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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 Thermodynamic integration ： ウィキペディア英語版 Thermodynamic integration Thermodynamic integration is a method used to compare the difference in free energy between two given states (e.g., A and B) whose potential energies $U\_A$ and $U\_B$ have different dependences on the spatial coordinates. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead a function of the Boltzmann-weighted integral over phase space (i.e. partition function), the free energy difference between two states cannot be calculated directly. In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the Kirkwood's coupling parameter method.〔J. G. Kirkwood. Statistical mechanics of fluid mixtures, J. Chem. Phys., 3:300-313,1935〕 == Derivation == Consider two systems, A and B, with potential energies $U\_A$ and $U\_B$. The potential energy in either system can be calculated as an ensemble average over configurations sampled from a molecular dynamics or Monte Carlo simulation with proper Boltzmann weighting. Now consider a new potential energy function defined as: :$U(\backslash lambda)\; =\; U\_A\; +\; \backslash lambda(U\_B\; -\; U\_A)$ Here, $\backslash lambda$ is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of $\backslash lambda$ varies from the energy of system A for $\backslash lambda\; =\; 0$ and system B for $\backslash lambda\; =\; 1$. In the canonical ensemble, the partition function of the system can be written as: :$Q(N,\; V,\; T,\; \backslash lambda)\; =\; \backslash sum\_\; \backslash exp\; ()$ In this notation, $U\_s(\backslash lambda)$ is the potential energy of state $s$ in the ensemble with potential energy function $U(\backslash lambda)$ as defined above. The free energy of this system is defined as: :$F(N,V,T,\backslash lambda)=-k\_T\; \backslash ln\; Q(N,V,T,\backslash lambda)$, If we take the derivative of F with respect to λ, we will get that it equals the ensemble average of the derivative of potential energy with respect to λ. :$\backslash Delta\; F(A\; \backslash rightarrow\; B)$ = \int_0^1 d\lambda \frac = -\int_0^1 d\lambda \frac \frac = \int_0^1 d\lambda \frac \sum_ \frac \exp(U_s(\lambda)/k_T ) \frac = \int_0^1 d\lambda \left\langle\frac\right\rangle_ The change in free energy between states A and B can thus be computed from the integral of the ensemble averaged derivatives of potential energy over the coupling parameter $\backslash lambda$.〔Frenkel, Daan and Smit, Berend. ''Understanding Molecular Simulation: From Algorithms to Applications.'' Academic Press, 2007〕 In practice, this is performed by defining a potential energy function $U(\backslash lambda)$, sampling the ensemble of equilibrium configurations at a series of $\backslash lambda$ values, calculating the ensemble-averaged derivative of $U(\backslash lambda)$ with respect to $\backslash lambda$ at each $\backslash lambda$ value, and finally computing the integral over the ensemble-averaged derivatives. Umbrella sampling is a related free energy method. It adds a bias to the potential energy. In the limit of an infinite strong bias it is equivalent to thermodynamic integration. 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「Thermodynamic integration」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.