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

$\left.h-c\right.=y-1$

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

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

or

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

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 \cite{PRA_1984_30_000999})

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 $-1 < \gamma (r) \leq 1$ arising from the logarithmic term \cite{JCP_2002_116_08517}. The HNC and PY are from the age of {\it `complete ignorance'} (Martynov Ch. 6) with respect to bridge functionals. A critical look at the PY was undertaken by Zhou and Stell in \cite{JSP_1988_52_1389_nolotengoSpringer}.

References

1. [RPP_1965_28_0169]
2. [P_1963_29_0517_nolotengoElsevier]
3. [PR_1958_110_000001]
4. [MP_1983_49_1495]