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 in Ref. 2)
(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>
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:<math>\left.h-c\right.=y-1</math>
:<math>\left.h-c\right.=y-1</math>


The ''PY'' closure can be written as (Ref. 3  Eq. 61)
The Percus-Yevick [[Closure relations | closure relation]] can be written as (Ref. 3  Eq. 61)


:<math>\left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]</math>
:<math>\left.f [ \gamma (r) ]\right. = [e^{-\beta \Phi} -1][\gamma (r) +1]</math>
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:<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>
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Note: the restriction <math>-1 < \gamma (r) \leq 1</math> arising from the logarithmic term Ref. 6.
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.
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==
==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)]
#[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)]
#[http://dx.doi.org/
# 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/10.1103/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/
#[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/
#[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/
#[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/
#[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)]
#[http://dx.doi.org/
 
 
 
#[P_1963_29_0517_nolotengoElsevier]


#[MP_1983_49_1495]
#[PRA_1984_30_000999]
#[JCP_2002_116_08517]
#[JSP_1988_52_1389_nolotengoSpringer]


[[Category: Integral equations]]
[[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)