Nonexponential models

A modified Maxwell model and a nonexponential model were used by Nussinovitch et al. (1989) for characterization of the stress relaxation of agar and alginate gels as described by the equation... 

The pathway model makes a number of key predictions, including (a) a substantial role for hydrogen bond mediation of tunnelling, (b) a difference in mediation characteristics as a function of secondary and tertiary stmcture, (c) an intrinsically nonexponential decay of rate witlr distance, and (d) patlrway specific Trot and cold spots for electron transfer. These predictions have been tested extensively. The most systematic and critical tests are provided witlr mtlrenium-modified proteins, where a syntlretic ET active group cair be attached to the protein aird tire rate of ET via a specific medium stmcture cair be probed (figure C3.2.5). 

Figure4.9 shows the best Freundlich quenching model plots. While more complex models could fit the data, this simple two-parameter model fits all the data within experimental error. Since the decays are highly nonexponential, the parameters represent lumped parameters that are some average over the available sites, lifetimes, and A sv s. A detailed discussion of the different models tried is given elsewhere.

This model permits xR to be determined using information on the fluorescence decay in a very simple way. If unrelaxed fluorophores are excited, the decay is exponential beyond the relaxation range and, in this range, consists of two components t, and r2. These components will be simple functions of xR and t>. If we assume that emission on the short-wavelength side occurs only from the unrelaxed state and that the simultaneous loss of emitting quanta occurs due to relaxation, then the longer component, t, equals xF, and the shorter one, t2, equals 1(1/t + jxF). Unfortunately, this approach is difficult to apply when the decay is nonexponential, which is almost always the case with proteins (see Section 2.3.1.). 

Koppel [180] has performed exact time-dependent quantum wave-packet propagations for this model, the results of which are depicted in Fig. 2A. He showed that the initially excited C state decays irreversibly into the X state within 250 fs. The decay is nonexponential and exhibits a pronounced beating of the C and B state populations. This model will allow us to test mixed quantum-classical approaches for multistate systems with several conical intersections. 

Later, Edmund Phelps and Robert Poliak (1968) offered a different approach to nonexponential time discounting. In their model, discounted utility is a sum of utility from consumption in the current period and... 

Engstrdm et al. [112] used molecular dynamics simulations to study quadrupole relaxation mechanism for Li+, Na+, and Cl ions in dilute aqueous solutions. They found that NMR relaxation rate for these ions was determined by the relaxation of water molecules in the first solvation shell. The simulations show nonexponential solvation dynamics which can be modeled by two relaxation time constants < 0.1 ps and x2 lps (see Table 4). 

The hydration dynamics were also studied by Karim et al. [58], These authors used the TIP4P model of water in their molecular dynamics simulations. The observed hydration dynamics was nonexponential with average time constant in the range of 0.4-0.7ps. In this case simulated relaxation of the first solvation shell was also faster than that of the other shells. 

Table I consists of a compilation of r /4> ratios as a function of X. Our results and those presented for p-GaP and n-ZnO are in rough agreement with this simple model (8,9,30,31,32). Construction of a more refined model awaits incorporation of other data (nonexponential lifetimes, electroabsorption, carrier properties, intensity effects, quantitative evaluation of 4>nr by photothermal spectroscopy, e.g.) and examination of other systems.

Often, one defines nonexponential relaxations in terms of a time-depen-dent rate coefficient k(t) through p(t) = exp(—k(t)t). For the fractional Kramers model one therefore obtains the rate coefficient k(t) = ln a(—r ta) /t which leads to two limiting cases, the short-time self-simi-... 

It has been claimed that reactions in proteins can, as an approximation, be formulated within the Kramers reaction theory of barrier crossing [106]. The highly nonexponential relaxation pattern can now be explained by our model,... 

The stochastic description of barrierless relaxations by Bagchi, Fleming, and Oxtoby (Ref. 195 and Section IV.I) was first applied by these authors to TPM dyes to explain the observed nonexponential fluorescence decay and ground-state repopulation kinetics. The experimental evidence of an activation energy obs < Ev is also in accordance with a barrierless relaxation model. The data presented in Table IV are indicative of nonexponential decay, too. They were obtained by fitting the experiment to a biexponential model, but it can be shown50 that a fit of similar quality can be obtained with the error-function model of barrierless relaxations. Thus, r, and t2 are related to r° and t", but, at present, we can only... 

Fig. 5.16. Nonexponential decay curves of crystal violet (excited with synchrotron radiation from BESSY) in glycerol at six different temperatures. Within experimental error limits, the decays can be fitted by a biexponential model. The two lifetime components at 13°C are 330 and 850 ps, respectively. Upon reducing the temperature, both fitted decay times increase, as well as the relative weight of the slower decay component. At -72°C, a monoexponential decay is observed (2.71 ns).

While polydisperse model systems can nicely be resolved, the reconstruction of a broad and skewed molar mass distribution is only possible within certain limits. At this point, experimental techniques in which only a nonexponential time signal or some other integral quantity is measured and the underlying distribution is obtained from e.g. an inverse Laplace transform are inferior to fractionating techniques, like size exclusion chromatography or the field-flow fractionation techniques. The latter suffer, however, from other problems, like calibration or column-solute interaction. 

In the context of chemical reactions that are subject to dispersive kinetics as a result of structural disorder, the above model suggests that a widening of the intermediate region between the Arrhenius law and low-temperature plateau should occur. The distribution of barrier heights should also lead to nonexponential kinetic curves (see Section 6.5). 

The initial idea is to use the differential equations of a probabilistic transfer model with hazard rates varying with the age of the molecules, i.e., to enlarge the limiting hypothesis (9.2). The objective is to find nonexponential families of survival distributions that are mathematically tractable and yet sufficiently flexible to fit the observed data. In the simplest case, the differential equation (9.7) links hazard rates and survival distributions. Nevertheless, this relation was at the origin of an erroneous use of the hazard function. In fact, substituting in this relation the age a by the exogenous time t, we obtain... 

This section proposes the use of a semi-Markov model with Erlang- and phase-type retention-time distributions as a generic model for the kinetics of systems with inhomogeneous, poorly stirred compartments. These distributions are justified heuristically on the basis of their shape characteristics. The overall objective is to find nonexponential retention-time distributions that adequately describe the flow within a compartment (or pool). These distributions are then combined into a more mechanistic (or physiologically based) model that describes the pattern of drug distribution between compartments. The new semi-Markov model provides a generalized compartmental analysis that can be applied to compartments that are not well stirred. 

The phase-type distribution has an interpretation in terms of the compartmental model. Indeed, if the phenomenological compartment in the model, which is associated with a nonexponential retention-time distribution, is considered as consisting of a number of pseudocompartments (phases) with movement... 

The rebinding dynamics to microperoxidase are nonexponential due to a solvent cage effect. To deduce the time constant for rebinding CO from the solvent cage, the recombination kinetics were modeled according to... 

However, these results are highly model dependent (64-66). For example, a model based on a random distribution of dipole-dipole interactions gives an exponential decay in the slow modulation limit (64). As another example, a nonexponential frequency correlation decay gives different intermediate results than the Kubo-Anderson model (67,68). 

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