= Jarzynski equality
{c}
{title2=$\langle e^{-\beta W}\rangle_F=e^{-\beta\Delta F}$}
Integrating the <Crooks fluctuation theorem> over <work> and using normalization of the reverse <probability distribution> gives
$$
\langle e^{-\beta W}\rangle_F=e^{-\beta\Delta F}.
$$
Consequently <Jensen's inequality> gives $\langle W\rangle_F\ge\Delta F$. 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. <The Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences by Gavin E. Crooks>[Crooks, 1999, Eq. (3) and derivation from Eqs. (10)–(11)]{ref}
Back to article page
