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

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

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

$\left.h-c\right.=y-1$

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

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

or

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

or (Eq. 10 in Ref. 4)

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

or (Eq. 2 of Ref. 5)

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

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 Ref. 6. A critical look at the PY was undertaken by Zhou and Stell in Ref. 7.

References

1. J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28 pp. 169-199 (1965)
2. [http://dx.doi.org/
3. Jerome K. Percus and George J. Yevick "Analysis of Classical Statistical Mechanics by Means of Collective Coordinates", Physical Review 110 pp. 1 - 13 (1958)
4. [http://dx.doi.org/
5. [http://dx.doi.org/
6. [http://dx.doi.org/
7. [http://dx.doi.org/
8. [http://dx.doi.org/

1. [P_1963_29_0517_nolotengoElsevier]

1. [MP_1983_49_1495]
2. [PRA_1984_30_000999]
3. [JCP_2002_116_08517]
4. [JSP_1988_52_1389_nolotengoSpringer]