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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 Boltzmann-Matano analysis ： ウィキペディア英語版 Boltzmann-Matano analysis The Boltzmann-Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration. Ludwig Boltzmann worked on Fick's second law to convert it into an ordinary differential equation, whereas Chujiro Matano performed experiments with diffusion couples and calculated the diffusion coefficients as a function of concentration in metal alloys.〔Matano, Chujiro. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). Japanese Journal of Physics. Jan. 16, 1933.〕 Specifically, Matano proved that the diffusion rate of A atoms into a B atom crystal lattice is a function of the amount of A atoms already in the B lattice. The importance of the classic Boltzmann-Matano method consists in the ability to extract diffusivities from concentration-distance data. These methods, also known as ''inverse methods'', have both proven to be reliable, convenient and accurate with the assistance of modern computational techniques. == Boltzmann’s Transformation == Boltzmann’s Transformation converts Fick's second law into an easily solvable ordinary differential equation. Assuming a diffusion coefficient ''D'' that is in general a function of concentration ''c'', Fick's second law is: :$\backslash frac\; =\; \backslash frac\backslash overbrace\; \backslash right\; )\}^\backslash text$ where ''t'' is time and ''x'' is distance. Boltzmann's transformation consists in introducing a variable ''ξ'', defined as a combination of ''t'' and ''x'': :$\backslash xi=\backslash frac\; =\; -\backslash frac$ :$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; \backslash frac\; =\; -\backslash frac\; \backslash frac$ :$\backslash frac\; =\; \backslash frac\; \backslash frac\; =\; \backslash frac$ Inserting these expressions into Fick's law produces the following modified form: :$-\backslash frac\; \backslash frac\; =\; \backslash frac\; \backslash left(D(c)\; \backslash frac\; \backslash right)$ Note how the time variable in the right-hand side could be taken outside of the partial derivative, since the latter regards only variable ''x''. It is now possible to remove the last reference to ''x'' by using again the same chain rule used above to obtain ''∂ξ/∂x'': :$-\backslash frac\; \backslash frac\; =\; \backslash frac\; \backslash frac\; \backslash left(D(c)\; \backslash frac\; \backslash right)$ Because of the appropriate choice in the definition of ''ξ'', the time variable ''t'' can now also be eliminated, leaving ''ξ'' as the only variable in the equation, which is now an ordinary differential equation: :$-2\backslash xi\; \backslash frac\; =\; \backslash frac\; \backslash xi\}\; \backslash left(D(c)\; \backslash frac\; \backslash right)$ This form is significantly easier to solve numerically, and one only needs to perform a back-substitution of ''t'' or ''x'' into the definition of ''ξ'' to find the value of the other variable. 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「Boltzmann-Matano analysis」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.