Density fluctuations, correlation time

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 ... 

We have performed an analysis of the entire MoBbauer line shape taking into aecount a density fluctuation correlation function of the Kohlrausch type 0(f) = aexp( r /r) with P = 1/2 and the relaxation time t. We could thus also follow the onset of the structural relaxation a process related to the dynamic glass transition. The relaxation times decrease from about 10 s at to 5 x 10 s at 300 K. Details of these studies especially concerning the behaviour of swollen polymers will be given elsewhere. 

In order for relaxation to occur through Wj, the magnetic field fluctuations need to correspond to the Larmor precession frequency of the nuclei, while relaxation via requires field fluctuations at double the Larmor frequency. To produce such field fluctuations, the tumbling rate should be the reciprocal of the molecular correlation time, i.e., f), so most efficient relaxation occurs only when voT, approaches 1. In very small, rapidly tumbling molecules, such as methanol, the concentration of the fluctuating magnetic fields spectral density) at the Larmor frequency is very low, so the relaxation processes Wj and do not occur efficiently and the nuclei of such molecules can accordingly relax very slowly. Such molecules have... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Let us consider now behaviour of the gas-liquid system near the critical point. It reveals rather interesting effect called the critical opalescence, that is strong increase of the light scattering. Its analogs are known also in other physical systems in the vicinity of phase transitions. In the beginning of our century Einstein and Smoluchowski expressed an idea, that the opalescence phenomenon is related to the density (order parameter) fluctuations in the system. More consistent theory was presented later by Omstein and Zemike [23], who for the first time introduced a concept of the intermediate order as the spatial correlation in the density fluctuations. Later Zemike [24] has applied this idea to the lattice systems. 

The only problem necessary for developing the condensation theory is to add to the above-mentioned equation of the state the equation defining the function x(r)- Unfortunately, it turns out that the exact equation for the joint correlation function, derived by means of basic equations of statistical physics, contains f/iree-particle correlation function x 3), which relates the correlations of the density fluctuations in three points of the reaction volume. The equation for this three-particle correlations contains four-particle correlation functions and so on, and so on [9], This situation is quite understandable, since the use of the joint correlation functions only for description of the fluctuation spectrum of a system is obviously not complete. At the same time, it is quite natural to take into account the density fluctuations in some approximate way, e.g., treating correlation functions in a spirit of the mean-field theory (i.e., assuming, in particular, that three-particle correlations could be expanded in two-particle ones). 

The temporal evolution of spatial correlations of both similar and dissimilar particles for d = 1 is shown in Fig. 6.15 (a) and (b) for both the symmetric, Da = Dft, and asymmetric, Da = 0 cases. What is striking, first of all, is rapid growth of the non-Poisson density fluctuations of similar particles e.g., for Dt/r = 104 the probability density to find a pair of close (r ro) A (or B) particles, XA(ro,t), by a factor of 7 exceeds that for a random distribution. This property could be used as a good aggregation criterion in the study of reactions between actual defects in solids, e.g., in ionic crystals, where concentrations of monomer, dimer and tetramer F centres (1 to 3 electrons trapped by anion vacancies which are 1 to 3nn, respectively) could be easily measured by means of the optical absorption [22], Namely in this manner non-Poissonian clustering of F centres was observed in KC1 crystals X-irradiated for a very long time at 4 K [23],... 

It has been established that the volume phase transition of gels is an universal phenomenon [17]. Dynamic light scattering studies indicate that the dynamic fluctuations of the density correlation diverges in the vicinity of the volume phase transition point of the gel. It has also been shown that the time scale of the density fluctuations become slow in the vicinity of the volume phase transition... 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

Both Ti and T2 relaxations of water protons are mainly due to fluctuating dipole-dipole interactions between intra- and inter-molecular protons [62]. The fluctuating magnetic noise from all the magnetic moments in the sample (these moments are collectively tamed the lattice) includes a specific range of frequency components which depends on the rate of molecular motion. The molecular motion is usually represented by the correlation time, xc, i.e., the average lifetime staying in a certain state. A reciprocal of the correlation time corresponds to the relative frequency (or rate) of the molecular motion. The distribution of the motional frequencies is known as the spectral density function. 

These correlated fluctuations themselves ride on a further set of coherent fluctuations taking place at a much lower frequency scale and normally attributed to the phonons, the traditional exchange Bosons associated with superconductivity. Real systems are never devoid of ionic or nuclear motion, and at the very least it is now Hamiltonian (3) (and eventually its extension to alloys) that applies for a full discussion of superconductivity density fluctuations in the nuclear coordinates are omnipresent and of course their effects on electronic ordering have been evident for quite some time. An elementary estimate of the relative importance of (monopole) polarization arising from phonons and the (multipole) equivalents arising from internal fluctuations, primarily of a dipole character, can now be easily given. 

Since the collective orientational correlation time depends on the structure of a liquid, it is plausible that the rate of structural evolution of the liquid is proportional to this quantity. Thus, at lower temperatures rcon is longer and therefore the structural fluctuations are slower. As a result, motional narrowing is less effective as the temperature is lowered. While less motional narrowing would normally lead to a slower decay in the time domain, in this case the spectral density goes down to zero frequency. Thus, motional narrowing can reduce the spectral density at low frequencies and thereby decrease the intermediate relaxation time. 

Whereas FCS measures fluorescence fluctuations over time, a related technique, ICS, measures fluorescence fluctuations over space, in particular from images collected using a laser-scanning microscope (56). ICS analysis of pixels within a single image provides information about protein clustering and density. A variation of ICS known as image cross-correlation spectroscopy evaluates the interactions of molecules labeled with different fluorescent probes. ICS can also be performed on stacks of... 

Since the correlation time of these density fluctuations is given by... 

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