Percus Yevick: Difference between revisions

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})

\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 \cite{PR_1958_110_000001}

$h-c=y-1$

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

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

or

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

or (Eq. 10 \cite{MP_1983_49_1495})

\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})

\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 \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]

@@ Line 1: / Line 1: @@
-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})
-\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
-\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}
-\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)
-\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
-\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})
-\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})
-\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
-\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}.
 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]

Percus Yevick: Difference between revisions

Revision as of 12:09, 23 February 2007

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