Overdamped Langevin dynamics: γẋ = F(x,t) + √(2γk_BT) η(t). A driven Brownian particle in a tilted harmonic trap accumulates entropy at rate σ = ⟨F·ẋ⟩/T. The Jarzynski equality: ⟨e^{-W/k_BT}⟩ = e^{-ΔF/k_BT}.
σ = 0.00 k_B/s | ⟨e^{-W}⟩ = 1.000 | n = 0
Entropy production rate σ = F·v/T (second law: σ≥0). Jarzynski 1997: ⟨exp(-W/k_BT)⟩ = exp(-ΔF/k_BT) holds far from equilibrium — an exact nonequilibrium identity. Crooks fluctuation theorem: P(+W)/P(-W) = e^{W/k_BT}. The work histogram shows the probability of observing negative entropy fluctuations (second law violations for individual trajectories, never on average).