Stretched exponential behavior

In order to understand the stretched exponential behavior of DCF (23), let us discuss Gibb s phase exponent r G = —lnp(p,q-,t). This quantity plays a special role in statistical mechanics and relates to the entropy of the system. If Gibb s exponent obeys the fractional evolution equation... 

Fig. 6. 29. The decay of the normalized light-induced defect density at different annealing temperatures, showing the stretched exponential behavior (Jackson and Kakalios 1988).

We have interpreted the observation of the stretched exponential behavior as being caused by the manifold of transition rates involved in the substrate turnover. This has been observed for all cases where the single enzyme catalysis has been studied by the product turnover [43-45] (Fig. 4.19). 

Hassler et al. [40, 41] observed that the stretched exponential behavior was related to the active phase of the enzyme (busy phase) while the less active phase (lazy phase) exhibited a single exponential phase (Fig. 4.20). The behavior which was originally observed by Edman et al. 1999 [39] at a much lower sensitivity had been observed again, however, with a much better intensity and time resolution. As can be seen from the trace, the turnover frequency is more than 4-fold higher in the busy periods. It is tempting to relate this to the memory effect decribed in the next chapter (Fig. 4.21). 

This stretched exponential behavior known from spin-glass theory. [16] Each run produces an estimate of r, that depends on the energy minimum visited as the systems freezes (or perhaps on two where the double peak in t is observed).[35] The ensemble average autocorrelation function is then a sum over many terms of the form which yields... 

Equation (145) represents the generalization of the a.v.c.f. of the Ornstein Uhlenbeck [21] (inertia-corrected Einstein) theory of the Brownian motion to fractional dynamics. The long-time tail due to the asymptotic (t >> t) t -like dependence [72] of the ((j)(O)(j)(t))o is apparent, as is the stretched exponential behavior at short times (t t). Eor a > 1, ((j)(O)(t)(f))o exhibits oscillations (see Eig. 14) which is consistent with the large excess absorption occurring at high frequencies. 

Stretched Exponential Behavior. The solid lines in figure 1 were actual fittings of the data to the stretched exponential function ... 

Here Xc is a characteristic condation time and the exponent p is less than unity. The occurrence of a stretched exponential behavior is also pointed out by Ngai, Mashimo and Fytas [95] in a recent comparison of rdaxation times for local... 

This is another example of a stretched exponential behavior at long times. In principle, one can apply the same calculations to any network built from domains of arbitrary internal architecture, as long as the relaxation spectrum inside the domains obeys a power-law form, see Eq. 155. For instance, the networks may belong to any type of regular lattice topology (bcc, fee, tetrahedral, triangular, hexagonal) or even be fractal structures. 

Transition concentrations ct[ /] are 60, 50, and 25 for ethylbenzene, CCI4, and ethylacetate, respectively. For c> q one sees upwards deviations from stretched-exponential behavior, perhaps consistent with a larger-concentration power law. However, with only a few points showing this behavior in any system it is diflBcult to be precise as to the deviation s functional form. 

Where does one not find stretched-exponential behavior In dilute solution, very modest deviations - concentration dependences weaker than expected - are sometimes seen. A few cases of re-entrant behavior, in which Ds c)t] c) 7 D (0) y(0) over some limited concentration range, have been noted. For melts, extensive reviews of the literature(2,3) generally find scaling behavior for tj and D, at least for adequately large polymers. It is then reasonable to expect that as the melt is approached there should be a transition to power-law behavior. Experiments of Tao, et al. are consistent with this expectation(4). 

There are broader exceptions to simple stretched-exponential behavior. These exceptions serve to test the generalization that stretched-exponential behavior is donfinant. For electrophoresis, as discussed in Chapter 3, a transition to power-law behavior with increasing P and E appears to correspond to the onset of nonlinear transport in which /i depends on the applied field. For the low-shear viscosity, the solufionfike-meltlike transition is sometimes seen as discussed in Section 9.10 and Chapter 11, this transition occurs simultaneously with the appearance of a light... 

Several paths exist for improving the original renormalization group calculation. Merriam and Phillies(7) have since extended the author s original calculation(6) to determine the five-point chain-chain-chain-chain-chain hydrodynamic interaction tensor. The deviation of the observed stretched-exponential behavior from simple calculations yielding pure-exponential behavior was predicted to arise from the concentration dependence of the chain radius. Dielectric relaxation measures both a relaxation time and a chain radius. Analysis demonstrated that chain contraction accounts quantitatively for the form of the stretched-exponential concentration dependence of the dielectric relaxation time(8). 

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