# Percus Yevick

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

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

one has the exact integral equation

${\displaystyle y(r_{12})-D(r_{12})=1+n\int (f(r_{13})y(r_{13})+D(r_{13}))h(r_{23})~dr_{3}}$

The Percus-Yevick integral equation sets D(r)=0. Percus-Yevick (PY) proposed in 1958 Ref. 3

${\displaystyle \left.h-c\right.=y-1}$

The Percus-Yevick closure relation can be written as (Ref. 3 Eq. 61)

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

or

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

or (Eq. 10 in Ref. 4)

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

or (Eq. 2 of Ref. 5)

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

where ${\displaystyle \Phi (r)}$ is the intermolecular pair potential.

In terms of the bridge function

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

Note: the restriction ${\displaystyle -1<\gamma (r)\leq 1}$ arising from the logarithmic term Ref. 6. A critical look at the PY was undertaken by Zhou and Stell in Ref. 7.