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| If one defines a class of [[cluster diagrams | diagrams]] by the linear combination (Eq. 5.18 Ref.1) | | If one defines a class of diagrams by the linear combination (Eq. 5.18 \cite{RPP_1965_28_0169}) |
| (See G. Stell in Ref. 2) | | (See G. Stell \cite{P_1963_29_0517_nolotengoElsevier}) |
| | | \begin{equation} |
| :<math>\left.D(r)\right. = y(r) + c(r) -g(r)</math>
| | D(r) = y(r) + c(r) -g(r) |
| | | \end{equation} |
| one has the exact [[integral equations | integral equation]] | | one has the exact integral equation |
| | | \begin{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>
| | 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 |
| | | \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 Ref. 3 | | Percus-Yevick (PY) proposed in 1958 \cite{PR_1958_110_000001} |
| | | \begin{equation} |
| :<math>\left.h-c\right.=y-1</math>
| | h-c=y-1 |
| | | \end{equation} |
| The Percus-Yevick [[Closure relations | closure relation]] can be written as (Ref. 3 Eq. 61) | | The {\bf PY} closure can be written as (\cite{PR_1958_110_000001} Eq. 61) |
| | | \begin{equation} |
| :<math>\left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]</math>
| | f [ \gamma (r) ] = [e^{-\beta \Phi} -1][\gamma (r) +1] |
| | | \end{equation} |
| or | | or |
| | | \begin{equation} |
| :<math>\left.c(r)\right.= {\rm g}(r)(1-e^{\beta \Phi})</math>
| | c(r)= {\rm g}(r)(1-e^{\beta \Phi}) |
| | | \end{equation} |
| or (Eq. 10 in Ref. 4) | | or (Eq. 10 \cite{MP_1983_49_1495}) |
| | | \begin{equation} |
| :<math>\left.c(r)\right.= \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)</math>
| | c(r)= \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega) |
| | | \end{equation} |
| or (Eq. 2 of Ref. 5) | | or (Eq. 2 of \cite{PRA_1984_30_000999}) |
| | | \begin{equation} |
| :<math>\left.g(r)\right. = e^{-\beta \Phi} (1+ \gamma(r))</math>
| | {\rm g}(r) = e^{-\beta \Phi} (1+ \gamma(r)) |
| | | \end{equation} |
| where <math>\Phi(r)</math> is the [[intermolecular pair potential]].
| | or in terms of the bridge function |
| | | \begin{equation} |
| In terms of the [[bridge function]]
| | B(r)= \ln (1+\gamma(r) ) - \gamma(r) |
| | | \end{equation} |
| :<math>\left.B(r)\right.= \ln (1+\gamma(r) ) - \gamma(r)</math>
| | 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. |
| 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 \cite{JSP_1988_52_1389_nolotengoSpringer}. |
| 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==
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| #[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)]
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| # G. Stell "PERCUS-YEVICK EQUATION FOR RADIAL DISTRIBUTION FUNCTION OF A FLUID", Physica '''29''' pp. 517- (1963)
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| #[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)]
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| #[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)]
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| #[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)]
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| #[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)]
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| #[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)]
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| [[Category: Integral equations]]
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