# 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

where is the linear transport coefficient and is the flux.

## Shear viscosity[edit]

The shear viscosity is related to the pressure tensor via

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

## Fluctuation theorem[edit]

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

## References[edit]

- ↑ Melville S. Green "Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena. II. Irreversible Processes in Fluids", Journal of Chemical Physics
**22**pp. 398-413 (1954) - ↑ Ryogo Kubo "Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems", Journal of the Physical Society of Japan
**12**PP. 570-586 (1957) - ↑ Debra J. Searles and Denis J. Evans "The fluctuation theorem and Green–Kubo relations", Journal of Chemical Physics
**112**pp. 9727-9735 (2000)

**Related reading**

- Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press, 3rd Edition (2006) ISBN 0-12-370535-5 (chapter 7)
- Denis J. Evans and Gary Morriss "Statistical Mechanics of Nonequilibrium Liquids", Cambridge University Press, 2nd Edition (2008) ISBN 9780521857918 (Chapter 4)