Percus Yevick

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

$D(r)=y(r)+c(r)-g(r)$

one has the exact integral equation

$y(r_{12})-D(r_{12})=1+n\int (f(r_{13})y(r_{13})+D(r_{13}))h(r_{23})d{r}_3$

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

$h-c=y-1$

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

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

or

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

or (Eq. 10 in Ref. 4)

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

or (Eq. 2 of Ref. 5)

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

or in terms of the bridge function

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

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

References

1. [RPP_1965_28_0169]
2. [P_1963_29_0517_nolotengoElsevier]
3. [PR_1958_110_000001]
4. [MP_1983_49_1495]
5. [PRA_1984_30_000999]
6. [JCP_2002_116_08517]
7. [JSP_1988_52_1389_nolotengoSpringer]