Crooks fluctuation theorem ($P_F(W)/P_R(-W)=e^{\beta (W-\Delta F)}$)

For a system coupled to a thermal reservoir at fixed temperature $T$, prepare the forward process in the initial canonical ensemble and the reverse process in the final control parameter's canonical ensemble. Reverse the protocol, assume microscopic reversibility and matching path probability support, and count $W$ as work done on the system. With Helmholtz free energy difference $\Delta F=F_{\mathrm{final}}-F_{\mathrm{initial}}$ and $\beta =1/(k_{B}T)$,The final forward distribution need not already be in thermal equilibrium. Its reverse preparation is prescribed equilibrium preparation, rather than the actual forward final distribution used in finite trajectory entropy production. Thus the paired path logarithm $\beta (W-\Delta F)$ is dissipated work in thermal units, not automatically the instantaneous total stochastic entropy production at the end of an arbitrary driven protocol. ^{[ref] [ref]}

Integrating the Crooks fluctuation theorem over work and using normalization of the reverse probability distribution givesConsequently Jensen's inequality gives $⟨W{⟩}_F\ge \Delta F$. Some realizations may nevertheless have $W<\Delta F$. This is a work relation with equilibrium endpoint preparations, while the integral fluctuation theorem for total entropy production uses the actual final forward distribution. ^[ref]