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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 Nakajima–Zwanzig equation ： ウィキペディア英語版 Nakajima–Zwanzig equation The Nakajima–Zwanzig equation (named after the physicists Sadao Nakajima and Robert Zwanzig) is an integral equation describing the time evolution of the "relevant" part of a quantum-mechanical system. It is formulated in the density matrix formalism and can be regarded a generalization of the Master equation. The equation belongs to the Mori–Zwanzig theory within the statistical mechanics of irreversible processes (named after Hazime Mori). By means of a projection operator the dynamics is split into a slow, collective part (''relevant part'') and a rapidly fluctuating ''irrelevant'' part. The goal is to develop dynamical equations for the collective part. == Derivation == The starting point〔A derivation analogous to that presented here is found, for instance, in Breuer, Petruccione ''The theory of open quantum systems'', Oxford University Press 2002, S.443ff〕 is the quantum mechanical Liouville equation (von Neumann equation) :$\backslash partial\_t\; \backslash rho\; =\; \backslash frac()\; =\; L\; \backslash rho,$ where the Liouville operator $L$ is defined as $L\; A\; =\; \backslash frac()$. The density operator (density matrix) $\backslash rho$ is split by means of a projection operator $\backslash mathcal$ into two parts $\backslash rho\; =\backslash left(\; \backslash mathcal+\backslash mathcal\; \backslash right)\backslash rho$, where $\backslash mathcal\backslash equiv\; 1-\backslash mathcal$. The projection operator $\backslash mathcal$ projects onto the aforementioned ''relevant'' part, for which an equation of motion is to be derived. The Liouville – von Neumann equation can thus be represented as :$\backslash left(\; \backslash begin$ \mathcal \\ \mathcal \\ \end \right)\rho =\left( \begin \mathcal \\ \mathcal \\ \end \right)L\left( \begin \mathcal \\ \mathcal \\ \end \right)\rho +\left( \begin \mathcal \\ \mathcal \\ \end \right)L\left( \begin \mathcal \\ \mathcal \\ \end \right)\rho. The second line is formally solved as〔To verify the equation it suffices to write the function under the integral as a derivative, ''deQLt'QeL(t-t') = -eQLt'QLPeL(t-t')dt.〕 :$\backslash mathcal\backslash rho\; =Q\backslash rho\; (t=0)+\backslash int\_^dt\text{'}^\backslash mathcalL\backslash mathcal\backslash rho\; (t-\text{'}).$ By plugging the solution into the first equation, we obtain the Nakajima–Zwanzig equation: :$\backslash partial\_t\; \backslash mathcal\backslash rho\; =\backslash mathcalL\backslash mathcal\backslash rho\; +\backslash underbrace\_+\backslash mathcalL\backslash int\_^\backslash rho\; (t-\text{'})\}.$ Under the assumption that the inhomogeneous term vanishes〔Such an assumption can be made if we assume that the irrelevant part of the density matrix is 0 at the initial time, so that the projector for t=0 is the identity.〕 and using :$\backslash mathcal\backslash left(\; t\; \backslash right)\backslash equiv\backslash mathcalL\backslash mathcalL\backslash mathcal,$ :$\backslash mathcal\backslash rho\; \backslash equiv\; ,$ as well as :$\backslash mathcal^2=\backslash mathcal,$ we obtain the final form :$\backslash partial\_t\_\backslash mathrm=\backslash mathcalL+\backslash int\_^\text{'})(t-\text{'})\}.$ 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「Nakajima–Zwanzig 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.