Crooks fluctuation theorem

The Crooks fluctuation theorem was developed by Gavin E. Crooks. It is also known as the Crooks Identity or the Crooks fluctuation relation. It is given by (Ref. 1 Eq. 2):

${\frac {P_{F}(+\omega )}{P_{R}(-\omega )}}=\exp({+\omega })$

where $\omega$ is the entropy production, $P_{F}(\omega )$ is the "forward" probability distribution of this entropy production, and $P_{R}(-\omega )$, time-reversed. This expression can be written in terms of work ($W$) (Ref. 1 Eq. 11):

${\frac {P_{F}(+\beta W)}{P_{R}(-\beta W)}}=\exp(-\Delta A)\exp(+\beta W)$

where $\beta :=1/(k_{B}T)$ where $k_{B}$ is the Boltzmann constant and $T$ is the temperature, and $A$ is the Helmholtz energy function.

References

1. Gavin E. Crooks "Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences", Physical Review E 60 pp. 2721 - 2726 (1999)