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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 Crooks fluctuation theorem ： ウィキペディア英語版 Crooks fluctuation theorem The Crooks Fluctuation Theorem (CFT), sometimes known as the Crooks Equation, 〔 G. Crooks, "Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences", ''Physical Review E'', 60, 2721 (1999) 〕 is an equation in statistical mechanics that relates the work done on a system during a non-equilibrium transformation to the free energy difference between the final and the initial state of the transformation. During the non equilibrium transformation the system is at constant volume and in contact with a heat reservoir. The CFT is named after the chemist Gavin E. Crooks (then at University of California) who discovered it in 1998. The most general statement of the CFT relates the probability of a space-time trajectory $x(t)$ to the time-reversal of the trajectory $\backslash tilde(t)$. The theorem says if the dynamics of the system satisfy microscopic reversibility, then the forward time trajectory is exponentially more likely than the reverse, given that it produces entropy, $\backslash frac(t)"\; TITLE="\backslash tilde(t)">)\}\; =\; e^$. If we define a generic reaction coordinate of the system as a function of the Cartesian coordinates of the constituent particles ( e.g. , a distance between two particles), we can characterize every point along the reaction coordinate path by a parameter $\backslash lambda$, such that $\backslash lambda\; =\; 0$ and $\backslash lambda\; =\; 1$ correspond to two ensembles of microstates for which the reaction coordinate is constrained to different values. A dynamical process where $\backslash lambda$ is externally driven from zero to one, according to an arbitrary time scheduling, will be referred as forward transformation , while the time reversal path will be indicated as backward transformation. Given these definitions, the CFT sets a relation between the following four quantities: * $P(A\; \backslash rightarrow\; B)$, i.e. the joint probability of taking a microstate $A$ from the canonical ensemble corresponding to $\backslash lambda\; =\; 0$ and of performing the forward transformation to the microstate B corresponding to $\backslash lambda\; =\; 1$; *$P(A\; \backslash leftarrow\; B)$, i.e. the joint probability of taking the microstate $B$ from the canonical ensemble corresponding to $\backslash lambda\; =\; 1$ and of performing the backward transformation to the microstate $A$ corresponding to $\backslash lambda\; =\; 0$; *$\backslash beta\; =\; (k\_B$ T)^, where $k\_B$ is the Boltzmann constant and $T$ the temperature of the reservoir; * $W\_$, i.e. the work done on the system during the forward transformation (from $A$ to $B$); * $\backslash Delta\; F\; =\; F(B)\; -\; F(A)$, i.e. the Helmholtz free energy difference between the state $A$ and $B$, represented by the canonical distribution of microstates having $\backslash lambda\; =\; 0$ and $\backslash lambda\; =\; 1$, respectively). The CFT equation reads as follows: $$ \frac = \exp (\beta ( W_ - \Delta F ) ). In the previous equation the difference $W\_\; -\; \backslash Delta\; F$ corresponds to the work dissipated in the forward transformation, $W\_d$. The probabilities $P(A\; \backslash rightarrow\; B)$ and $P(A\; \backslash leftarrow\; B)$ become identical when the transformation is performed at infinitely slow speed, i.e. for equilibrium transformations. In such case $W\_\; =\; \backslash Delta\; F$ and $W\_d$ = 0. Using the time reversal relation $W\_\; =\; -W\_$, and grouping together all the trajectories yielding the same work (in the forward and backward transformation), we can write the above equation in terms of the work distribution functions as follows $$ P_ (W) = P_(- W) ~ \exp((W - \Delta F) ). Note that for the backward transformation, the work distribution function must be evaluated by taking the work with the opposite sign. The two work distributions for the forward and backward processes cross at $W=\backslash Delta\; F$. This phenomenon has been experimentally verified using optical tweezers for the process of unfolding and refolding of a small RNA hairpin and an RNA three-helix junction () The CFT implies the Jarzynski equality. ==Notes== 〔 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「Crooks fluctuation theorem」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.