Percus Yevick

If one defines a class of diagrams by the linear combination (Eq. 5.18 Ref.1) (See G. Stell in Ref. 2)

$\left.D(r)\right.=y(r)+c(r)-g(r)$

one has the exact integral equation

$y(r_{12})-D(r_{12})=1+n\int (f(r_{13})y(r_{13})+D(r_{13}))h(r_{23})~dr_{3}$

The Percus-Yevick integral equation sets D(r)=0. Percus-Yevick (PY) proposed in 1958 Ref. 3

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.h-c\right.=y-1}

The PY closure can be written as (Ref. 3 Eq. 61)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.c(r)\right.= {\rm g}(r)(1-e^{\beta \Phi})}

or (Eq. 10 in Ref. 4)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.c(r)\right.= \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)}

or (Eq. 2 of Ref. 5)

$\left.g(r)\right.=e^{-\beta \Phi }(1+\gamma (r))$

or in terms of the bridge function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.B(r)\right.= \ln (1+\gamma(r) ) - \gamma(r)}

Note: the restriction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 < \gamma (r) \leq 1} arising from the logarithmic term Ref. 6. A critical look at the PY was undertaken by Zhou and Stell in Ref. 7.

References

1. J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28 pp. 169-199 (1965)
2. G. Stell "PERCUS-YEVICK EQUATION FOR RADIAL DISTRIBUTION FUNCTION OF A FLUID", Physica 29 pp. 517- (1963)
3. Jerome K. Percus and George J. Yevick "Analysis of Classical Statistical Mechanics by Means of Collective Coordinates", Physical Review 110 pp. 1 - 13 (1958)
4. G. A. Martynov and G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics 49 pp. 1495-1504 (1983)
5. Forrest J. Rogers and David A. Young "New, thermodynamically consistent, integral equation for simple fluids", Physical Review A 30 pp. 999 - 1007 (1984)
6. Niharendu Choudhury and Swapan K. Ghosh "Integral equation theory of Lennard-Jones fluids: A modified Verlet bridge function approach", Journal of Chemical Physics, 116 pp. 8517-8522 (2002)
7. Yaoqi Zhou and George Stell "The hard-sphere fluid: New exact results with applications", Journal of Statistical Physics 52 1389-1412 (1988)