⟨e^(−βW)⟩ = e^(−βΔF) — recovering free energy from irreversible work
The Jarzynski equality (1997) ⟨e^(−βW)⟩ = e^(−βΔF) is an exact nonequilibrium relation: by exponentially averaging work over many irreversible trajectories, we recover the equilibrium free energy difference ΔF. The average work always exceeds ΔF (Second Law), but rare trajectories with W < ΔF (violation of Clausius) dominate the exponential average.