Percus Yevick: Difference between revisions
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The ''PY'' closure can be written as (Ref. 3 Eq. 61) | 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> | :<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> | :<math>\left.c(r)\right.= {\rm g}(r)(1-e^{\beta \Phi})</math> | ||
or (Eq. 10 | or (Eq. 10 in Ref. 4) | ||
:<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> | ||
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A critical look at the PY was undertaken by Zhou and Stell in \cite{JSP_1988_52_1389_nolotengoSpringer}. | A critical look at the PY was undertaken by Zhou and Stell in \cite{JSP_1988_52_1389_nolotengoSpringer}. | ||
==References== | ==References== | ||
#[RPP_1965_28_0169] | #[RPP_1965_28_0169] | ||
#[P_1963_29_0517_nolotengoElsevier] | #[P_1963_29_0517_nolotengoElsevier] | ||
#[PR_1958_110_000001] | #[PR_1958_110_000001] | ||
#[ | #[MP_1983_49_1495] | ||
Revision as of 12:13, 23 February 2007
If one defines a class of diagrams by the linear combination (Eq. 5.18 Ref.1) (See G. Stell in Ref. 2)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.D(r)\right. = y(r) + c(r) -g(r)}
one has the exact integral equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.h-c\right.=y-1}
The PY closure can be written as (Ref. 3 Eq. 61)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]}
or
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.c(r)\right.= {\rm g}(r)(1-e^{\beta \Phi})}
or (Eq. 10 in Ref. 4)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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})
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.g(r)\right. = e^{-\beta \Phi} (1+ \gamma(r))}
or in terms of the bridge function
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
- [RPP_1965_28_0169]
- [P_1963_29_0517_nolotengoElsevier]
- [PR_1958_110_000001]
- [MP_1983_49_1495]