Concentration fluctuations time correlation functions

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

Two applications of the fluctuating diffusion equation are made here to illustrate the additional information the fluctuations provide over and beyond the deterministic behaviour. Consider an infinite volume with an initial concentration, c, that is constant, Cq, everywhere. The solution to the averaged diffusion equation is then simply = Cq for all t. However, the two-time correlation function may be shown [26] to be... 

Our chief emphasis will be on electrolyte solutions in which one of the ionic species is a polyelectrolyte and the remaining species are small ions. The spectrum will then be proportional to the time-correlation function of the polyelectrolyte concentration fluctuation alone. 

One of the most important principles of linear response theory relates the system s response to an externally imposed perturbation, which causes it to depart from equilibrium, to its equilibrium fluctuations. Indeed, the system response to a small perturbation should not depend on whether this perturbation is a result of some external force, or whether it is just a random thermal fluctuation. Spontaneous concentration fluctuations, for instance, occur all the time in equilibrium systems at finite temperatures. If the concentration c at some point of a liquid at time zero is (c) + 3c(r, t), where (c) is an average concentration, concentration values at time t + 8t dXt and other points in its vicinity will be affected by this. The relaxation of the spontaneous concentration fluctuation is governed by the same diffusion equation that describes the evolution of concentration in response to the external imposition of a compositional heterogeneity. The relationship between kinetic coefficients and correlations of the fluctuations is derived in the framework of linear response theory. In general, a kinetic coefficient is related to the integral of the time correlation function of some relevant microscopic quantity. 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

A chemical relaxation technique that measures the magnitude and time dependence of fluctuations in the concentrations of reactants. If a system is at thermodynamic equilibrium, individual reactant and product molecules within a volume element will undergo excursions from the homogeneous concentration behavior expected on the basis of exactly matching forward and reverse reaction rates. The magnitudes of such excursions, their frequency of occurrence, and the rates of their dissipation are rich sources of dynamic information on the underlying chemical and physical processes. The experimental techniques and theory used in concentration correlation analysis provide rate constants, molecular transport coefficients, and equilibrium constants. Magde" has provided a particularly lucid description of concentration correlation analysis. See Correlation Function... 

Correlation functions are powerful tools in statistical physics, and in the above example they permit one to examine the behavior of a fluctuating system from a reference time back to previous times. Such fluctuations can occur in the concentration of two (or more) interconverting chemical species in dynamic equilibrium, and the technique of concentration correlation analysis permits one to determine the forward and reverse rate constants for their interconversion. See Concentration Correlation Analysis... 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

First, all time series of the five elements were univariate cross-correlated, e.g. each time series from the first sampling point in the river was cross-correlated with the time series from the second sampling point. Single time series of trace concentrations of metals in the river show a distinctly scattered pattern. There is a large fluctuation in the univariate cross-correlation functions for the five elements and, therefore, no useful information is obtained (Fig. 6-19). 

This ansatz can be rationalized by some theoretical considerations [325,326]. It is also supported by the experimental data at very low concentrations of the component A where the study is reduced to the dynamics of the probe A in host B. Each probe molecule experiences the same environment, which eliminates the complications from concentration fluctuations. We have mentioned in Section III, paragraph 4, that the probe rotational correlation function indeed has the Kohlrausch form. The differential between the probe rotational time xA and the host a-relaxation time xaB is gauged by their ratio, xA/xaB. As expected, the slower the host B compared with the probe A, the larger the coupling parameter, nA = (1 — pA), obtained from the stretch exponent (3A of the measured probe correlation function. The experimental data are shown in Fig. 52. For more details, see Ref. 172. 

As a result of time averaging, several new terms appear in the GDE. The fourth term on the left-hand side, the tiiictuating growth term, depends on the correlation between the fluctuating size distribution function n and the local concentrations of the gaseous species converted to aerosol. It result.s in a tendency for spread to occur in the particle size range—-a turbulent diffusion through v space (Levin and Sedunov, 1968). 

The nonequilibrium concentration fluctuations can be measured experimentally by dynamic light scattering. Fluctuating hydrodynamics predicts that the time-dependent correlations function C(k,t) ofthe scattered light is given by... 

The scattered light intensity correlation function C (/) has been measured over a very wide range of temperature and wave vectors for various systems. Typical intensity correlation functions C-(/) are depicted in Fig. 4 for the sake of illustration. These graphs show systematic deviations from the usual exponential decay, which are also observed for most of the systems we report in this part. As a remark, nonexponential decays that are small at low concentration, close to the critical point, become large for dense systems. From the initial slope of the time-dependent intensity correlation function, one can deduce the first cumulant F, which is the relaxation rate of the order parameter fluctuations. 

Probe diffusion was determined using quasi-elastic light scattering spectroscopy. QELSS monitors the temporal evolution of concentration fluctuations by measuring the intensity I(q,t) of the light scattered at time t, and calculating the intensity-intensity correlation function... 

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