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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_F(\omega)} is the "forward" probability distribution of this entropy production, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_R(-\omega)} , time-reversed. This expression can be written in terms of work (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} ) (Ref. 1 Eq. 11):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{P_F(+\beta W)}{P_R(- \beta W)}= \exp (- \Delta A) \exp (+\beta W)}

where $\beta :=1/(k_{B}T)$ where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)