# Crooks涨落定理

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.

The most general statement of the CFT relates the probability of a space-time trajectory $\displaystyle{ x(t) }$ to the time-reversal of the trajectory $\displaystyle{ \tilde{x}(t) }$. 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,

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,

$\displaystyle{ \frac{P[x(t)]}{\tilde{P}[\tilde{x}(t)]} = e^{\sigma[x(t)]}. }$
\frac{P[x(t)]}{\tilde{P}[\tilde{x}(t)]} = e^{\sigma[x(t)]}.
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 $\displaystyle{ \lambda }$, such that $\displaystyle{ \lambda = 0 }$ and $\displaystyle{ \lambda = 1 }$ correspond to two ensembles of microstates for which the reaction coordinate is constrained to different values. A dynamical process where $\displaystyle{ \lambda }$ 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:

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

• 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:

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

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

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.

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

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

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 $\displaystyle{ W=\Delta F }$. 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.

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.

The CFT implies the Jarzynski equality.

The CFT implies the Jarzynski equality.