantaneously but take place on a time scale characterizing the interactions between the particle and its surrounding can be described by a generalized Langevin equation (GLE),158,187... 

One recognizes here the previously mentioned time-scale separation. 1. tlr < 1 ... 

To readily make clear the conditions of the time-scale separation in the previously described dynamics, it is suggested that one examine the variation of the survival probability p(0) in the absence of spontaneous decay (kT = 0) versus 0 = i/t" for various values of the parameter W = 2t /t" = U(x0)lkBT [see Eq. (4.221)]. From the curves in Fig. 4.11 one sees that for W>1 no time-scale separation is apparent, and if Wtime scales are clearly distinguishable. These conclusions are corroborated by examining the curves on Fig. 4.12 where the plots of In p(0) versus 0 are shown for typical values of W and different values of kr compared to 1/t". 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

Because of the time scale separation of 3 to 4 decades between both modes, it is of advantage to convolute the heat mode gT(t) = T, 1 exp(-t/Tth) into the effective excitation ... 

Frequently, a flat power spectrum is not the best choice for a given problem, and it might be of advantage to focus the excitation energy into certain frequency ranges of interest. One example is a system with fast and slow relaxation processes of comparable amplitude and a large time scale separation of several decades. In such cases, a white power spectrum with a constant power density is inferior to a power spectrum, where the discrete frequencies are evenly spaced on a logarithmic instead of a linear frequency scale. 

Dynamical Self-Organization. When the parameter X passes slowly through X (l),the bifurcation picture of the previous section accurateiy describes the system. However, in Fucus, and probably in many other examples, this time scale separation between the characteristic time on which X varies and the time to obtain the patterned state does not hold. Thus a dynamical theory allowing for the interplay of these two time scales is required to characterize the developmental scenario. A natural formalism to describe this process is that of time dependent Ginzburg-Landau (tdgl) equations used successfully in other contexts of nonequilibrium phase transitions (27). 

Typically developing systems are in a situation of dynamical patterning. A bifurcation picture is not relevant unless there is time scale separation thus a TDGL formalism is more appropriate as discussed above in dynamicsl self-organization. We are developing such a theory to study lability in Fucus-like polarization phenomena. 

Delgado, J. Liao, J. C. Control of metabolic pathways by time-scale separation. Biosystems 1995, 36 55-70. 

This chapter addressed the dynamics and control of process systems with material recycling. We established that whenever the flow rate of the recycle stream is significantly larger than the flow rates of the feed/product streams, the overall process exhibits a time-scale separation in its dynamics. 

A similar result was reported earlier (Georgakis 1986), when an eigenvalue analysis was used to prove that a time-scale separation is present in the transient evolution of the states 91 and 92. It is noteworthy, however, that, in contrast to the approach presented in this chapter, an eigenvalue analysis does not provide a means by which to derive physically meaningful reduced-order models for the dynamics in each time scale. 

Equations (7.15) are in the general form of Equations (7.1), with 2 playing the role of the small parameter in the sense used in (7.1). On the other hand, E captures the presence of material streams of vastly different flow rates, which, as we saw in Section 3.5, leads to a time-scale separation in the dynamics of the material-balance variables. 

Kumar, A., Christofides, P. D., and Daoutidis, P. (1998). Singular perturbation modeling of nonlinear processes with non-explicit time-scale separation. Chem. Eng. Sci., 53, 1491-1504. 

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