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, (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} ), can be reconstructed by averaging the external 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} , performed in many non-equilibrium realizations of the process (Eq. 2a in ):

$\exp \left({\frac {-\Delta A}{k_{B}T}}\right)=\left\langle \exp \left({\frac {-W}{k_{B}T}}\right)\right\rangle$

or can be trivially re-written as (Eq. 2b)

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 \Delta A = - k_BT \ln \left\langle \exp \left( \frac{-W}{k_BT} \right) \right\rangle }

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. The only assumption in the proof of this relation is that of a weak coupling between the system and the reservoir.

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)