Jarzynski Equality

Free energy differences from nonequilibrium work distributions: ⟨e−W/kT⟩ = e−ΔF/kT

2.0 (fast)
kT = 1.0
N = 80

The Equality

Jarzynski (1997) proved that for ANY nonequilibrium protocol connecting two states, an exponential average of the work W done on the system equals the equilibrium free energy difference ΔF:

⟨e−βW⟩ = e−βΔF, β = 1/kT

This holds regardless of how fast the protocol is. Slow protocols cluster near ΔF; fast ones spread widely. Crucially, rare trajectories with W < ΔF (entropy-consuming) dominate the average and pull it down to the correct equilibrium value.