Jarzynski Equality — Nonequilibrium Work

Temperature β⁻¹ = kT: 1.0 Protocol speed v: 1.0 Trajectories N: 200

⟨W⟩: —

ΔF (exact): —

⟨e^−βW⟩: —

−β⁻¹ ln⟨e^−βW⟩: —

2nd law OK: —

Violations: —

Jarzynski 1997:

⟨e^−βW⟩ = e^−βΔF

Exact for any protocol speed.
By Jensen's inequality:

⟨W⟩ ≥ ΔF

(Second Law of Thermodynamics)

Work distribution P(W) shifts right for fast protocols but rare low-W events dominate the exponential average.