Jarzynski Equality & Work Fluctuations

The Jarzynski equality ⟨e^{-βW}⟩ = e^{-βΔF} connects nonequilibrium work to equilibrium free energy. Even for fast (irreversible) processes, free energy can be extracted from the work distribution. This fundamentally extends the Clausius inequality.

Protocol

ΔF (exact)—
⟨W⟩ (mean work)—
⟨W⟩ - ΔF (dissipation)—
-ln⟨e^{-βW}⟩/β—
Jarzynski error—
Model: Harmonic oscillator; spring constant switched k₁→k₂ over finite time T.

Exact: ΔF = (1/2β)·ln(k₂/k₁)
Quasi-static work W_rev = ΔF
Fast switching: W_irrev > ΔF (dissipation)

Crooks fluctuation theorem:
P_F(W)/P_R(-W) = e^{β(W-ΔF)}

The tails of P(W) are crucial — the exponential average is dominated by rare trajectories with W < ΔF ("lucky" fluctuations).