Crooks (1999): P_F(W)/P_R(−W) = exp[β(W−ΔF)] — the crossing point P_F(W*)=P_R(−W*) gives W*=ΔF exactly.
Jarzynski (1997): ⟨e^{−βW}⟩_F = e^{−βΔF} — exponential average over nonequilibrium work recovers equilibrium free energy.
Second law: ⟨W⟩ ≥ ΔF (Jensen's inequality on Jarzynski). Dissipated work W_diss = ⟨W⟩ − ΔF ≥ 0.