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Jarzynski equality: ⟨e^{−βW}⟩ = e^{−βΔF} — exact free energy from nonequilibrium work!
Crooks fluctuation theorem: P_F(W) / P_R(−W) = e^{β(W−ΔF)} relates forward/reverse work distributions.
Even when W > ΔF on average (2nd Law), rare trajectories with W < ΔF allow exact free energy recovery.
Left: particle trajectories in a driven double-well. Right: work distributions P_F(W) and P_R(−W).