GREEN-KUBO RELATIONS

Transport Coefficients from Equilibrium Fluctuations
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D = —
The Green-Kubo formula relates transport coefficients to equilibrium time-correlation functions: D = ∫₀^∞ ⟨v(0)·v(t)⟩ dt. For Brownian particles: ⟨v(0)v(t)⟩ = (k_BT/m)·e^{-γt}, giving Einstein's relation D = k_BT/γ. Top left: particle trajectories. Top right: velocity autocorrelation C(t). Bottom: running integral (diffusion coefficient).