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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ornstein-Zernike equation ： ウィキペディア英語版 Ornstein–Zernike equation In statistical mechanics the Ornstein–Zernike equation (named after Leonard Ornstein and Frits Zernike) is an integral equation for defining the direct correlation function. It basically describes how the correlation between two molecules can be calculated. Its applications can mainly be found in fluid theory. == Derivation == The derivation below is heuristic in nature: rigorous derivations require extensive graph analysis or functional techniques. The interested reader is referred to〔Frisch, H. & Lebowitz J.L. ''The Equilibrium Theory of Classical Fluids'' (New York: Benjamin, 1964)〕 for the full derivation. It is convenient to define the total correlation function: : $h(r\_)=g(r\_)-1\; \backslash ,$ which is a measure for the "influence" of molecule 1 on molecule 2 at a distance $r\_$ away with $g(r\_)$ as the radial distribution function. In 1914 Ornstein and Zernike proposed 〔Ornstein, L. S. and Zernike, F. Accidental deviations of density and opalescence at the critical point of a single substance. Proc. Acad. Sci. Amsterdam 1914, 17, 793-806〕 to split this influence into two contributions, a direct and indirect part. The direct contribution is ''defined'' to be given by the direct correlation function, denoted $c(r\_)$. The indirect part is due to the influence of molecule 1 on a third molecule, labeled 3, which in turn affects molecule 2, directly and indirectly. This indirect effect is weighted by the density and averaged over all the possible positions of particle 3. This decomposition can be written down mathematically as : $h(r\_)=c(r\_)\; +\; \backslash rho\; \backslash int\; d\; \backslash mathbf\_\; c(r\_)h(r\_)\; \backslash ,$ which is called the Ornstein–Zernike equation. Its interest is that, by eliminating the indirect influence, $c(r)$ is shorter-ranged than $h(r)$ and can be more easily described. If we define the distance vector between two molecules $\backslash mathbf\_:=\; \backslash mathbf\_-\backslash mathbf\_$ for $i,j=1,2,3$, the OZ equation can be rewritten using a convolution. : $h(\backslash mathbf\_)=\; c(\backslash mathbf\_)\; +\; \backslash rho\; \backslash int\; d\; \backslash mathbf\_\; c(\backslash mathbf\_-\backslash mathbf\_)h(-\backslash mathbf\_)\; =\; c(\backslash mathbf\_)\; +\; \backslash rho\; \backslash int\; d\; \backslash mathbf\_\; c(\backslash mathbf\_-\backslash mathbf\_)h(\backslash mathbf\_)$ = c(\mathbf_) + \rho (c\, *\, h)(\mathbf_) \, . If we then denote the Fourier transforms of $h(\backslash mathbf)$ and $c(\backslash mathbf)$ by $\backslash hat(\backslash mathbf)$ and $\backslash hat(\backslash mathbf)$, respectively, and use the convolution theorem we obtain : $\backslash hat(\backslash mathbf)=\backslash hat(\backslash mathbf)\; +\; \backslash rho\; \backslash hat(\backslash mathbf)\backslash hat(\backslash mathbf),\; \backslash ,$ which yields : $\backslash hat(\backslash mathbf)=\backslash frac)\})\}\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; \backslash hat(\backslash mathbf)=\backslash frac)\})\}.\; \backslash ,$ One needs to solve for both $h(r)$ and $c(r)$ (or, equivalently, their Fourier transforms). This requires an additional equation, known as a closure relation. The Ornstein–Zernike equation can be formally seen as a definition of the direct correlation function $c(r)$ in terms of the total correlation function $h(r)$. The details of the system under study (most notably, the shape of the interaction potential $u(r)$) are taken into account by the choice of the closure relation. Commonly used closures are the Percus–Yevick approximation, well adapted for particles with an impenetrable core, and the hypernetted-chain equation, widely used for "softer" potentials. More information can be found in.〔D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ornstein–Zernike equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.