Density fluctuations variables

The non-consen>ed variable (.t,0 is a broken symmetry variable, it is the instantaneous position of the Gibbs surface, and it is the translational synnnetry in z direction that is broken by the inlioinogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach 121, it is related to the number density fluctuation 3p(x,z,t) as... 

As indicated, the power law approximations to the fS-correlator described above are only valid asymptotically for a —> 0, but corrections to these predictions have been worked out.102,103 More important, however, is the assumption of the idealized MCT that density fluctuations are the only slow variables. This assumption breaks down close to Tc. The MCT has been augmented by coupling to mass currents, which are sometimes termed inclusion of hopping processes, but the extension of the theory to temperatures below Tc or even down to Tg has not yet been successful.101 Also, the theory is often not applied to experimental density fluctuations directly (observed by neutron scattering) but instead to dielectric relaxation or to NMR experiments. These latter techniques probe reorientational motion of anisotropic molecules, whereas the MCT equation describes a scalar quantity. Using MCT results to compare with dielectric or NMR experiments thus forces one to assume a direct coupling of orientational correlations with density fluctuations exists. The different orientational correlation functions and the question to what extent they directly couple to the density fluctuations have been considered in extensions to the standard MCT picture.104-108... 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

This is a well-known expression obtained by many authors [3, 30, 67]. This expression can be derived when the subspace orthogonal to the hydrodyna-mical variables is constructed only by the bilinear products of the density fluctuation. In the language used by Kadanoff and Swift, we can say that the two density mode contribution to r (q, t) is given by Rlpp. 

The front is inherently unstable, however, and this is often studied by a linear stability analysis. Infinitesimal perturbations are applied to all of the variables to simulate reservoir heterogeneities, density fluctuations, and other effects. Just as in the Buckley-Leverett solution, the perturbed variables are governed by force and mass balance equations, and they can be solved for a perturbation of any given wave number. These solutions show whether the perturbation dies out or if it grows with time. Any parameter for which the perturbation grows indicates an instability. For water flooding, the rate of growth, B, obeys the proportionality... 

The probability density function, written as pif), describes the fraction of time that the fluctuating variable/ takes on a value between/ and/ + A/. The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side while p(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of/. Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as... 

To illustrate a FT, we use the Dyson equation that is characteristic of helds and we focus on the local density fluctuations that is the most natural variable in this approach. Two systems are considered a homogeneous Yukawa fluid and an ionic soluhon near a hard wall. 

Another kind of dissimilarities both from simple liquids and a binary mixtures of neutral particles is caused by the factor B(k = 0) = Bo. Let us use the results obtained above for deriving the last two hydrodynamic equations [see (41) and (42)], describing the longitudinal dynamics of the model in small k limit. It can be done directly from the Eqs. (46) and (47), taking into account the existing relations between two subsets of dynamic variables, describing the density fluctuations, namely, pk, i k and pk, 7k [see Eq. (51)]. In small k... 

It projects any variable into the space perpendicular to linear density fluctuations. Introducing it into (5c) is straightforward because couplings to linear density can not arise in it anyway. One obtains... 

The fluctuating variables aie thereby projected onto pair-density fluctuations, whose time-dependence follows from that of the transient density correlators q(,)(z), defined in (12). Tliese describe the relaxation (caused by shear, interactions and Brownian motion) of density fluctuations with equilibrium amplitudes. Higher order density averages are factorized into products of these correlators, and the reduced dynamics containing the projector Q is replaced by the full dynamics. The entire procedure is written in terms of equilibrium averages, which can then be used to compute nonequilibrium steady states via the ITT procedure. The normalization in (10a) is given by the equilibrium structure factors such that the pair density correlator with reduced dynamics, which does not couple linearly to density fluctuations, becomes approximated to ... 

Re3Tiolds decomposition and time averaging were then applied to the instantaneous variables in the volume average model equations. However, it was assumed that none of the densities fluctuate. The terms of fluctuating quantities with order higher than two were considered small compared to those of first and second order and thus neglected. 

In a homogeneous equilibrium system the ensemble average (p(r)) = p is independent of r, and the difference /)(r) = /)(r) — p is a random function of position that measures local fluctuations from the average density. Obviously (5p(r)) = 0, while a measure of the magnitude of density fluctuations is given by The density-density spatial correlation function measures the correlation between the random variables p(r ) and <5p(r"), that is, C(r, r") = (5/)(r )<5p(r"))- In a homogeneous system it depends only on the distance r — r",... 

Fig. 2.4.1. Schematic of apparatus for measuring the spectral density of a fluctuating variable.

The last equation represents the fluctuation of the energy of a subsystem within a control volume. In order for the system to be stable, the fluctuation in the energy and the number of the molecules inside the system must be finite and hence Cy and — dP/d/V)f j must be positive and finite. In both of the above equations the definitions of and imply that these variables are positive, therefore Cy and — dP/dV)f jZXt positive. Adding heat to a reahstic material always increases the temperature, therefore Cy is always positive. However, in some conditions, the value of — dP/dV) j. can become small and very close to zero, which increases the fluctuation in the number of the molecules in the control volume. Therefore, the liquid density fluctuates in small control volume. This shows the fact that in some conditions, although the material is stable and the requirements for stability are met, fluctuations in some properties of the material may be large enough to shift the substance into a new phase. 

The density fluctuation of the total sample is the superposition of local fluctuations in a dynamic, time dependent, equilibrium. We now consider, as a subsystem of the total liquid sample, a fixed (but arbitrary) volume AV which contains a variable number of molecules. The local fluctuations are considered random with no correlation between the various AV s. This subsystem can be considered as a grand canonical ensemble in equilibrium with its surroundings (the balance of the sample). Then,... 

Vitreous silica has a unique set of properties. It is produced either from natural quartz by fusion or, if extreme purity is required, by chemical vapor deposition or via a sol-gel routes. Depending on the manufacturing process, variable quantities impurities are incorporated in the ppm or ppb range, such as Fe, Mg, Al, Mn, Ti, Ce, OH, Cl, and F. These impurities and radiation-induced defects, as well as complexes of impurities and defects, and also overtones, control the UV and IR transmittance. In the visible part of the spectrum, Rayleigh scattering from thermod)uiamically caused density fluctuations dominates. Defects are also responsible for the damage threshold under radiation load, and for fluorescence. The refractive index n and the absorption... 

The local mobility A in general depends on the environment. In the simplest model (local coupling approximation), A is assumed to be a constant related to an effective diffusion coefficient D . This assumption is valid when the segment density fluctuations are small compared to the average densities o. Another simple modd is to assume that A depends on the segment density as A =Ao< (r,t), where Aq is a constant. The formal theory of time evolution of density variables with general kinetic coeffidents was developed by Kawasaki and Sddmoto, but this is rather difficult to implement in practice. 

This equation gives the time evolution of density fluctuations resolved in their Fourier modes. The Fourier variable k is actually the scattering wave vector... 

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