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In [[statistical mechanics]] the '''Percus–Yevick approximation'''<ref>Percus, Jerome K. and Yevick, George J. Analysis of Classical Statistical Mechanics by Means of Collective Coordinates. Phys. Rev. 1958, 110, 1, [[Digital object identifier|DOI]]: [http://dx.doi.org/10.1103/PhysRev.110.1 10.1103/PhysRev.110.1]</ref> is a [[Closure (mathematics)|closure]] relation to solve the [[Ornstein–Zernike equation]]. It is also referred to as the '''Percus–Yevick equation'''. It is commonly used in fluid theory to obtain e.g. expressions for the [[radial distribution function]].
 
==Derivation==
The direct correlation function represents the direct correlation between two particles in a system containing ''N''&nbsp;&minus;&nbsp;2 other particles. It can be represented by
 
: <math> c(r)=g_{\rm total}(r) - g_{\rm indirect}(r) \, </math>
 
where <math>g_{\rm total}(r)</math> is the [[radial distribution function]], i.e. <math>g(r)=\exp[-\beta w(r)]</math> (with ''w''(''r'') the [[potential of mean force]]) and <math>g_{\rm indirect}(r)</math> is the radial distribution function without the direct interaction between pairs <math>u(r)</math> included; i.e. we write <math>g_{\rm indirect}(r)=\exp^{-\beta[w(r)-u(r)]}</math>. Thus we ''approximate'' ''c''(''r'') by
 
: <math> c(r)=e^{-\beta w(r)}- e^{-\beta[w(r)-u(r)]}. \, </math>
 
If we introduce the function <math>y(r)=e^{\beta u(r)}g(r)</math> into the approximation for ''c''(''r'') one obtains
 
: <math> c(r)=g(r)-y(r)=e^{-\beta u}y(r)-y(r)=f(r)y(r). \, </math>
 
This is the essence of the Percus–Yevick approximation for if we substitute this result in the [[Ornstein–Zernike equation]], one obtains the '''Percus–Yevick equation''':
 
: <math> y(r_{12})=1+\rho \int f(r_{13})y(r_{13})h(r_{23}) d \mathbf{r_{3}}.   \, </math>
 
The approximation was defined by Percus and Yevick in 1958. For [[hard spheres]], the equation has an analytical solution.<ref>Wertheim, M. S. Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres. Phys. Rev. Lett. 1963, 10, 321-323, [[Digital object identifier|DOI]]: [http://dx.doi.org/10.1103/PhysRevLett.10.321 10.1103/PhysRevLett.10.321]</ref>
 
==See also==
 
* [[Hypernetted chain equation]] &mdash; another closure relation
 
==References==
{{Reflist}}
 
== External links ==
* [http://www.sklogwiki.org/SklogWiki/index.php/Exact_solution_of_the_Percus_Yevick_integral_equation_for_hard_spheres Exact solution of the Percus Yevick integral equation for hard spheres]
 
{{DEFAULTSORT:Percus-Yevick approximation}}
[[Category:Statistical mechanics]]

Latest revision as of 22:24, 21 June 2013

In statistical mechanics the Percus–Yevick approximation[1] is a closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function.

Derivation

The direct correlation function represents the direct correlation between two particles in a system containing N − 2 other particles. It can be represented by

c(r)=gtotal(r)gindirect(r)

where gtotal(r) is the radial distribution function, i.e. g(r)=exp[βw(r)] (with w(r) the potential of mean force) and gindirect(r) is the radial distribution function without the direct interaction between pairs u(r) included; i.e. we write gindirect(r)=expβ[w(r)u(r)]. Thus we approximate c(r) by

c(r)=eβw(r)eβ[w(r)u(r)].

If we introduce the function y(r)=eβu(r)g(r) into the approximation for c(r) one obtains

c(r)=g(r)y(r)=eβuy(r)y(r)=f(r)y(r).

This is the essence of the Percus–Yevick approximation for if we substitute this result in the Ornstein–Zernike equation, one obtains the Percus–Yevick equation:

y(r12)=1+ρf(r13)y(r13)h(r23)dr3.

The approximation was defined by Percus and Yevick in 1958. For hard spheres, the equation has an analytical solution.[2]

See also

References

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External links

  1. Percus, Jerome K. and Yevick, George J. Analysis of Classical Statistical Mechanics by Means of Collective Coordinates. Phys. Rev. 1958, 110, 1, DOI: 10.1103/PhysRev.110.1
  2. Wertheim, M. S. Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres. Phys. Rev. Lett. 1963, 10, 321-323, DOI: 10.1103/PhysRevLett.10.321