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 ( Eq. 2):

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

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 \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} ) (Eq. 11):

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

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 \beta := 1/(k_BT)} 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.

See also

• Jarzynski equality

References

Related reading

• L. Y. Chen "On the Crooks fluctuation theorem and the Jarzynski equality", Journal of Chemical Physics 129 091101 (2008)
• Riccardo Chelli "Nonequilibrium work relations for systems subject to mechanical and thermal changes", Journal of Chemical Physics 130 054102 (2009)