What does this equation mean?

This is a striking claim on its face: it says a quantity defined only for reversible, quasistatic processes — a free energy difference — can be computed from measurements of an irreversible process, no matter how fast or violent the driving, provided enough repetitions are averaged. Two years later, Gavin Crooks derived a closely related and in some respects more general result, the Crooks fluctuation theorem, in “Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences,” published in Physical Review E in 1999 [ 4 ] . Crooks’s relation compares the probability of observing work W in a forward protocol to the probability of observing -W in the corresponding time-reversed protocol: $\frac{P_F(W)}{P_R(-W)} = e^{(W-\Delta F)/k_B T}$. The Jarzynski equality follows from the Crooks relation as a special case, and both are instances of a broader family of fluctuation theorems that had been developing since Denis Evans, E. G. D. Cohen, and G. P. Morriss’s 1993 work on fluctuation theorems in shearing steady states. What made Jarzynski’s and Crooks’s results different from that broader family was specificity and testability: they made a sharp, falsifiable numerical prediction about a quantity — work performed on a driven system — that single-molecule experimentalists were, by the late 1990s, on the verge of being able to measure directly with laser tweezers and atomic force microscopes.