⟨e^{−W/kT}⟩ = e^{−ΔF/kT} — free energy differences can be recovered from nonequilibrium work measurements, even for irreversible processes. Crooks fluctuation theorem: P_F(W)/P_R(−W) = e^{(W−ΔF)/kT}.
Jarzynski: ⟨e^{-W}⟩ = ? | ΔF = ?
A Brownian particle is dragged through a harmonic potential V(x,λ)=½k(x−λ)² as λ moves from 0 to L at speed v. Work W=∫(∂V/∂λ)dλ varies across trajectories. The Jarzynski equality ⟨e^{-βW}⟩=e^{-βΔF} holds exactly. Slow pulling → near-reversible, narrow W distribution. Fast pulling → irreversible, wide distribution skewed right, but equality still holds.