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A Transport coefficient \gamma can be expressed via a Green-Kubo relation:

\( \gamma= \int_0^\infty \langle \dot{A}(t) \dot{A}(0) \rangle dt, \)

where A is an observable occurring in a perturbed Hamiltonian, \( \langle \cdot \rangle \) is an ensemble average and the dot above the A denotes the time derivative.[1] For times t that are greater than the correlation time of the fluctuations of the observable the transport coefficient obeys a generalized Einstein relation:

\( 2t\gamma=\langle |A(t)-A(0)|^2 \rangle \)

Examples

Diffusion constant, relates the flux of particles with the negative gradient of the concentration (see Fick's laws of diffusion)
Heat transport coefficient
Mass transport coefficient
Shear Viscosity \eta = \frac{1}{k_BT V}\int_0 ^\infty dt \langle \sigma(0)_{xy} \sigma_{xy} (t) \rangle
Electrical conductivity