Green-Kubo relations: Difference between revisions
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The '''Green-Kubo relations''' | The '''Green-Kubo relations''' <ref>[http://dx.doi.org/10.1063/1.1740082 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)]</ref> | ||
[[time correlation functions]]. | <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> | ||
The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]]. | are expressions that relate macroscopic [[transport coefficients]] to integrals of microscopic | ||
[[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>L</math> is the linear transport coefficient and <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> | |||
==References== | ==References== | ||
<references/> | |||
'''Related reading''' | |||
*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press, 3rd Edition (2006) ISBN 0-12-370535-5 ([http://dx.doi.org/10.1016/B978-012370535-8/50009-4 chapter 7]) | |||
* Denis J. Evans and Gary Morriss "Statistical Mechanics of Nonequilibrium Liquids", Cambridge University Press, 2nd Edition (2008) ISBN 9780521857918 (Chapter 4) | |||
[[Category: Non-equilibrium thermodynamics]] | [[Category: Non-equilibrium thermodynamics]] | ||
Latest revision as of 15:33, 22 December 2009
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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
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)