Percus Yevick: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
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) -...)
 
m (Added internal link)
 
(13 intermediate revisions by 2 users not shown)
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
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]]

Latest revision as of 12:53, 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)

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 Percus-Yevick closure relation 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)

or (Eq. 2 of Ref. 5)

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))}

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

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 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 -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.

See also[edit]

References[edit]

  1. J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28 pp. 169-199 (1965)
  2. G. Stell "PERCUS-YEVICK EQUATION FOR RADIAL DISTRIBUTION FUNCTION OF A FLUID", Physica 29 pp. 517- (1963)
  3. Jerome K. Percus and George J. Yevick "Analysis of Classical Statistical Mechanics by Means of Collective Coordinates", Physical Review 110 pp. 1 - 13 (1958)
  4. G. A. Martynov and G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics 49 pp. 1495-1504 (1983)
  5. Forrest J. Rogers and David A. Young "New, thermodynamically consistent, integral equation for simple fluids", Physical Review A 30 pp. 999 - 1007 (1984)
  6. 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)
  7. Yaoqi Zhou and George Stell "The hard-sphere fluid: New exact results with applications", Journal of Statistical Physics 52 1389-1412 (1988)
Retrieved from "http://www.sklogwiki.org/SklogWiki/index.php?title=Percus_Yevick&oldid=6190"