Crooks fluctuation theorem

The Crooks fluctuation theorem (CFT), sometimes known as the Crooks equation, 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 formula_1 to the time-reversal of the trajectory formula_2. 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,
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 formula_4, such that formula_5 and formula_6 correspond to two ensembles of microstates for which the reaction coordinate is constrained to different values. A dynamical process where formula_4 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:
The CFT equation reads as follows:
In the previous equation the difference formula_30 corresponds to the work dissipated in the forward transformation, formula_31. The probabilities formula_8 and formula_13 become identical when the transformation is performed at infinitely slow speed, " i.e. " for equilibrium transformations. In such cases, formula_34 and formula_35
Using the time reversal relation formula_36, and grouping together all the trajectories yielding the same work (in the forward and backward transformation), i.e. determining the probability distribution (or density) formula_37 of an amount of work formula_38 being exerted by a random system trajectory from formula_9 to formula_11, we can write the above equation in terms of the work distribution functions as follows
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 formula_42. 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.