Editing Percus Yevick
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If one defines a class of | If one defines a class of diagrams by the linear combination (Eq. 5.18 Ref.1) | ||
(See G. Stell in Ref. 2) | (See G. Stell in Ref. 2) | ||
:<math>\left.D(r)\right. = y(r) + c(r) -g(r)</math> | :<math>\left.D(r)\right. = y(r) + c(r) -g(r)</math> | ||
one has the exact | one has the exact integral equation | ||
:<math>y(r_{12}) - D(r_{12}) = 1 + n \int (f(r_{13})y(r_{13})+D(r_{13})) h(r_{23})~dr_3</math> | :<math>y(r_{12}) - D(r_{12}) = 1 + n \int (f(r_{13})y(r_{13})+D(r_{13})) h(r_{23})~dr_3</math> | ||
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:<math>\left.h-c\right.=y-1</math> | :<math>\left.h-c\right.=y-1</math> | ||
The | The ''PY'' closure can be written as (Ref. 3 Eq. 61) | ||
<math>\left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]</math> | |||
or | or | ||
<math>c(r)= {\rm g}(r)(1-e^{\beta \Phi})</math> | |||
or (Eq. 10 | or (Eq. 10 \cite{MP_1983_49_1495}) | ||
:<math>\left.c(r)\right.= \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)</math> | :<math>\left.c(r)\right.= \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)</math> | ||
or (Eq. 2 of | or (Eq. 2 of \cite{PRA_1984_30_000999}) | ||
:<math>\left.g(r)\right. = e^{-\beta \Phi} (1+ \gamma(r))</math> | :<math>\left.g(r)\right. = e^{-\beta \Phi} (1+ \gamma(r))</math> | ||
or in terms of the bridge function | |||
:<math>\left.B(r)\right.= \ln (1+\gamma(r) ) - \gamma(r)</math> | :<math>\left.B(r)\right.= \ln (1+\gamma(r) ) - \gamma(r)</math> | ||
Note: the restriction | Note: the restriction $-1 < \gamma (r) \leq 1$ arising from the logarithmic term \cite{JCP_2002_116_08517}. | ||
A critical look at the PY was undertaken by Zhou and Stell in | 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==\cite{PR_1958_110_000001} | |||
[[ | #[RPP_1965_28_0169] | ||
#[P_1963_29_0517_nolotengoElsevier] | |||
#[PR_1958_110_000001] | |||
#[\cite{PR_1958_110_000001} |