Green–Kubo relations

The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for a transport coefficient ${\displaystyle \gamma }$ in terms of the integral of the equilibrium time correlation function of the time derivative of a corresponding microscopic variable ${\displaystyle A}$ (sometimes termed a "gross variable", as in ):

$${\displaystyle \gamma =\int _{0}^{\infty }\left\langle {\dot {A}}(t){\dot {A}}(0)\right\rangle \;{\mathrm {d} }t.}$$

One intuitive way to understand this relation is that relaxations resulting from random fluctuations in equilibrium are indistinguishable from those due to an external perturbation in linear response.

Green-Kubo relations are important because they relate a macroscopic transport coefficient to the correlation function of a microscopic variable. In addition, they allow one to measure the transport coefficient without perturbing the system out of equilibrium, which has found much use in molecular dynamics simulations.