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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{R}(-\omega )} , time-reversed. This expression can be written in terms of work ($W$) (Ref. 1 Eq. 11):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {P_{F}(+\beta W)}{P_{R}(-\beta W)}}=\exp(-\Delta A)\exp(+\beta W)}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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)
2. L. Y. Chen "On the Crooks fluctuation theorem and the Jarzynski equality", Journal of Chemical Physics 129 091101 (2008)
3. Riccardo Chelli "Nonequilibrium work relations for systems subject to mechanical and thermal changes", Journal of Chemical Physics 130 054102 (2009)