Percus Yevick

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.D(r)\right.=y(r)+c(r)-g(r)}

one has the exact integral equation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.h-c\right.=y-1}

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

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

or

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.c(r)\right.={\rm {g}}(r)(1-e^{\beta \Phi })}

or (Eq. 10 in Ref. 4)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)

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.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. [http://dx.doi.org/
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. [http://dx.doi.org/
5. [http://dx.doi.org/
6. [http://dx.doi.org/
7. [http://dx.doi.org/
8. [http://dx.doi.org/

1. [P_1963_29_0517_nolotengoElsevier]

1. [MP_1983_49_1495]
2. [PRA_1984_30_000999]
3. [JCP_2002_116_08517]
4. [JSP_1988_52_1389_nolotengoSpringer]