Green-Kubo relations: Difference between revisions

Revision as of 14:55, 22 December 2009

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_{B}T}}\int _{0}^{\infty }\left\langle {J(0)J(s)}\right\rangle _{0}~{\mathrm {d} }s

where $J$ is the flux.

\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.

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 (2006) (Third Edition) ISBN 0-12-370535-5 (chapter 7)

@@ Line 3: / Line 3: @@
 <ref>[http://dx.doi.org/10.1143/JPSJ.12.570 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)]</ref>
 are expressions that relate  macroscopic [[transport coefficients]] to integrals of microscopic
-[[time correlation functions]].
+[[time correlation functions]]. In general one has
+:<math> L(F_e  = 0) =\frac{V}{k_BT} \int_0^\infty  \left\langle {J(0)J(s)} \right\rangle _{0}  ~{\mathrm{d}} s</math>
+where <math>J</math> is the flux.
+==Shear viscosity==
+The [[Viscosity |shear viscosity]] is related to the [[Pressure |pressure tensor]] via
+:<math>\eta = \frac{V}{k_BT}\int_0^{\infty} \langle   p_{xy}(0) p_{xy}(t) \rangle ~{\mathrm{d}} t</math>
+i.e. the integral of the autocorrelation of the off-diagonal components of the microscopic [[Stress| stress tensor]].
 ==Fluctuation theorem==
 The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]]<ref>[http://dx.doi.org/10.1063/1.481610 Debra J. Searles and Denis J. Evans "The fluctuation theorem and Green–Kubo relations", Journal of Chemical Physics '''112''' pp. 9727-9735 (2000)]</ref>