# Green-Kubo relations

The Green-Kubo relations [1] [2] are expressions that relate macroscopic transport coefficients to integrals of microscopic time correlation functions. In general one has

$L(F_e = 0) =\frac{V}{k_BT} \int_0^\infty \left\langle {J(0)J(s)} \right\rangle _{0} ~{\mathrm{d}} s$

where $L$ is the linear transport coefficient and $J$ is the flux.

## Shear viscosity

The shear viscosity is related to the pressure tensor via

$\eta = \frac{V}{k_BT}\int_0^{\infty} \langle p_{xy}(0) p_{xy}(t) \rangle ~{\mathrm{d}} t$

i.e. the integral of the autocorrelation of the off-diagonal components of the microscopic stress tensor.

## Fluctuation theorem

The Green-Kubo relations can be derived from the Evans-Searles transient fluctuation theorem[3]