Correlation function defined

Although stratification, according to the plot in Fig. 10, occurs continuously as increases, it is accompanied by a curious structural reorganization in transverse directions (i.e., parallel to the planar substrate). A suitable measure of transverse structure is the pair correlation function defined in Eq. (62). However, for simplicity we are concerned only with the in-plane pair correlation function defined as [see Eq. (62)]... 

In homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

Similar Fourier transform pairs relate the spatial correlation functions defined in (3.40) and (3.41) to corresponding cospectra t) and t), respectively. 

While writing the above expression, the facts that have been used are GsD = CSC, GsBD = CqC0, and (G ) 1 = (CsylC l. Note that the definition of GsBD is different from the previous formulation [9] this follows from the difference in definition of CB, which has been discussed before. Here Cs and C are the phase space correlation functions defined as... 

Table III. Minus the total Si crystal valence electron energy per atom with relaxation energy and pseudopotential corrections included, along with the equilibrium lattice constant, bulk modulus, and cohesive energy calculated with four different exchange-correlation functionals (defined in the caption of Table I) are compared with experimental values. The experimental total energy is the sum of Acoh plus the four-fold ionization energy.

Next we extend our understanding of the distribution of time-averaged intensity to the time averaged correlation functions defined in Eq. (12). 

We notice that the correlation function defined by Eq. (147) is stationary. Thus, it fits the Onsager principle [101], which establishes that the regression to equilibrium of an infinitely aged system is described by the unperturbed correlation function. The authors of Ref. 102 have successfully addressed this issue, using the following arguments. According to an earlier work [96] the GME of infinite age has the same time convoluted structure as Eq. (59), with the memory kernel T(t) replaced by (1>,XJ (f). They proved that the Laplace transform of Too is... 

If one takes g n gir) as the pair correlation function defined in Eq. (5.2), the radial distribution function represents the number of particles of 5 in a shell up to rj around A. If r, is then (see Fig. 5.7), one can regard Eq. (5.2) as giving the coordination number of A in the liquid. In the example chosen for simplicity, species A is the same as species B but this of course is only true for radial distributions of monatomics, e.g., sodium. It is found in practice that in a liquid, g B r) settles to unity by the third or fourth atom away from the reference atom A. 

Finally, introduce c(r,r ), the Ornstein-Zernike direct correlation function defined by x (r, r ), and derive... 

In order to apply the WDA in the relativistic regime, a fully covariant extension of the concept of the pair correlation function would be desirable. To our knowledge this is, however, not available. Nevertheless, if one restricts the discussion to the (instantaneous) longitudinal limit, one can express xc[ ]> Eq. (3.21), via a relativistic pair correlation function defined in analogy to the nonrelativistic case as... 

Characteristic reorientational times can be calculated from MD simulations using time correlation functions defined by ... 

The correlation function Cp(t x) for various values of x is reported in Figs. 10a and 10b for N = 1000, and the relative error of correlation function, defined as... 

The second equality holds for a homogeneous and isotropic fluid. A related quantity is the locational pair correlation function, defined in terms of the locational pair distribution function, i.e.,... 

As in the T, V, N ensemble, one may introduce correlation functions in the T, V, ji ensemble. Of particular importance is the pair correlation function defined by... 

Different normalizations of G (f, t + x) can be applied in the analysis of photon-number correlations. Here, we analyze the normalized two-time second-order intensity correlation functions defined as... 

Our catalog of exact representations of e for the rigid-particle model would be incomplete without the expression for e in terms of site-site (i.e., atom-atom) correlation functions, defined in terms of h( 2) by... 

In the limit X = (a/L) /r with L < a/2, the system should consist of dipolar dumb-bells. The as>miptotic form of the direct correlation function (defined through the Omstein-Zemike equation) for this system (in the absence of a solvent) is given by... 

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