Green–Kubo relations

The Green–Kubo relations (<a href="/facts/Melville_S._Green/S3BmA6eM">Melville S. Green</a> 1954, <a href="/facts/Ryogo_Kubo/hAHo0S85">Ryogo Kubo</a> 1957) give the exact mathematical expression for a <a href="/facts/Transport_coefficient/qbLoUSGa">transport coefficient</a> 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 in terms of the integral of the equilibrium <a href="/facts/Correlation_function/8mmhdp8J">time correlation function</a> of the time derivative of a corresponding microscopic variable 
 
 
 
 A
 
 
 {\displaystyle A}
 
 (sometimes termed a "gross variable", as in ):

 
 
 
 γ
 =
 
 ∫
 
 0
 
 
 ∞
 
 
 
 ⟨
 
 
 
 
 A
 ˙
 
 
 
 (
 t
 )
 
 
 
 A
 ˙
 
 
 
 (
 0
 )
 
 ⟩
 
 
 
 
 d
 
 
 t
 .
 
 
 {\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.

Green–Kubo relations open-in-new

Green–Kubo relations