Jarzynski equality

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

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

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

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

where $k B$ is the Boltzmann constant and $T$ 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.