Percus Yevick: Difference between revisions

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If one defines a class of diagrams by the linear combination (Eq. 5.18 Ref.1)
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)


:<math>\left.D(r)\right. = y(r) + c(r) -g(r)</math>
:<math>\left.D(r)\right. = y(r) + c(r) -g(r)</math>


one has the exact integral equation
one has the exact [[integral equations | integral 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>
:<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>


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


<math>h-c=y-1</math>
:<math>\left.h-c\right.=y-1</math>


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)


<math>f [ \gamma (r) ] = [e^{-\beta \Phi} -1][\gamma (r) +1]</math>
:<math>\left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]</math>


or
or


<math>c(r)= {\rm g}(r)(1-e^{\beta \Phi})</math>
:<math>\left.c(r)\right.= {\rm g}(r)(1-e^{\beta \Phi})</math>


or (Eq. 10 \cite{MP_1983_49_1495})
or (Eq. 10 in Ref. 4)


:<math>\left.c(r)\right.=  \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)</math>
:<math>\left.c(r)\right.=  \left( e^{-\beta \Phi } -1\right) e^{\omega}= g - \omega - (e^{\omega} -1 -\omega)</math>


or (Eq. 2 of \cite{PRA_1984_30_000999})
or (Eq. 2 of Ref. 5)


:<math>\left.g(r)\right. = e^{-\beta \Phi} (1+ \gamma(r))</math>
:<math>\left.g(r)\right. = e^{-\beta \Phi} (1+ \gamma(r))</math>


or in terms of the bridge function
where <math>\Phi(r)</math> is the [[intermolecular pair potential]].
 
In terms of the [[bridge function]]


:<math>\left.B(r)\right.= \ln (1+\gamma(r) ) - \gamma(r)</math>
:<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}.
Note: the restriction <math>-1 < \gamma (r) \leq 1</math> arising from the logarithmic term Ref. 6.
The HNC and PY are from the age of {\it `complete ignorance'} (Martynov Ch. 6) with
A critical look at the PY was undertaken by  Zhou and Stell in Ref. 7.
respect to bridge functionals.
==See also==
A critical look at the PY was undertaken by  Zhou and Stell in \cite{JSP_1988_52_1389_nolotengoSpringer}.
*[[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)]


==References==


#[RPP_1965_28_0169]
[[Category: Integral equations]]

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)

one has the exact integral equation

The Percus-Yevick integral equation sets D(r)=0. Percus-Yevick (PY) proposed in 1958 Ref. 3

The Percus-Yevick closure relation can be written as (Ref. 3 Eq. 61)

or

or (Eq. 10 in Ref. 4)

or (Eq. 2 of Ref. 5)

where is the intermolecular pair potential.

In terms of the bridge function


Note: the restriction 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)
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