Density autocorrelation function

Since j-c-v, the electrical current density autocorrelation function and the velocity autocorrelation function are proportional to each other. The latter function, however, can be expressed with the help of the time derivative of the decaying pro-... 

Using linear response theory and noting (according to the results at the end of Section 5.1.3) that the (complex) electrical conductivity a is the Fourier transform of the current density autocorrelation function, we obtain from Eqn. (5.75) (see the equivalent Eqn. (5.21))... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

To obtain an approximate expression for the density autocorrelation function, first we consider that the density fluctuation is coupled only to the longitudinal current fluctuation, and its coupling to the temperature fluctuation and other higher-order components are neglected. 

Experimentally the density correlations fire most important, and we therefore exemplify the construction of grand canonical correlation functions with the density autocorrelation function or the density cimiulant. Recall the definition (3.14) of the local segment density of the m-th chain ... 

In the summation we omit the endpoint nm as usual.) The density autocorrelation function is defined as... 

To give another example we consider the density autocorrelation function Ida (q) defined as Fourier transform of the segment density correlations within a chain (cf, Eq. (5.17)). According to Eq. (5.24) this function obeys the sum rule... 

As a final example we consider the density autocorrelation function, where we first discuss the dilute limit... 

Time- and space-resolved major component concentrations and temperature in a turbulent gas flow can be obtained by observation of Raman scattering from the gas. (1, 2) However, a continuous record of the fluctuations of these quantities is available only in those most favorable cases wherein high Raman scattering rate and/or slow rate of time variation of the gas allow many scattered photons (> 100) to be detected during a time resolution period which is sufficiently short to resolve the turbulent fluctuations. (2, 3 ) Fortunately, in other cases, time-resolved information still can be obtained in the forms of spectral densities, autocorrelation functions and probability density functions. (4 5j... 

It is known that the density autocorrelation function or the intermediate scattering function F(q, ai) in the supercooled fluid phase can well be described by a stretched exponential function of the form F q,u>) A exp[—(t/computer simulations. This particular relaxation is called a relaxation, and such a characteristic decay manifests itself in a slower decay of the dynamical structure factor S q,u>) and of the a peak of the general... 

Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor ire experimentally measured, while in the present MD simulation only the self pairt was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self pMt is more easily obtained than the full autocorrelation the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation. 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

Bryk, T., and Mryglod, I. Charge density autocorrelation functions of molten salts Analytical treatment in long-wavelength limit. J. Cond. Matt. Phys., 2004, 16, p. L463-L469. 

Based on data of this type, the density autocorrelation function between pairs of primary particles can be defined as follows For the three-dimensional case of a suspended... 

A fractal is an object that displays scale invariant symmetry that is, it looks the same when viewed at different scales. Any real fractal object will have this scale invariance over only a finite range of scales. One important consequence of this symmetry is that the density autocorrelation function will have a power law dependence, which can be written as... 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

The superscripts c and m designate cluster and monomer, respectively. S q) is the static structure factor of the cluster which is the Fourier transform of the cluster, density autocorrelation function, and hence it contains information regarding the cluster morphology. The stracture factor has the asymptotic forms S(0) = 1 and S q) for q >... 

The structure factor and the density autocorrelation function are Fourier transform pairs thus. 

The statistical fluctuation or noise level of a toner image is also by these postulates dependent on the particle size distribution. The function which relates the statistical fluctuations to spatial frequency is the Wiener spectrum, which is the Fourier transform of the optical density autocorrelation function. In terms of toner images, it is a measure of the dimensional extent over which the presence or absence of a particular toner particle will contribute to density. The density fluctuations can be measured as a function of position, normally with a slit aperture. This is schematically represented in Figure 6 where the left-hand sketch is related to large particles and the right-hand one to small particles. The density data can be used to calculate noise power or Wiener spectrum (8). Formally, the Wiener spectrum is ... 

In analogy to the density autocorrelation function Tp(r) introduced in Section 1.5.2, we can define the concentration correlation function Tap(r) as... 

The double differential scattering cross section, according to (8.15), is proportional to the space-time Fourier transform of G(r,f). This is analogous to the fact, discussed in Section 1.5.2, that in the static approximation the intensity I(q) (or the differential scattering cross section ds/dQ) is given by the spatial Fourier transform of the density-density autocorrelation function defined in (1.79). In the special case of t = 0, G(r,0) denotes the probability of finding a particle at r when there is already a particle at position 0. G(r,0) is therefore related to the pair distribution function g(r) discussed in Section 4.1.1, as in... 

Form and structure factors, and the density autocorrelation function... 

To calculate the structure factor we need information about the distances between the particles i and y comprising the aggregate. This is encapsulated in the density autocorrelation function g ( ), which describes the distribution of separations between pairs of particles. This fundamental descriptor of aggregate structure will also make an appearance in our later discussion of aggregate hydrodynamics. It represents the average density of particles at a distance from any other particle in the aggregate ... 

Cai, J., Lu, N. and Sorensen, C. M. (1995). Analysis of fractal cluster morphology parameters structural coefficient and density autocorrelation function cutoff. J. Colloid Interface ScL, 171, 470-473. 

The intensity can be found from the X-ray diffraction experiment and the result compared with calculated diffraction pattern that is angular spectrum of the scattered X-ray intensity. To this effect, we should make a Fourier transform F(q) of the density function p(r) i.e. find the scattering amplitude and then take square of it, 7(q) = IF (q)l. This works well for solid crystals, but is not always convenient for liquids, liquid crystals and other soft matter materials in which the thermal fluctuations play a very substantial role. In such cases, the so-called density autocorrelation function appears to be more convenient. However, before to proceed along that way, we should separate two sources of scattering. 

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