Percus Yevick: Difference between revisions

Latest revision as of 11:53, 14 March 2008

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 Percus-Yevick closure relation 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))

where $\Phi (r)$ is the intermolecular pair potential.

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[edit]

@@ 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 [[cluster diagrams | diagrams]] by the linear combination (Eq. 5.18 Ref.1)
-(See G. Stell \cite{P_1963_29_0517_nolotengoElsevier})
+(See G. Stell in Ref. 2)
-\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 equations | 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 Ref. 3
-\begin{equation}
-h-c=y-1
+:<math>\left.h-c\right.=y-1</math>
-\end{equation}
-The {\bf PY} closure can be written as (\cite{PR_1958_110_000001} Eq. 61)
+The Percus-Yevick [[Closure relations | closure relation]] can be written as (Ref. 3  Eq. 61)
-\begin{equation}
-f [ \gamma (r) ] = [e^{-\beta \Phi} -1][\gamma (r) +1]
+:<math>\left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]</math>
-\end{equation}
 or
-\begin{equation}
-c(r)= {\rm g}(r)(1-e^{\beta \Phi})
+:<math>\left.c(r)\right.= {\rm g}(r)(1-e^{\beta \Phi})</math>
-\end{equation}
-or (Eq. 10 \cite{MP_1983_49_1495})
+or (Eq. 10 in Ref. 4)
-\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 Ref. 5)
-\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
+where <math>\Phi(r)</math> is the [[intermolecular pair potential]].
-\begin{equation}
-B(r)= \ln (1+\gamma(r) ) - \gamma(r)
+In terms of the [[bridge function]]
-\end{equation}
-Note: the restriction $-1 < \gamma (r) \leq 1$ arising from the logarithmic term \cite{JCP_2002_116_08517}.
+:<math>\left.B(r)\right.= \ln (1+\gamma(r) ) - \gamma(r)</math>
-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}.
+Note: the restriction <math>-1 < \gamma (r) \leq 1</math> arising from the logarithmic term Ref. 6.
+A critical look at the PY was undertaken by  Zhou and Stell in Ref. 7.
+==See also==
+*[[Exact solution of the Percus Yevick integral equation for hard spheres]]
+==References==
+#[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
+# G. Stell "PERCUS-YEVICK EQUATION FOR RADIAL DISTRIBUTION FUNCTION OF A FLUID", Physica '''29''' pp. 517- (1963)
+#[http://dx.doi.org/10.1103/PhysRev.110.1  Jerome K. Percus and George J. Yevick "Analysis of Classical Statistical Mechanics by Means of Collective Coordinates", Physical Review '''110''' pp. 1 - 13 (1958)]
+#[http://dx.doi.org/10.1080/00268978300102111 G. A. Martynov and G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics '''49''' pp. 1495-1504 (1983)]
+#[http://dx.doi.org/10.1103/PhysRevA.30.999  Forrest J. Rogers and David A. Young "New, thermodynamically consistent, integral equation for simple fluids", Physical Review A '''30''' pp. 999 - 1007 (1984)]
+#[http://dx.doi.org/10.1063/1.1467894      Niharendu Choudhury and Swapan K. Ghosh "Integral equation theory of Lennard-Jones fluids: A modified Verlet bridge function approach", Journal of Chemical Physics, '''116''' pp. 8517-8522 (2002)]
+#[http://dx.doi.org/10.1007/BF01011655 Yaoqi Zhou and George Stell "The hard-sphere fluid: New exact results with applications", Journal of Statistical Physics '''52''' 1389-1412 (1988)]
+[[Category: Integral equations]]

Percus Yevick: Difference between revisions

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