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The '''Jarzynski equality''' | The '''Jarzynski equality''' is also known as the ''work relation'' or ''non-equilibrium work relation''. | ||
According to this equality, the equilibrium [[Helmholtz energy function]] of a process, | According to this equality, the ''equilibrium'' [[Helmholtz energy function]] of a process, <math>\Delta A</math>, can be reconstructed by averaging the external [[work]], <math>W</math>, performed in many nonequilibrium realizations of the process (Ref. 1 Eq. 2a): | ||
:<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> | ||
or can be trivially re-written as (Eq. 2b) | or can be trivially re-written as (Ref. 1 Eq. 2b) | ||
:<math>\Delta A = - k_BT \ln \left\langle \exp \left( \frac{-W}{k_BT} \right) \right\rangle </math> | :<math>\Delta A = - \frac{1}{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 | where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]]. The proof of this equation is given in Ref. 1 and the only assumption is that of a weak coupling between the system and the reservoir. | ||
==References== | |||
#[http://dx.doi.org/10.1103/PhysRevLett.78.2690 C. Jarzynski "Nonequilibrium Equality for Free Energy Differences", Physical Review Letters '''78''' 2690-2693 (1997)] | |||
'''Related reading''' | '''Related reading''' | ||
*[http://dx.doi.org/10.1073/pnas.071034098 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)] | *[http://dx.doi.org/10.1073/pnas.071034098 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)] | ||
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*[http://dx.doi.org/10.1063/1.3132747 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)] | *[http://dx.doi.org/10.1063/1.3132747 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)] | ||
*[http://dx.doi.org/10.1088/0143-0807/31/5/012 Humberto Híjar and José M Ortiz de Zárate "Jarzynski's equality illustrated by simple examples", European Journal of Physics '''31''' pp. 1097 (2010)] | *[http://dx.doi.org/10.1088/0143-0807/31/5/012 Humberto Híjar and José M Ortiz de Zárate "Jarzynski's equality illustrated by simple examples", European Journal of Physics '''31''' pp. 1097 (2010)] | ||
[[category: Non-equilibrium thermodynamics]] | [[category: Non-equilibrium thermodynamics]] | ||
[[category: fluctuation theorem]] | [[category: fluctuation theorem]] |