Percus Yevick

If one defines a class of diagrams by the linear combination (Eq. 5.18 \cite{RPP_1965_28_0169}) (See G. Stell \cite{P_1963_29_0517_nolotengoElsevier}) \begin{equation} D(r) = y(r) + c(r) -g(r) \end{equation} one has the exact integral equation \begin{equation} y(r_{12}) - D(r_{12}) = 1 + n \int (f(r_{13})y(r_{13})+D(r_{13})) h(r_{23})~{\rm d}{\bf r}_3 \end{equation} The Percus-Yevick integral equation sets $D(r)=0$.\\ Percus-Yevick (PY) proposed in 1958 \cite{PR_1958_110_000001} \begin{equation} h-c=y-1 \end{equation} The {\bf PY} closure can be written as (\cite{PR_1958_110_000001} Eq. 61) \begin{equation} f [ \gamma (r) ] = [e^{-\beta \Phi} -1][\gamma (r) +1] \end{equation} or \begin{equation} c(r)= {\rm g}(r)(1-e^{\beta \Phi}) \end{equation} or (Eq. 10 \cite{MP_1983_49_1495}) \begin{equation} c(r)= \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega) \end{equation} or (Eq. 2 of \cite{PRA_1984_30_000999}) \begin{equation} {\rm g}(r) = e^{-\beta \Phi} (1+ \gamma(r)) \end{equation} or in terms of the bridge function \begin{equation} B(r)= \ln (1+\gamma(r) ) - \gamma(r) \end{equation} 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}.