Editing Percus Yevick
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If one defines a class of | If one defines a class of 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 | one has the exact 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 | The ''PY'' closure 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 | |||
:<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. | ||
==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/ | ||
#[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/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ | ||
#[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]] |