Laplace transform case studies

A study of electron scavenging in multipair clusters [21] has shown that the total scavenging probability decreases with increasing number of ion pairs in the cluster. However, the Laplace transform relationship [Eq. (28)] between the scavenging probability and the recombination kinetics was found to work reasonably well also in the multipair case. 

The time dependence of the yield of hydrated electrons during the chemical stage of water radiolysis (ca. 10 to 10 sec) can be obtained from scavenging studies and by direct observation using pulse radiolysis. In the former case, the scavenger dependence G(S) is converted into the time dependence using the inverse Laplace transform of F(S) to F(t), which is given by Eqs. (19) and (20) ... 

In the case of reaction kinetics described by eqn. (3) and eqn. (3) of Chap. 4, the problem under study, as in refs. 16 and 17, can be solved by using the inverse Laplace transform. Actually, differentiating both sides of eqn. (3) of Chap. 4 with respect to t and using the notation x = ve exp ( 2Rjae) we have... 

When the environment of the particle is itself out-of-equilibrium, as is the case for a particle evolving in an aging medium, we showed how the study of both the mobility and the diffusion of the particle allows one to obtain the effective temperature of the medium. We derived an out-of-equilibrium generalized Stokes-Einstein relation finking the Laplace transform of the mean-square displacement and the z-dependent mobility. This relation provides an efficient way of deducing the effective temperature from the experimental results. 

Both the case where the Laplace transform of K(t) of Eq. (24) diverge (superdiffusion) or vanish (subdiffusion) must be treated with caution. These conditions will be the main subject under study in this review. The existence of environment fluctuations makes it possible for us to interpret the electron transport as resulting from random jumps, without involving the notion of wave-function collapse, but this is limited to the case of Poisson statistics. Anderson... 

Boutis and coworkers report measurements of the relaxation times of water hydrated N. clavipes and A. aurantia spider silks as a function of temperature by deuterium 2D T1-T2 inverse Laplace transform (ILT) NMR to study the distribution, population and dynamics of hydration water at different temperatures finding correlation times much longer than those for free water and in some cases increasing with increased T. MD simulations reveal that peptides prepared from a number of repeating motifs show inverse temperature transition behavior found for example in protein elastin. 

In this chapter another usejul tool will be described for model analysis linearization of model equations. When combined with Laplace or z-transform, the process model can be decomposed into linear transfer functions and the dynamic response of the variable of interest can easily be obtained and compared with the solution from a simulation study. The advantage of working with linear transfer functions is that process variable interaction can be visualized and better understood than in the case of a simulation study. 

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