Green–Kubo relations

Remarks

The proof is a short computation. By stationarity, $\mathbb{E}[(X_t - X_0)^2] = \int_0^t \int_0^t C(s - u)\, du\, ds = 2\int_0^t (t - s) C(s)\, ds$. Dividing by $2t$ gives $\int_0^t (1 - s/t) C(s)\, ds$, which tends to $\int_0^\infty C(s)\, ds$ by dominated convergence. For the Ornstein–Uhlenbeck velocity with $C(t) = \tfrac{k_B T}{m} e^{-\gamma t}$, the formula gives $D = k_B T/(m\gamma)$, which is the Einstein relation with mobility $1/(m\gamma)$. The general Green–Kubo relations have the same form. A transport coefficient equals the time integral of the equilibrium autocorrelation of the corresponding current, times a prefactor. For the electrical conductivity of a system of volume $V$, the prefactor is $1/(V k_B T)$, applied to one component of the total current. Without integrability of $C$ the limit can be infinite, which happens in anomalous diffusion.

Going deeper

For a student

Green–Kubo relations are the time-integrated form of linear response theory. They are how molecular dynamics simulations compute transport coefficients: one records currents in an equilibrium run and integrates their autocorrelation. The diffusion case above is G. I. Taylor's lemma on dispersion by continuous movements, and the conductivity case follows from the Kubo formula at zero frequency. In low dimensions the current correlations of interacting fluids can decay so slowly that the integral diverges, so a transport coefficient need not exist.