The Extended Exponential

This is an extended exponential. It operates within the remit of linear viscoelastic theory. So for example for a simple exponential we can show that the integral under the relaxation function gives the low shear viscosity  

The same applies to the extended exponential. The integration is slightly more difficult but gives 

In general it is fair to say that rheologists have been conservative in their use of non-exponential kernels. One particular form clearly stands out as a candidate for describing experimental data, at least for a limited range of relaxation times. This is the power law equation, often applied to 

Therefore using the Boltzmann Superposition Principle (Equation 4.61) we have the gel equation  

This approach was pioneered by Winter5 in terms of structural relaxation. The transforms already given in this chapter can be applied to this expression to give the following relationships  

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The Extended Exponential Vasicek model (Brigo and Mercurio) is... 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

It should be clear that most of the methods discussed in the present section extend readily to three dimensions—the random geometric partitioning of a volume. For example, the linear exponential expression corresponding to (8.59) is... 

Figure 5.16 Typical stress relaxation data for concentrated charge dispersions. Two models are shown, one based on a model for the relaxation spectra (Equation 5.59) and one based on an extended exponential (Equation 5.51)...

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

The variation of the signs of intermolecular. b. the absolute values of which decrease with distance R in an exponential manner, can be explained by the extended McConnell model (20). 

An approach suggested in USEPA (1998) is to supplement the empirical distribution with an exponential tail (the mixed exponential approach ). An approach not mentioned is to use a smoothed empirical distribution (a continuous nonparametric distribution). The most likely approach would be to use a kernel smoother, e.g., as sometimes used in flood prediction to provide a distribution for flood magnitudes (review in Tail 1995). These procedures have the effect of adding a continuous tail to the distribution, extending beyond the largest observed value. 

Formally, the sum of random electromagnetic-field fluctuations in any set of bodies can be Fourier (frequency) decomposed into a sum of oscillatory modes extending through space. The "shaky step" in this derivation, already mentioned, is that we treat the modes extending over dissipative media as though they were pure sinusoidal oscillations. Implicitly this treatment filters all the fluctuations and dissipations to imagine pure oscillations only then does the derivation transform these oscillations into the smoothed, exponentially decaying disturbances of random fluctuation. 

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