Green-Kubo relations: Difference between revisions

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<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>
<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   
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>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==
==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>
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>
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<references/>
<references/>
'''Related reading'''
'''Related reading'''
*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 ([http://dx.doi.org/10.1016/B978-012370535-8/50009-4 chapter 7])
*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]]

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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]

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)