Percus Yevick

If one defines a class of diagrams by the linear combination (Eq. 5.18 Ref.1) (See G. Stell \cite{P_1963_29_0517_nolotengoElsevier})

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

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 \cite{PR_1958_110_000001}

$h-c=y-1$

The {\bf PY} closure can be written as (\cite{PR_1958_110_000001} Eq. 61)

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

or

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

or (Eq. 10 \cite{MP_1983_49_1495})

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

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

1. [RPP_1965_28_0169]