Jarzynski equality: Difference between revisions
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The '''Jarzynski equality''' | The '''Jarzynski equality''', also known as the ''work relation'' or ''non-equilibrium work relation'' was developed by Chris Jarzynski. | ||
According to this equality, the equilibrium [[Helmholtz energy function]] of a process, (<math>A</math>), can be reconstructed by averaging the external [[work]], <math>W</math>, performed in many [[Non-equilibrium thermodynamics | non-equilibrium]] realizations of the process (Eq. 2a in <ref>[http://dx.doi.org/10.1103/PhysRevLett.78.2690 | According to this equality, the equilibrium [[Helmholtz energy function]] of a process, (<math>A</math>), can be reconstructed by averaging the external [[work]], <math>W</math>, performed in many [[Non-equilibrium thermodynamics | non-equilibrium]] realizations of the process (Eq. 2a in <ref>[http://dx.doi.org/10.1103/PhysRevLett.78.2690 Chris Jarzynski "Nonequilibrium Equality for Free Energy Differences", Physical Review Letters '''78''' 2690-2693 (1997)]</ref>): | ||
:<math>\exp \left( \frac{-\Delta A}{k_BT}\right)= \left\langle \exp \left( \frac{-W}{k_BT} \right) \right\rangle</math> | :<math>\exp \left( \frac{-\Delta A}{k_BT}\right)= \left\langle \exp \left( \frac{-W}{k_BT} \right) \right\rangle</math> | ||
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:<math>\Delta A = - k_BT \ln \left\langle \exp \left( \frac{-W}{k_BT} \right) \right\rangle </math> | :<math>\Delta A = - k_BT \ln \left\langle \exp \left( \frac{-W}{k_BT} \right) \right\rangle </math> | ||
where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]]. The only assumption in the proof of this relation is that of a weak coupling between the system and the reservoir. | where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]]. The only assumption in the proof of this relation is that of a weak coupling between the system and the reservoir. More recently Jarzynski has re-derived this formula, dispensing with this assumption <ref>[http://dx.doi.org/10.1088/1742-5468/2004/09/P09005 Chris Jarzynski "Nonequilibrium work theorem for a system strongly coupled to a thermal environment", Journal of Statistical Mechanics: Theory and Experiment P09005 (2004)]</ref>. | ||
==See also== | ==See also== | ||
*[[Crooks fluctuation theorem]] | *[[Crooks fluctuation theorem]] | ||
Revision as of 11:41, 5 July 2011
The Jarzynski equality, also known as the work relation or non-equilibrium work relation was developed by Chris Jarzynski. According to this equality, the equilibrium Helmholtz energy function of a process, (), can be reconstructed by averaging the external work, , performed in many non-equilibrium realizations of the process (Eq. 2a in [1]):
or can be trivially re-written as (Eq. 2b)
where is the Boltzmann constant and is the temperature. The only assumption in the proof of this relation is that of a weak coupling between the system and the reservoir. More recently Jarzynski has re-derived this formula, dispensing with this assumption [2].
See also
References
Related reading
- Gerhard Hummer and Attila Szabo "Free energy reconstruction from nonequilibrium single-molecule pulling experiments", Proceedings of the National Academy of Sciences of the United States of America 98 pp. 3658-3661 (2001)
- E. G. D. Cohen and D. Mauzerall "The Jarzynski equality and the Boltzmann factor", Molecular Physics 103 pp. 2923 - 2926 (2005)
- L. Y. Chen "On the Crooks fluctuation theorem and the Jarzynski equality", Journal of Chemical Physics 129 091101 (2008)
- Eric N. Zimanyi and Robert J. Silbey "The work-Hamiltonian connection and the usefulness of the Jarzynski equality for free energy calculations", Journal of Chemical Physics 130 171102 (2009)
- Humberto Híjar and José M Ortiz de Zárate "Jarzynski's equality illustrated by simple examples", European Journal of Physics 31 pp. 1097 (2010)