Stretched exponential relaxation time distribution function

Both forms have a distribution of relaxation times about t0 which contribute to s(Laplace transform can be obtained numerically, and in this case C(t) can be well fit to a stretched exponential function [46] ... 

Rb and 1H SLR rate as a function of temperature is a very important parameter which shows the suppression of phase transition and reveals the frustration in the mixed system. Temperature dependence of Ti in any ordered system can be described by the well known Bloembergen-Purcell-Pound (BPP) type expression. However, disordered systems show deviations from BPP behaviour, showing a broad distribution of relaxation times. The magnetization recovery shows a stretched exponential recovery of magnetization following M(t)=Mo(1 — 2 exp (— r/Ti) ) where a is the stretched exponent. 

The two intensity levels correspond to the two life times of the excited state of the TMR molecule which is due to electron transfer from neighboring Guanosin bases to TMR [27], [31]. We have chosen to calculate the correlation function (Fig. 4.8c) which shows the chemical relaxation rate of the conformational transition. The correlation function cannot be fitted with a single exponential as would be normal for first order kinetics. Instead, a stretched exponential of type exp-(kt) gave the best representation of the measured data. The stretched exponential can be envisaged as a distribution of relaxation rates with a mean relaxation rate k and a distribution defined by the... 

Figure 12 displays the Mittag-Leffler function as well as the two asymptotes, the dashed curve being the stretched exponential and the dotted curve the inverse power law. What is apparent from this discussion is the long-time memory associated with the fractional relaxation process, being an inverse power law rather than the exponential of ordinary relaxation. It is apparent that the Mittag-Leffler function smoothly joins these two empirically determined asymptotic distributions. 

Here tq is the relaxation time at equilibrium (Tf = T) at high temperatures, x is a structural parameter and measure of nonlinearity, with values 0 < x < 1, and AE is the activation energy for the relaxation processes and has an Arrhenius temperature dependence. The models also use the stretched exponential function of Kohlrausch, Williams, and Watts [1970] (KWW) to describe the distribution of relaxation times as... 

The stretched-exponential temporal response of Eq. (63), Section 2.1, a versatile and theoretically plausible correlation function, is one whose corresponding frequency behavior is now called Kohlrausch-Williams-Watts or just Kohlrausch [1854] model response, denoted here by Kk. It is also now customary to replace the a of the stretched-exponential equation by P or P, with A =D or 0. The k=D choice may be related to KD-model dispersive frequency response involving a distribution of dielectric relaxation (properly retardation ) times, and the A = 0 and 1 choices to two different distributions of resistivity relaxation times and thus to KO and K1-model responses, respectively. Note that the P parameter of the important K1 model is not directly related to stretched exponential temporal response, as are the other Kohlrausch models, but the DRTs of the KO and K1 models are closely related (Macdonald [1997a]). Further, although the KD and KO models are identical in form, they apply at different immittance levels and so represent distinct response behaviors. 

This anomalous relaxation behavior is often observed in complex systems including glass-forming materials and interpreted in terms of a wide distribution of the relaxation time. Using the distribution function g(ln Tj(), the observed stretched exponential function can be described by... 

It is well known that the a-process has a wide distribution of the relaxation time and hence there is a possibility that the E-process is a tail of the a-process. Due to the wide distribution of the relaxation time the intermediate scattering function due to the a-process is described by a stretched exponential function [exp(-(t/T) ) 0< < ]. In order to check this possibility, we fitted a dynamic scattering law, Sjjn(Q>(o)> derived from the Havriliak-Negami (HN) function (see Eqs. 32 and 33) to the observed S(Q,co). The HN function (co) and SjjN(Q.ro) are given by... 

The molecular dynamics associated with the glass transition of polymers are cooperative segmental dynamics. The relaxation process of the cooperative segmental motions is known as the a-relaxation process. At the glass transition, the length scale of a cooperative segmental motion is believed to be 1-4 nm, and the average a-relaxation time is 100 s [56]. The a-relaxation process is represented by a distribution of relaxation times. In time-domain measurements, the a-relaxation is non-exponential and can be described by a stretched-exponential function. The most common function used to describe the a-process is that of the Kohlrausch-Williams-Watts (KWW) [57, 58] equation ... 

The weakness of this approach as first introduced by Narayanaswamy [3] is the assumption that a single relaxation time can describe the physical aging process accurately. Some improvement can be achieved by introducing the Kohlrausch-Williams-Watts (KWW) stretched exponential function [8] to describe the distribution of relaxation times eq>ressed as ... 

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