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

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)

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

or (Eq. 10 \cite{MP_1983_49_1495})

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 \cite{PRA_1984_30_000999})

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

or in terms of the bridge function

$\left.B(r)\right.=\ln(1+\gamma (r))-\gamma (r)$

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==\cite{PR_1958_110_000001}

1. [RPP_1965_28_0169]
2. [P_1963_29_0517_nolotengoElsevier]
3. [PR_1958_110_000001]
4. [\cite{PR_1958_110_000001}