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, $\Delta A$, can be reconstructed by averaging the external work, $W$, performed in many nonequilibrium realizations of the process (Ref. 1 Eq. 2a):

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

or can be trivially re-written as (Ref. 1 Eq. 2b)

$\Delta A=-{\frac {1}{k_{B}T}}\ln \left\langle \exp \left({\frac {-W}{k_{B}T}}\right)\right\rangle$

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 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

1. C. Jarzynski "Nonequilibrium Equality for Free Energy Differences", Physical Review Letters 78 2690-2693 (1997)
2. E. G. D. Cohen; D. Mauzerall "The Jarzynski equality and the Boltzmann factor", Molecular Physics 103 pp. 2923 - 2926 (2005)