= Green-Kubo formula for diffusion
{c}
{title2=$D=\int_0^\infty\chi(u)\,du$}
For a stationary zero-<mean> <velocity> with an absolutely integrable <velocity autocorrelation function> $\chi$, the <diffusion coefficient> is its time <integral>. Indeed, <differentiation> of the <mean squared displacement> identity gives $\tfrac12\frac{d}{dt}\mathbb E[(X(t)-X(0))^2]=\int_0^t\chi(u)\,du$, and the right side tends to the displayed <integral>. In the equilibrium <generalized Langevin equation>, this gives $D=\Theta/\widetilde\kappa(0)$ whenever stable relaxation and the required <integrability> hold. The formula is named after <Melville S. Green> and <Ryogo Kubo>.
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