Average density approximation for the radial distribution function of a fluid

J. L. Lebowitz, Eigil Praestgaard

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The radial distribution function of a fluid, g(r), is directly related to the difference between the work W (r, ρ) necessary to introduce an extra particle at a distance r from the known position of a particle, and the work W (ρ) of introducing a particle at an arbitrary point in the fluid. Now a system in which one particle is kept fixed can be thought of as a nonuniform system with density ρg(r). This suggests that W(r, ρ) can be written as W(ρ*) where ρ*(r) is some appropriate effective local density. We have utilized this idea to approximate the radial distribution function using for ρ* the average density p̄, of the non-uniform system, with the Mayer f-function as a weighing factor. This leads to an integral equation for g (r) which we have investigated in detail for a fluid of hard spheres. When this equation is solved in a virial expansion it yields reasonably good results for the first few terms. A linearization of this equation leads to an integral equation of the type considered by Kirkwood who was first to consider this type of approximation. We obtain the solution of this linearized equation. A different approximation for ρ*, using the direct correlation function as the weighing factor is obtained by the use of functional Taylor expansion. It leads to an improvement in the virial coefficients and is shown to lead upon appropriate linearizations to either the hypernetted chain or the Percus-Yevick equations for g(r). We also discuss the rigorous form of ρ*(r).

Original languageEnglish (US)
Pages (from-to)2951-2958
Number of pages8
JournalThe Journal of Chemical Physics
Issue number10
StatePublished - 1964
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)
  • Physical and Theoretical Chemistry


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