Green-Kubo relations

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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[edit]

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[edit]

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


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)