Percus Yevick: Difference between revisions

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m (New page: If one defines a class of diagrams by the linear combination (Eq. 5.18 \cite{RPP_1965_28_0169}) (See G. Stell \cite{P_1963_29_0517_nolotengoElsevier}) \begin{equation} D(r) = y(r) + c(r) -...)
 
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If one defines a class of diagrams by the linear combination (Eq. 5.18 \cite{RPP_1965_28_0169})
If one defines a class of diagrams by the linear combination (Eq. 5.18 Ref.1)
(See G. Stell \cite{P_1963_29_0517_nolotengoElsevier})
(See G. Stell \cite{P_1963_29_0517_nolotengoElsevier})
\begin{equation}
 
D(r) = y(r) + c(r) -g(r)
:<math>\left.D(r)\right. = y(r) + c(r) -g(r)</math>
\end{equation}
 
one has the exact integral equation
one has the exact integral equation
\begin{equation}
 
y(r_{12}) - D(r_{12}) = 1 + n \int (f(r_{13})y(r_{13})+D(r_{13})) h(r_{23})~{\rm d}{\bf r}_3
<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>
\end{equation}
 
The Percus-Yevick integral equation sets $D(r)=0$.\\
The Percus-Yevick integral equation sets ''D(r)=0''.
Percus-Yevick (PY) proposed in 1958 \cite{PR_1958_110_000001}
Percus-Yevick (PY) proposed in 1958 \cite{PR_1958_110_000001}
\begin{equation}
 
h-c=y-1
<math>h-c=y-1</math>
\end{equation}
 
The {\bf PY} closure can be written as (\cite{PR_1958_110_000001} Eq. 61)
The {\bf PY} closure can be written as (\cite{PR_1958_110_000001} Eq. 61)
\begin{equation}
 
f [ \gamma (r) ] = [e^{-\beta \Phi} -1][\gamma (r) +1]
<math>f [ \gamma (r) ] = [e^{-\beta \Phi} -1][\gamma (r) +1]</math>
\end{equation}
 
or
or
\begin{equation}
 
c(r)= {\rm g}(r)(1-e^{\beta \Phi})
<math>c(r)= {\rm g}(r)(1-e^{\beta \Phi})</math>
\end{equation}
 
or (Eq. 10 \cite{MP_1983_49_1495})
or (Eq. 10 \cite{MP_1983_49_1495})
\begin{equation}
 
c(r)=  \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)
:<math>\left.c(r)\right.=  \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)</math>
\end{equation}
 
or (Eq. 2 of \cite{PRA_1984_30_000999})
or (Eq. 2 of \cite{PRA_1984_30_000999})
\begin{equation}
 
{\rm g}(r) = e^{-\beta \Phi} (1+ \gamma(r))
:<math>\left.g(r)\right. = e^{-\beta \Phi} (1+ \gamma(r))</math>
\end{equation}
 
or in terms of the bridge function
or in terms of the bridge function
\begin{equation}
 
B(r)= \ln (1+\gamma(r) ) - \gamma(r)
:<math>\left.B(r)\right.= \ln (1+\gamma(r) ) - \gamma(r)</math>
\end{equation}
 
 
Note: the restriction $-1 < \gamma (r) \leq 1$ arising from the logarithmic term \cite{JCP_2002_116_08517}.
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
The HNC and PY are from the age of {\it `complete ignorance'} (Martynov Ch. 6) with
respect to bridge functionals.
respect to bridge functionals.
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==
#[RPP_1965_28_0169]

Revision as of 12:09, 23 February 2007

If one defines a class of diagrams by the linear combination (Eq. 5.18 Ref.1) (See G. Stell \cite{P_1963_29_0517_nolotengoElsevier})

one has the exact integral equation

The Percus-Yevick integral equation sets D(r)=0. Percus-Yevick (PY) proposed in 1958 \cite{PR_1958_110_000001}

The {\bf PY} closure can be written as (\cite{PR_1958_110_000001} Eq. 61)

or

or (Eq. 10 \cite{MP_1983_49_1495})

or (Eq. 2 of \cite{PRA_1984_30_000999})

or in terms of the bridge function


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

  1. [RPP_1965_28_0169]