Large-scale fluctuations

Dfc Damkohler number characterizing Kolmogorov-scale fluctuations D Damkohler number characterizing large-scale fluctuations I integral length scale... 

Under the conditions just discussed, large time-scale fluctuations of the density of pairs were neglected. In order to take into account such large-scale fluctuations we introduce an average of the pair operator over a small volume, p. This small volume is much smaller than the total volume of the system, however, it is large enough to smear out small-scale fluctuations. We call the quantity p the smoothed pair operator. Then it is useful to define... 

If we have only small-scale fluctuations, obviously Spa(3 is zero. Therefore, 8pafj describes the large-scale fluctuations. Thus the entire fluctuation (5.3) can be divided into two parts ... 

The usual collision integral, Ia(pt), is defined by (4.62). The additional term, which is produced by the large-scale fluctuations, is determined by the following expression ... 

The quantity 8Ia is the linearized collision operator corresponding to (4.62). The subscript 0 in the second term means that the operator acts only on the function fa and not on the delta distribution. The contribution Aap in (5.14) is determined by /a/3, that is by the large-scale fluctuations. We consider only the approximation that Aa/ is equal to zero. 

As discussed in Section IV, we may derive from (5.9) a kinetic equation for the distribution function of the electrons, which takes into account reactions and large-scale fluctuations. From such an equation we get a Langevin-type rate equation for the concentrations of electrons, ions, and atoms, which reads in the case of local equilibrium35... 

Large-scale fluctuations of an animal virus particle are important for its ability to infect cells, but another type of particle dynamic behavior has been known for decades and was first identified in a plant virus. Brome mosaic virus (BMV), a single-stranded RNA (ssRNA) virus with T=3 icosahedral... 

The spectroscopic data have been compared with the theoretical predictions of the Doi- Edwards model. In the time scale of our experiments, a quantitative agreement between experiment and theory is obtained if chain length fluctuations, retraction and reptation are taken into account. In the case of star polymers, the large scale fluctuation mechanism as proposed by Pearson and Helfand associated with the retraction process is accounting for... 

This coarse-graining can also be used to justify the "drumhead model of an interface, cf. fig. 6, where on a more macroscopic scale the internal structure of the interface is disregarded, and one is more interested in large scale fluctuations of the local position z = h(x,y) of this interface. In this sharp kink approximation the interface is described similarly to an elastically deformable membrane. 

Figure 7. The meandering Gaussian plume. The cross-section of the time-averaged tracer concentration has a normal or Gaussian distribution, as does the hazard area envelope within which the actual plume is to be found. The latter is obtained by including the large-scale fluctuations in the time-average of the tracer concentration (after Gifford, 1959)...

W. L. Grosshandler and P. Joulain, The Effect of Large-Scale Fluctuations on Flame Radiation, Progress in Astronautics and Aeronautics, 105, Part II, AIAA, Washington, pp. 123-152,1986. 

Figure 3.2. Fluctuation of the direction of the dipole moment vector of a tagged water molecule in the course of its motion through the liquid. The places where large-scale fluctuations occur are indicated by arrows. Adapted wifli permission from/. Phys. Chem. B., 112 (2008), 14230. Copyright (2008) American Chemical Society.

The range of dynamic anomalies shown by liquid water at room temperature and pressure is truly amazing. It is not just the extended hydrogen-bond network that is responsible for this but also the small lifetime of an HB that allows large-scale fluctuations in liquid water (as discussed in Chapter 1). 

It is interesting to note that the useful properties of supercritical water arise from the breakdown of the extensive HB network that is at least partly responsible for many of the anomalies of liquid water. We have discussed how the use of the idea inherent in the Widom line helps in understanding the large-scale fluctuations observed in supercritical water. Because of the large separation of timescales between vibrational relaxation and density relaxation, the vibrational line widths are influenced significantly by the transient density inhomogeneity present near the critical temperature. 

The transition from the Eq. (46) to the law (2.47) atDMDAACh polymerization reaetion description is realized within the range of c 0.727-0.808 kg/1. As it is known [50], at the critical value ds=4 large-scale fluctuations appear in the system. Their influence on DMDAACh synthesis process will be considered lower. 

Pitzer, K. S. (1986) Large-Scale Fluctuations and the Critical Behavior of Dilute NaCl in H20, Journal of Physical Chemistry 90, 1502-1504... 

It is necessary to mark, that the singular dependence on time in the equation (2) appears simultaneously with large-scale fluctuations (inhomogeneity) of traps density. If the traps ean move, their mobility averages the influence of spatial heterogeneity, so the assumptions resulting to (1) will be carried out better. It was shown, that in this case the concentration of the particles drops imder the combined law [4] ... 

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