Pair distribution function Percus-Yevick

One of us has shown that the Percus-Yevick (PY) equation for the pair distribution function, g r), can be solved analytically for the intermolecular potential, (r), given by ... 

The intra-chain pair density functions obtained from both the bulk simulation and the continuous unperturbed chains were used as input to the polymer-RISM integral equation for estimating the intermolecular pair distribution function g(r) (using a soft-Percus Yevick closure). We found that PRISM underpredicts the first peak in g(r), while also overpredicting the steefmess of the rise to the first peak. 

The interpolation between the low and high density limits, which is inherent to this variational approach, leads in a very natural way to the scaled particle theory for the structure and thermodynamics of isotropic fluids of hard particles. This unifies, for the first time the Percus Yevick theory, which is based on diagram expansions, and the scaled particle theory of Reiss, Frisch and Lebowitz, and, at the same time yields the analytical expressions of the dcf conformal to those of the hard spheres. It provides an unified derivation of the most comprehensive analytic description available of the hard sphere thermodynamics and pair distribution functions as given by the Percus Yevick and scaled particle theories, and yields simple explicit expressions for the higher direct direct correlation functions of the uniform fluid. 

In the physical picture ion-pairs are just consequences of large values of the Mayer /-functions that describe the ion distribution [22], The technical consequence, however, is a major complication of the theory the high-temperature approximations of the /-functions applied, e.g. in the mean spherical approximation (MSA) or the Percus-Yevick approximation (PY) [25], suffice in simple fluids but not in ionic systems. 

A fundamental approach to liquids is provided by the integral equation methods (sometimes called distribution function methods), initiated by Kirkwood and Yvon in the 1930s. As we shall show below, one starts by writing down an exact equation for the molecular distribution function of interest, usually the pair function, and then introduces one or more approximations to solve the problem. These approximations are often motivated by considerations of mathematical simplicity, so that their validity depends on a posteriori agreement with computer simulation or experiment. The theories in question, called YBG (Yvon-Bom-Green), PY (Percus-Yevick), and the HNC (hypemetted chain) approximation, provide the distribution functions directly, and are thus applicable to a wide variety of properties. 

There are three approximate theories of the liquid state in frequent use (see, for example, Enderby and March (1965)). Their common feature is that they attempt to relate the radial distribution function g(r) to the interatomic pair potential 0(r). For convenience we list the theories and the relevant equations below as applied to a pure liquid. The generalisation to include multi-component liquids is straightforward Percus-Yevick (PY) ... 

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