en>Rememberlands: math formatting

2013-06-21T21:24:23Z

math formatting

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	~~Hi there~~. ~~My name~~ is ~~Garland even though it~~ is ~~not~~ the ~~name on my beginning certification~~. ~~One of his preferred hobbies~~ is ~~playing crochet but he hasn~~'~~t made a dime with it~~. ~~Bookkeeping~~ is ~~what she does~~. ~~Her husband~~ and ~~her chose to reside~~ in ~~Alabama~~.<br><br>~~My site~~ ... [http://~~C-Lug~~.~~com~~/~~Activity-Feed~~/~~My-Profile~~/~~UserId~~/~~249456 extended auto warranty~~]		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'' − 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]] — 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]]

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Hi there. My name is Garland even though it is not the name on my beginning certification. One of his preferred hobbies is playing crochet but he hasn't made a dime with it. Bookkeeping is what she does. Her husband and her chose to reside in Alabama.<br><br>My site ... [http://C-Lug.com/Activity-Feed/My-Profile/UserId/249456 extended auto warranty]

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