Fast fluctuations

Even without using numerical methods, one can analyse some physically sound limiting cases of the exact solution for the case of strong collisions (7.64). First of all, (7.64) evidently reduces to the results of Robert and Galatry in the quasi-static case, i.e. when tc — oo. An opposite limiting case of fast fluctuations... 

Having a closer look at the pyramid algorithm in Fig. 40.43, we observe that it sequentially analyses the approximation coefficients. When we do analyze the detail coefficients in the same way as the approximations, a second branch of decompositions is opened. This generalization of the discrete wavelet transform is called the wavelet packet transform (WPT). Further explanation of the wavelet packet transform and its comparison with the DWT can be found in [19] and [21]. The final results of the DWT applied on the 16 data points are presented in Fig. 40.44. The difference with the FT is very well demonstrated in Fig. 40.45 where we see that wavelet describes the locally fast fluctuations in the signal and wavelet a the slow fluctuations. An obvious application of WT is to denoise spectra. By replacing specific WT coefficients by zero, we can selectively remove... 

The Landau-Zener formula in the limit J2/ v - oo gives PLZ = 1 whereas Eq. (183) for the case of fast fluctuations in this limit gives P — At small values of J2/ u, both formulas give the same result, P = 2itJ2/ v. ... 

Liquid-like portion Fast fluctuation motions... 

An ACF is always an even function, symmetrical with respect to =0. The fast decreasing ACF of a very fast fluctuating stochastic signal can be considered as an impulse. 

One possible structure would consist of irregular forms among which spheres and cylinders with alternatively hydrophobic and hydrophilic outer surfaces are formed according to Figure 8. This suggestion has an immediate attraction its features are similar to critical phenomena [cf., two-dimensional Ising s model (40)]. The resemblance between micellar associations and the fast fluctuation aggregates before phase separations... 

In the pheromone orientation system it has been shown how important time aspects are. Several species will not be attracted to a pheromone source unless the stimulus arrives in a pulsed fashion, mimicking the filamentous structure of a natural odor plume. Correlates to this requisite have been found among AL neurons, where both fast neurons, able to code fast fluctuations in concentration, and slow neurons, seemingly only coding qualitative aspects of the plume, are present. 

Pure dephasing describes the adiabatic modulation of the vibrational energy levels of a transition caused by fast fluctuations of its environment (29,30). Measurement of this quantity, and how this quantity changes with temperature, solvent, viscosity, or other experimental parameter, provides detailed insight into the dynamics of the system. 

In conclusion, due to the fast fluctuations of X, the system of interest is driven by the mean value of this variable. 

The stochastic model of ion transport in liquids emphasizes the role of fast-fluctuating forces arising from short (compared to the ion transition time), random interactions with many neighboring particles. Langevin s analysis of this model was reviewed by Buck [126] with a focus on aspects important for macroscopic transport theories, namely those based on the Nernst-Planck equation. However, from a microscopic point of view, application of the Fokker-Planck equation is more fruitful [127]. In particular, only the latter equation can account for local friction anisotropy in the interfacial region, and thereby provide a better understanding of the difference between the solution and interfacial ion transport. 

A further factor influencing the vertical ozone distribution is the vertical turbulence. This is, however, not the cause of fast fluctuations. If, for instance, a considerable change should be caused by turbulence at an altitude of 30 km. against the tendency to re-establish the photochemical equilibrium, the exchange factor must temporarily show a value of 0.1 to 1 gram per cm.-second, which would be 10 to 100 times larger than the mean value at this altitude. 

We have chosen to encompass our methodology in the necessarily limited framework of rotational FPK operators for describing the solute molecule and the solvent cages (slow fluctuating solvent structures) with translational FPK operators for describing stochastic fields (fast fluctuating solvent structures). We are aware that a truly complete description... 

G2. Gachon, A. M., Ghaddah, M., Kitzis, A., Wacjman, H., and Dastugue, B., Fast fluctuations of glycosylated hemoglobins. 1. Implications for the preparation and storage of samples for hemoglobin Aj, determinations. Clin. Chim. Acta 121, 125-131... 

Polydispersity of simple bile salt micelles can only be assessed by modem QLS techniques employing the 2nd cumulant analysis of the time decay of the autocorrelation function [146,161]. These studies have shown, in the cases of the 4 taurine conjugates in 10 g/dl concentrations in both 0.15 M and 0.6 M NaCl, that the distribution in the polydispersity index (V) varies from 20% for small n values to 50% for large n values [6,146]. Others [112] have foimd much smaller V values (2-10%) for the unconjugated bile salts in 5% (w/v) solutions. Recently, the significance of QLS-derived polydispersities have been questioned on the basis of the rapid fluctuation in n of micellar assemblies hence V may not actually represent a micellar size distribution [167-169]. This argument is specious, since a micellar size distribution and fast fluctuations in aggregation number are identical quantities on the QLS time scale (jusec-msec) [94]. 

According to eq. (21), transverse field measurements allow, in principle, separation of the static field width (oc o) from the fluctuation rate (1/r) in an intermediate case. In practice, this is rather difficult and the combination of zero and longitudinal field measurements are more powerful in this respect, as will be shown further below. In the static limit (r —> oo) we can extract the second moment of the field distribution (see eq. 20). In the fast fluctuation limit only the product a r appears (eq. 24) and independent information on one of the two quantities is needed. 

Recall (above) that in ZF, fast field fluctuations in a Gaussian distribution result in muon spin relaxation at a rate determined solely by AVv, so that ZF (in that limit) cannot separate the distribution width from the fluctuation rate, but LF in principle can. For a Lorentzian distribution, the ZF fast-fluctuation limit relaxation rate A(0) 4a/3 depends... 

The first task is to formulate a theoretical signal fimction, which we denote as representing muon polarization as a fimction of time. For example, a transverse field measurement in the fast fluctuation limit should have the form... 

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