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# Transport coefficient

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

See also

Linear response theory

References

Water in Biology, Chemistry, and Physics: Experimental Overviews and Computational Methodologies, G. Wilse Robinson, ISBN 9789810224516, p. 80, Google Books

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