Jarzynski equality: ⟨e^(−βW)⟩ = e^(−βΔF) — an exact relation connecting nonequilibrium work to equilibrium free energy difference. The second law requires ⟨W⟩ ≥ ΔF, but rare trajectories with W<ΔF (violations of the second law) are essential to recover ΔF. The Crooks fluctuation theorem generalizes this: P_F(W)/P_R(−W) = e^(β(W−ΔF)). This is the foundation of single-molecule pulling experiments.