Compressibility equation, integral equations pair correlation function

In applying the compressibility equation (3.109), care must be exercised to use the pair correlation function g(R) as obtained in the grand canonical ensemble, rather than the corresponding function g(R) obtained in a closed system. Whenever this distinction is important, we use the notation gQ (R) and gc(R) for open and closed systems, respectively. Although the difference between the two is in a term of the order of AT 1 this small difference becomes important when integration over the entire volume is performed as in the definition of the quantity G (equation 3.110). 

As we have expressed the compressibility equation in terms of the integral over the direct correlation function in (C.16), one can write the KB theory in terms of Cy instead of G / the two are equivalent formulations. O Connel (1971) has expressed the view that the formulation in terms of Cy might be more useful for numerical work since the direct correlation function is considered to be shorter range than the pair correlation function. For further applications of this approach, see O Connel (1971), Perry and O Connel (1984), and Hamad et al. (1987, 1989, 1990a, b, 1993, 1997, 1998). 

Fluctuations in thermodynamics automatically imply the existence of an underlying structure that has created them. We know that such structure is comprised of molecules, and that their large number allows statistical studies, which, in turn, allow one to relate various statistical moments to macroscopic thermodynamic quantities. One of the purposes of the statistical theory of liquids (STL) is to provide such relations for liquids (Frisch and Lebowitz 1964 Gray and Gubbins 1984 Hansen and McDonald 2006). In such theories, many macroscopic quantities appear as limits at zero wave number of the Fourier transforms of statistical correlation functions. For example, the Kirkwood-Buff theory allows one to relate integrals of the pair density correlation functions to various thermo-physical properties such as the isothermal compressibility, the partial molar volumes, and the density derivatives of the chemical potentials (Kirkwood and Buff 1951). If one wants a connection between detailed correlations and integrated moments, one may ask about the nature of the wave-number dependence of these quantities. It turns out that the statistical theory of liquids allows an answer to such a question very precisely, which leads to new types of questions. The Ornstein-Zemike equation (Hansen and McDonald 2006), which is an exact equation of the STL, introduces the concept of correlation length which relates to the spatial extension of the density and/or concentration (the latter in the case of mixtures) fluctuations. This quantity cannot be accessed from pure... 

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