Omstein-Zemike

The direct correlation fimction c(r) of a homogeneous fluid is related to the pair correlation fimction tiirough the Omstein-Zemike relation... 

The second tenu in the Omstein-Zemike equation is a convolution integral. Substituting for h r) in the integrand, followed by repeated iteration, shows that h(r) is the sum of convolutions of c-fiinctions or bonds containing one or more c-fiinctions in series. Representing this graphically with c(r) = o-o, we see that... 

Combining tliis witli the Omstein-Zemike equation, we have two equations and tluee unknowns h(r),c(r) and B(r) for a given pair potential u r). The problem then is to calculate or approximate the bridge fiinctions for which there is no simple general relation, although some progress for particular classes of systems has been made recently. 

The compressibility equation can also be written in tenns of the direct correlation fiinction. Taking the Fourier transfomi of the Omstein-Zemike equation... 

From the Omstein-Zemike equation in Fourier space one finds that... 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

Theories based on the solution to integral equations for the pair correlation fiinctions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Furtlier improvements for simple fluids would require better approximations for the bridge fiinctions B(r). It has been suggested that these fiinctions can be scaled to the same fiinctional fomi for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation fiinction c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

In the limit k = (a/i) /i with L < all, the system should consist of dipolar dumb-bells. The asymptotic fonn of the direct correlation fiinction (defined tln-ough the Omstein-Zemike equation) for this system (in the absence of a solvent) is given by... 

Integral equation theories are widely used in the theoretical study of liquids. There are two broad classes of integral equation theories those based on the Bom-Green-Yvon (BGY) hierarchy and those based on the Omstein-Zemike (OZ) equation [88]. Although the formalism is exact in both classes, it is generally easier to fashion approximations in the case of the OZ-equation-based approach, and this type of theory has therefore been more popular. Surprisingly, the BGY approach has never been implemented for nonuniform polymers, and this section is therefore restricted to a discussion of the OZ-equation-based approach. 

Omstein-Zemike theory of the critical opalescence which operates also with a linear equation for the joint correlation function. 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

To calculate Rpp(t), the two-particle direct correlation function c 2(q) is required which is obtained from the nearly analytical expression given by Baus and Colot for a 2-D system [177]. The static structure factor S(q) has been calculated from the two-particle direct correlation function through the well-known Omstein-Zemike relation [21]. 

Here Si(k) is the equilibrium structure factor of the solvent, which we approximate by the Omstein-Zemike expression, given by (87)... 

Chandler and Andersen introduced the site-site Omstein-Zemike (SSOZ) integral equation for the radial correlation functions between pairs of interaction sites of polyatomic species [14,27],... 

If, being the Fourier transform of u x). From Eqs. (4.3) and (4.10), the structure factor for the fluctuation is of the usual Omstein-Zemike form. 

Differentiation of the Fourier transformed Omstein-Zemike equation with respect to T(temperature) gives the following expression. 

Figure 9. Profiles of the order parameter Q near a defect. Omstein-Zemike profiles exhibit longer-ranging tails than Gaussian profiles, implying a more extensive modification of the order parameter of the host stracture. Modified from Figure 5 in Salje (1995).

A replica Omstein-Zemike integral equation approach... 

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