Fluctuation decay

Very recently Tokita and Tanaka have performed a macroscopic measurement of the friction coefficient and have found its dramatic decrease with slight opacity near the critical point [86]. We conjecture that such a large anomaly was caused by stationary domain structures with large spatial scales and not by the thermal fluctuations decaying diffusively. 

It is obvious that, as p —> 0, the density fluctuations decay on a shorter scale than the charge fluctuations. 

Equation (237) shows that when the coupling constant is larger than the critical value Xc = 1 the ansatz [Eq. (234)] leads to an acceptable solution. This implies that for X < 1, density fluctuations decay to zero for a long time but for X > 1 they decay to a finite value/. The value off increases from/ = 1/2 for X — 1 to / = 1 for X —> oo. Thus the spectrum of density fluctuation exhibits a delta function peak at zero frequency, with strength / which is the characteristic of a glassy phase. Thus in the glass phase the translational motion is frozen in and the vibrational motion around the arrested position is described by (z). 

As shown by Leutheusser, the above analysis can be extended to show that at glass transition the density fluctuations decay with a long-time power law (0 t with a = 0.395. As one approaches the transition, the viscosity is predicted to diverge as e- 1 and ebelow and above the transition, respectively. e = 1 - X/Xc, p = (1 + a)/2a, and p = p — 1, where p 1.7-2. It is shown by Kirkpatrick [30] that the diffusion coefficient near glass transition goes to zero as e1. ... 

A proper closure strategy for the reaction rate thus depends on the ratio of the rate of reactions and the rate of mixing. In the previous case, considering very slow reactions the concentration fluctuations decay to zero before the reactions occur and no turbulence modeling is needed. The other extreme involves infinitely fast reactions so that local instantaneous chemical equilibrium prevails everywhere in the mixture. If the rate constant is very large k oo), the reaction rate can only be finite when c.4Cb - - c c 0. 

To clarify the dynamic character of the ring fluctuations, it is useful to introduce time correlation functions.210 211 The time correlation function CA(t) = < 4(s + t)A(s)) for a dynamical variable j4 is obtained by multiplying. <4(,s), the value of A at times, byA(s + t), the value taken by A after the system has evolved for an additional time, t, and averaging over the initial time s. If the averaging is done over a sufficiently long dynamical simulation of an equilibrated system, CA(t) will be independent of the initial time, s, used in the calculation it is then customary to write CA(t) = . If A is the fluctuation of a variable from its mean value, < y4(0) 2> is the mean-square fluctuation of the variable for an equilibrated system, while the time correlation function, CA(t), describes the average way in which the fluctuation decays. 

Not all fluctuations decay exponentially. We often want to have some parameter that typifies the time scale for the decay of the correlations. We therefore define the correlation time tc to be... 

In order to calculate a light-scattering spectrum we must have a model for the mechanism by which dielectric fluctuations decay. The remainder of this book is devoted primarily to the study of these fluctuations. 

Concentration fluctuations in polymer solutions that are in thermal equilibrium are well understood. The intensity of the polymer concentration fluctuations is proportional to the osmotic compressibility and the fluctuations decay exponentially with a decay rate determined by the mass-diffusion coefficient D. Probing these fluctuations with dynamic light scattering provides a convenient way for measuring this diffusion coefficient Z) [ 1],... 

Therefore, using the just formulated postulate in the isolated mixture from our material model—the regular linear fluid mixture (cf. end of Sect. 4.6), we expect that an arbitrary perturbed state (obtainable, say, by molecular fluctuations) decays back to the final uniform equilibrium state with maximum entropy [39,146] in which Eqs. (4.303), (4.302), (4.316)-(4.319), (4.321)-<4.326), (4.328)-(4.331), (4.336) are valid and gradients of pressures and chemical potentials are zero (see (4.327), (4.332), (4.333)). Let us denote by m° the whole mass of such an equilibrium mixture, by the mass of each constituent, by its total volume, by E° its total energy and by its total (and maximum) entropy. Therefore (cf. (3.240)-(3.242))... 

We start the discussion with crystallization under isothermal conditions. When the temperature drops below the melting temperature, density fluctuations increase to critical size. Below critical size, fluctuations decay and above critical size, they become crystallization nuclei that are able to grow to the new phase by attaching stems of adjacent chains (so-called nucleation and growth mechanism). The dynamics of crystallization is governed by the dynamic law (see Chapter 7, Equation (9)) ... 

The most commonly used technique for determining 5 is photon correlation spectroscopy (PCS) [also known as quasi-elastic light scattering (QELS)]. PCS has become one of the standard tools of the trade for the colloid chemist. In this technique concentration fluctuations arising from the diffusive motion of the dispersion particles give rise to fluctuations in the dielectric constant of the medium are monitored photometrically. These fluctuations decay exponentially with a time constant related to the diffusion coefficient, Ds, of the scatterer, which can in turn be related to its hydrodynamic radius through the Stokes-Einstein equation ... 

Before leaving the topic of Green-Kubo integrals for transport properties, we mention briefly the characteristics of the electric current correlation functions that are used to compute the electrical conductivity. Figure 18 shows the electric current and velocity autocorrelation functions for [C2mim][Cl] at 486 K and 1 bar. The current fluctuations decay rapidly and appear to vanish... 

The mixing time is defined as the time required for the normalized probe output to reach and remain between 95 and 105% ( 5%) of the final equilibrium value. This value is called the 95% mixing time. It can be difficult to identify the 95% mixing time accurately from the normalized probe outputs (Figure 4-16) because of the fine scale around the endpoint. Because the probe fluctuations decay exponentially, the data can be conveniently replotted in terms of a probe log variance as a function of time (Figure 4-17) ... 

In order for the particles to flow under applied shear, the packing must dilate, which implies particle motion perpendicular to the flow direction. There can be no average net motion, however, so the relevant information is embedded in the root mean square (rms) velocity fluctuations. The rms velocity fluctuations decay more... 

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