# Crooks fluctuation theorem Crooks fluctuation theorem Statistical mechanics Thermodynamics Kinetic theory show Particle statistics show Thermodynamic ensembles show Models show Potentials show Scientists vte 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, 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, {displaystyle {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(Arightarrow 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(Aleftarrow 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_{Arightarrow 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.

The CFT equation reads as follows: {displaystyle {frac {P(Arightarrow B)}{P(Aleftarrow B)}}=exp[beta (W_{Arightarrow B}-Delta F)].} In the previous equation the difference {displaystyle W_{Arightarrow B}-Delta F} corresponds to the work dissipated in the forward transformation, {displaystyle W_{d}} . The probabilities {displaystyle P(Arightarrow B)} and {displaystyle P(Aleftarrow B)} become identical when the transformation is performed at infinitely slow speed, i.e. for equilibrium transformations. In such cases, {displaystyle W_{Arightarrow B}=Delta F} and {displaystyle W_{d}=0.} Using the time reversal relation {displaystyle W_{Arightarrow B}=-W_{Aleftarrow 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_{Arightarrow 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 {displaystyle P_{Arightarrow B}(W)=P_{Aleftarrow 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. The CFT implies the Jarzynski equality.

Notes ^ G. Crooks, "Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences", Physical Review E, 60, 2721 (1999) ^ 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. This thermodynamics-related article is a stub. You can help Wikipedia by expanding it.

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