Moderate jump diffusion

Four different models for the molecular dynamics have been tested to simulate the experimental spectra. Brownian rotational diffusion and jump type diffusion [134, 135] have been used for this analysis, both in their pure forms and in two mixed models. Brownian rotational diffusion is characterized by the rotational diffusion constant D and jump type motion by a residence time t. The motions have been assumed to be isotropic. In the moderate jump model [135], both Brownian and jump type contributions to the motion are eou-pled via the condition Dx=. ... 

Fig. 10.13. Correlation time as a function of the inverse temperature Estimated from line width (O) estimated from extrema separation based on Brownian diffusion (A), moderate jump ( ) and strong Jump ( ) (Ref. )...

Figure 48 shows a histogram of the terrace widths calculated from the STM frame shown in the inset. The maxima of the terrace width distributions are separated by a = 2.2 A. Thus, one may conclude that step edge atoms spend the predominant part of time in fee compatible hollow sites, indicating that their jump rate is moderate compared to the sampling rate (here 2.5 kHz). With lower miscuts, that is, wider terraces, quantization is less pronounced. Analyses of terrace width distributions and of step correlation functions, extracted from STM frames, allow the determination of the step-step interaction potential and the dimensionality of the diffusion processes at steps [165, and references therein],... 

We plot in Fig. 6.5 the dimensionless front velocity vt/u vs the reaction rate r on a log-log scale. The front velocity increases with r. For the cases / = a and I = 2a, the slope is very similar, but for / = oo it is steeper. In all cases the front velocity increases as a power law of r, straight line in a log-log plot, for small and moderate values of r and saturates to 1 for larger values, the slope in the log-log plot tends to 0. This behavior is due to the fact that an increase of the reaction rate r leads to an increase of the front velocity. However, the front cannot travel faster than the jump velocity of the particles if all of them jump in the backbone direction, i.e., V < ajx. For I = a and / = 2a the transport is diffusive, and the diffusion coefficient is properly defined. If this transport is combined with a KPP reaction, a Fisher velocity is expected, i.e., in both cases v fr. Computing numerically the slope from a linear fit in Fig. 6.5 we obtain and for / = a and I = 2a, respectively. The case / oo is quite different, because the transport is anomalous. Equation (5.36) with y = 1/2 yields v while the linear fit of the numerical results yields Numerical and analytical results are in good agreement. 

While the experimental results are suggestive, simulation results are conclusive in their support for Lieb and Stein s hypothesis. Further, the simulations suggest refinement of their hypothesis. MD simulations of the passive diffusion within the bilayer of molecules of a range of sizes, including methane, benzene, adamantane, and nifedipine clearly show that small molecule (methane and benzene) movement includes discrete rapid jumps between voids within the bilayer. These voids are often as much as the volume of benzene, but seldom much larger. These jumps have been observed to be moderated by the torsional isomerization of the lipid hydrocarbon chains, whose motion creates passages between existing voids [4, 5,7,21]. These jumps are occasional for benzene, however they are very frequent for the much smaller molecule, methane. The movement of adamantane and nifedipine is... 

The dynamical processes described in the previous section take place when the Mossbauer atom physically moves during the lifetime of the Mossbauer nucleus. However, even when the temperature is moderately low in comparison with the melting point of the crystal and diffusive jumps are rather infrequent, there is another source of time-dependent effects which has a strong influence on the spectrum. These arise when the environment of the Mossbauer nucleus changes within its lifetime, thereby altering the frequency coq of the Mossbauer radiation. These processes are usually called relaxation and they occur when the hyperfine interactions, by which the Mossbauer nucleus senses its environment, undergo time-dependent fluctuations. 

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