Crooks fluctuation theorem: Difference between revisions
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The '''Crooks fluctuation theorem''' was developed by Gavin E. Crooks. It is also known as the ''Crooks Identity'' or the ''Crooks fluctuation relation''. | 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): | ||
:<math>\frac{P_F( | |||
:<math>\frac{P_F(+\omega)}{P_R(-\omega)}= \exp({+ \omega})</math> | |||
where <math>\omega</math> is the [[entropy]] production, <math>P_F(\omega)</math> is the "forward" probability distribution of this entropy production, and <math>P_R(-\omega)</math>, time-reversed. This expression can be written in terms of [[work]] (<math>W</math>) (Ref. 1 Eq. 11): | |||
:<math>\frac{P_F(+\beta W)}{P_R(- \beta W)}= \exp (- \Delta A) \exp (+\beta W)</math> | |||
where <math>\beta := 1/(k_BT)</math> where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]], and <math>A</math> is the [[Helmholtz energy function]]. | |||
==References== | ==References== | ||
#[http://dx.doi.org/10.1103/PhysRevE.60.2721 Gavin E. Crooks "Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences", Physical Review E '''60''' pp. 2721 - 2726 (1999)] | #[http://dx.doi.org/10.1103/PhysRevE.60.2721 Gavin E. Crooks "Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences", Physical Review E '''60''' pp. 2721 - 2726 (1999)] | ||
[[category:Non-equilibrium thermodynamics]] | [[category:Non-equilibrium thermodynamics]] | ||
[[category: fluctuation theorem]] | [[category: fluctuation theorem]] | ||
Revision as of 11:53, 6 February 2008
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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):
- 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(+\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}
) (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 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.
