Ornstein-Zernike equation total correlation functions

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

The correlation functions play an essential role in the static description of homogenous classical liquids whose particles are taken to interact through an effective pair potential. The starting point of the liquid-state theory, in terms of correlation functions, is the well-known Ornstein-Zernike equation [25]. The total correlation function h r) defined in Section II is actually a sum of two contributions that is illustrated by the following relationship... 

Assuming the pair potential known, the radial distribution function for two-dimensional systems can be calculated using the two-dimensional version of the Ornstein-Zernike equation, Eq. (22), and one of the closure relations. Although Eq. (22) does not relate one to one the radial distribution function with the pair potential, one might attempt to invert the procedure to get u(r) from the experimental values for g(r). Thus, by taking the Fourier-Bessel (FB) transform [43,44] of Eq. (22) an expression for c(k) is obtained in terms of the FB transform of the measured total correlation function, i.e. 

Most modern theories are based on the equation of Ornstein and Zernike (OZ). This equation was developed eighty years ago to describe light scattering in a fluid. Its utility in the theory of fluids was realized much later, about thirty-five years ago. The OZ equation is obtained by defining the total correlation function,... 

At the core of any integral equation approach we have the (exact) Ornstein-Zernike (OZ) equation [300] relating the total correlation function(s) of a given fluid to the so-called direct correlation function(s). For the replicated system at hand, the OZ equation is that of a multicomponent mixture [30],... 

This is the Ornstein-Zernike equation. It is an exact integral equation relating the two 2-particle correlation functions li2(l,2) and C2(l,2). It is possible to motivate this equation form purely physical arguments the idea is to interpret the total correlation function li2(l,2) as the sum of all possible direct correlations, thus C2(l, 2) is termed the direct correlation function. We imagine that 112(1,2) is the sum of the direct correlation between 1 and 2 (that is 2(1,2)), and all chains of direct correlations via a third, fourth etc., particle. The weakness of this heuristic derivation is that we do not know how to write down an expression for 2(1,2). The great advantage of the formal... 

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

It is the Ornstein-Zernike equation that has been used most popularly to describe the density fluctuations in liquids [37], [35]. The equation by itself is nothing more than the defining of a correlation function called the direct correlation function c(r, r ) in terms of the total correlation function h r,r ) defined in Eq. (1.7). 

Ornstein-Zernike equations for the total correlation functions... 

All measured SAXS data were analyzed by the generalized indirect Fourier transformation (GIFT) technique with the Boltzmann simplex simulated annealing (BSSA) algorithm [34, 35]. The GIFT calculation is based on the analytical or numerical solution of the Ornstein-Zernike (OZ) equation that describes the interplay between the total (h(r)) and direct (c(r)) correlation functions ... 

It is convenient now to introduce the direct correlation function of a uniform fluid c(rt2) which is defined by its relation to the total function h(ri2) by the integral equation of Ornstein and Zernike, ... 

The Percus-Yevick and hypernetted chain equations can be "derived in a similar manner. To do this, we must introduce one last function, the so-called direct correlation function, c(r). The quantity h(r 2) = g(ri2) - 1 is a measure of the total influence of a particle at on a particle at r. Ornstein and Zernike proposed a division of hCr ) two parts, a direct part and an indirect part ... 

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