Jarzynski Equality

⟨e−βW⟩ = e−βΔF — exact nonequilibrium free energy recovery
⟨e^{−βW}⟩ = e^{−βΔF}
ΔF = −kT·ln⟨e^{−βW}⟩
⟨W⟩ ≥ ΔF (2nd law)
W_diss = ⟨W⟩ − ΔF ≥ 0
Runs: 0
⟨W⟩ = — kT
ΔF (true) = — kT
ΔF (Jarz.) = — kT
W_diss = — kT
2nd law satisfied: —
Pulling speed v: 1.0
Temperature β⁻¹ (kT): 1.0
Barrier ΔF: 2.0 kT
Batch size: 20
The Jarzynski equality (1997) is an exact relation valid for processes of arbitrary speed. Slow (quasi-static) protocols give W≈ΔF with low variance; fast protocols give ⟨W⟩≫ΔF but the exponential average still recovers ΔF exactly. This works because rare trajectories with W<ΔF (that violate the 2nd law for a single trajectory) have exponentially large weight e−βW that compensates. Here we simulate Brownian particles pulled through a double-well potential.