Green-Kubo relations

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

$L(F_{e}=0)={\frac {V}{k_{B}T}}\int _{0}^{\infty }\left\langle {J(0)J(s)}\right\rangle _{0}~{\mathrm {d} }s$

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

Shear viscosity

The shear viscosity is related to the pressure tensor via

$\eta ={\frac {V}{k_{B}T}}\int _{0}^{\infty }\langle p_{xy}(0)p_{xy}(t)\rangle ~{\mathrm {d} }t$

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

References

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)