Crooks涨落定理

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模板:Statistical mechanics The Crooks fluctuation theorem (CFT), sometimes known as the Crooks equation,[1] 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, Berkeley) who discovered it in 1998.


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, Berkeley) who discovered it in 1998.

涨落定理,有时也被称为克鲁克斯方程,g。克鲁克斯,“自由能差异的产生熵涨落定理和非平衡功关系”,《物理评论 e 》 ,60,2721(1999)是统计力学中的一个方程,它将非平衡变换过程中系统所做的功与变换的最终状态和初始状态之间的自由能差联系起来。在非平衡转化过程中,系统处于定容状态,并与热源接触。CFT 是以化学家 Gavin e. Crooks (当时在加州大学伯克利分校)的名字命名的,他在1998年发现了 CFT。

The most general statement of the CFT relates the probability of a space-time trajectory [math]\displaystyle{ x(t) }[/math] to the time-reversal of the trajectory [math]\displaystyle{ \tilde{x}(t) }[/math]. The theorem says if the dynamics of the system satisfies microscopic reversibility, then the forward time trajectory is exponentially more likely than the reverse, given that it produces entropy,

The most general statement of the CFT relates the probability of a space-time trajectory x(t) to the time-reversal of the trajectory \tilde{x}(t). The theorem says if the dynamics of the system satisfies microscopic reversibility, then the forward time trajectory is exponentially more likely than the reverse, given that it produces entropy,

CFT 最一般的说法是将时空轨迹 x (t)的概率与轨迹波浪{ x }(t)的时间反转联系起来。这个定理说,如果系统的动力学满足微观可逆性,那么前进的时间轨迹比后者更有可能成指数增长,因为它产生了熵,

[math]\displaystyle{ \frac{P[x(t)]}{\tilde{P}[\tilde{x}(t)]} = e^{\sigma[x(t)]}. }[/math]
\frac{P[x(t)]}{\tilde{P}[\tilde{x}(t)]} = e^{\sigma[x(t)]}.
frac { p [ x (t)]}{ tilde { p }[ tilde { x }(t)]} = e ^ { sigma [ x (t)]}.

If one defines 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), one can characterize every point along the reaction coordinate path by a parameter [math]\displaystyle{ \lambda }[/math], such that [math]\displaystyle{ \lambda = 0 }[/math] and [math]\displaystyle{ \lambda = 1 }[/math] correspond to two ensembles of microstates for which the reaction coordinate is constrained to different values. A dynamical process where [math]\displaystyle{ \lambda }[/math] 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 five quantities:

  • [math]\displaystyle{ P(A \rightarrow B) }[/math], i.e. the joint probability of taking a microstate [math]\displaystyle{ A }[/math] from the canonical ensemble corresponding to [math]\displaystyle{ \lambda = 0 }[/math] and of performing the forward transformation to the microstate [math]\displaystyle{ B }[/math] corresponding to [math]\displaystyle{ \lambda = 1 }[/math];
  • [math]\displaystyle{ P(A \leftarrow B) }[/math], i.e. the joint probability of taking the microstate [math]\displaystyle{ B }[/math] from the canonical ensemble corresponding to [math]\displaystyle{ \lambda = 1 }[/math] and of performing the backward transformation to the microstate [math]\displaystyle{ A }[/math] corresponding to [math]\displaystyle{ \lambda = 0 }[/math];
  • [math]\displaystyle{ \beta = (k_B T)^{-1} }[/math], where [math]\displaystyle{ k_B }[/math] is the Boltzmann constant and [math]\displaystyle{ T }[/math] the temperature of the reservoir;
  • [math]\displaystyle{ W_{A \rightarrow B} }[/math], i.e. the work done on the system during the forward transformation (from [math]\displaystyle{ A }[/math] to [math]\displaystyle{ B }[/math]);
  • [math]\displaystyle{ \Delta F = F(B) - F(A) }[/math], i.e. the Helmholtz free energy difference between the state [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math], represented by the canonical distribution of microstates having [math]\displaystyle{ \lambda = 0 }[/math] and [math]\displaystyle{ \lambda = 1 }[/math], respectively.

If one defines 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), one can characterize every point along the reaction coordinate path by a parameter \lambda, such that \lambda = 0 and \lambda = 1 correspond to two ensembles of microstates for which the reaction coordinate is constrained to different values. A dynamical process where \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 five quantities:

  • P(A \rightarrow B), i.e. the joint probability of taking a microstate A from the canonical ensemble corresponding to \lambda = 0 and of performing the forward transformation to the microstate B corresponding to \lambda = 1;
  • P(A \leftarrow B), i.e. the joint probability of taking the microstate B from the canonical ensemble corresponding to \lambda = 1 and of performing the backward transformation to the microstate A corresponding to \lambda = 0;
  • \beta = (k_B

T)^{-1}, where k_B is the Boltzmann constant and T the temperature of the reservoir;

  • W_{A \rightarrow B}, i.e. the work done on the system during the forward transformation (from A to B);
  • \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 \lambda = 0 and \lambda = 1, respectively.

如果把系统的一般反应坐标定义为组成粒子的笛卡尔坐标的函数(例如:。,两个粒子之间的距离) ,我们可以用一个参数 lambda 来表示反应坐标路径上的每一个点,例如,lambda = 0和 lambda = 1对应于两个微反应坐标的整合,对于这两个微态,它们的反应坐标被约束为不同的值。一个动态过程,其中的 lambda 是外部驱动从零到一,根据任意时间调度,将被称为前向变换,而时间反转路径将被指示为后向变换。根据这些定义,CFT 在以下五个量之间建立了一个关系:

  • p (a right tarrow b) ,即。从 λ = 0的正则系综中取出微状态 a,并将其转换为 λ = 1的微状态 b 的联合概率;
  • p (a leftarrow b) ,即。从对应于 lambda = 1的正则系综中获取微观状态 b 并执行向后转换到对应于 lambda = 0的微观状态 a 的联合概率;
  • beta = (k _ b t) ^ {-1} ,其中 k _ b 是波兹曼常数,t 是蓄水池的温度;
  • w { a right tarrow b } ,即。系统在前向转换(从 a 到 b)过程中所做的功;
  • Delta f = f (b)-f (a) ,即。状态 a 和状态 b 之间的亥姆霍兹自由能差,分别用 λ = 0和 λ = 1的微型状态的标准分布表示。

The CFT equation reads as follows:

The CFT equation reads as follows:

CFT 方程式如下:

[math]\displaystyle{ \frac{P(A \rightarrow B)}{P( A \leftarrow B)} = \exp [ \beta ( W_{A \rightarrow B} - \Delta F)]. }[/math]

\frac{P(A \rightarrow B)}{P( A \leftarrow B)} = \exp [ \beta ( W_{A \rightarrow B} - \Delta F)].


frac { p (a right tarrow b)}{ p (a leftarrow b)} = exp [ beta (w _ { a right tarrow b }-Delta f)].

In the previous equation the difference [math]\displaystyle{ W_{A \rightarrow B} - \Delta F }[/math] corresponds to the work dissipated in the forward transformation, [math]\displaystyle{ W_d }[/math]. The probabilities [math]\displaystyle{ P(A \rightarrow B) }[/math] and [math]\displaystyle{ P(A \leftarrow B) }[/math] become identical when the transformation is performed at infinitely slow speed, i.e. for equilibrium transformations. In such cases, [math]\displaystyle{ W_{A \rightarrow B} = \Delta F }[/math] and [math]\displaystyle{ W_d = 0. }[/math]

In the previous equation the difference W_{A \rightarrow B} - \Delta F corresponds to the work dissipated in the forward transformation, W_d. The probabilities P(A \rightarrow B) and P(A \leftarrow B) become identical when the transformation is performed at infinitely slow speed, i.e. for equilibrium transformations. In such cases, W_{A \rightarrow B} = \Delta F and W_d = 0.

在前一个方程中,差分 w _ { a right tarrow b }-Delta f 对应于正向变换耗散的功 w _ d。当变换以无限慢的速度执行时,p (a 右行 b)和 p (a 左行 b)的概率是相同的,即 p (a 左行 b)的概率是无限慢的。平衡变换。在这种情况下,w { a right tarrow b } = Delta f 和 w _ d = 0。

Using the time reversal relation [math]\displaystyle{ W_{A \rightarrow B} = -W_{A \leftarrow B} }[/math], and grouping together all the trajectories yielding the same work (in the forward and backward transformation), i.e. determining the probability distribution (or density) [math]\displaystyle{ P_{A\rightarrow B}(W) }[/math] of an amount of work [math]\displaystyle{ W }[/math] being exerted by a random system trajectory from [math]\displaystyle{ A }[/math] to [math]\displaystyle{ B }[/math], we can write the above equation in terms of the work distribution functions as follows

Using the time reversal relation W_{A \rightarrow B} = -W_{A \leftarrow B}, and grouping together all the trajectories yielding the same work (in the forward and backward transformation), i.e. determining the probability distribution (or density) P_{A\rightarrow B}(W) of an amount of work W being exerted by a random system trajectory from A to B, we can write the above equation in terms of the work distribution functions as follows

利用时间反转关系 w _ { a right tarrow b } =-w _ { a leftarrow b } ,将所有产生相同工作的轨迹(在向前和向后变换中)组合在一起,即:。确定从 a 到 b 的随机系统轨迹施加的工作量 w 的概率分布 p { a right tarrow b }(w) ,我们可以根据工作分布函数写出上述方程如下

[math]\displaystyle{ P_{A \rightarrow B} (W) = P_{A\leftarrow B}(- W) ~ \exp[\beta (W - \Delta F)]. }[/math]

P_{A \rightarrow B} (W) = P_{A\leftarrow B}(- W) ~ \exp[\beta (W - \Delta F)].


p _ { a right tarrow b }(w) = p _ { a leftarrow b }(- w) ~ exp [ beta (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 [math]\displaystyle{ W=\Delta F }[/math]. 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.[2]

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=\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.

注意,对于向后转换,功分布函数必须通过取相反符号的功来计算。正向和反向过程的两个功分布在 w = △ f 处交叉。这种现象已经被用于小 RNA 发夹和 RNA 三螺旋连接的展开和折叠过程的光镊实验所证实。

The CFT implies the Jarzynski equality.

The CFT implies the Jarzynski equality.

中国金融时报》暗示了中国 Jarzynski恆等式。

Notes

  1. G. Crooks, "Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences", Physical Review E, 60, 2721 (1999)
  2. Collin, D.; Ritort, F.; Jarzynski, C.; Smith, S. B.; Tinoco, I.; Bustamante, C. (8 September 2005). "Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies". Nature. 437 (7056): 231–234. arXiv:cond-mat/0512266. Bibcode:2005Natur.437..231C. doi:10.1038/nature04061. PMC 1752236. PMID 16148928.


Category:Non-equilibrium thermodynamics Category:Statistical mechanics theorems

类别: 非平衡态热力学类别: 统计力学定理


This page was moved from wikipedia:en:Crooks fluctuation theorem. Its edit history can be viewed at Crooks涨落定理/edithistory