= Crooks fluctuation theorem
{c}
{title2=$P_F(W)/P_R(-W)=e^{\beta(W-\Delta F)}$}
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 $\Delta F=F_{\rm final}-F_{\rm initial}$ and $\beta=1/(k_BT)$,
$$
\frac{P_F(W)}{P_R(-W)}=e^{\beta(W-\Delta F)}.
$$
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. <The Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences by Gavin E. Crooks>[Crooks, 1999, Eqs. (5)–(7), (9)–(11): reversal, equilibrium weights and work relation]{ref} <Entropy production along a stochastic trajectory and an integral fluctuation theorem by Udo Seifert>[Seifert, 2005, Eq. (19) and following paragraph: actual final distribution versus final equilibrium distribution]{ref}
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