Green-Kubo relations: Difference between revisions
Jump to navigation
Jump to search
Carl McBride (talk | contribs) No edit summary |
Carl McBride (talk | contribs) mNo edit summary |
||
| (5 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
The '''Green-Kubo relations''' | {{Stub-general}} | ||
[[time correlation functions]]. | 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> | ||
The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]]. | <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]]. 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
|
This article is a 'stub' page, it has no, or next to no, content. It is here at the moment to help form part of the structure of SklogWiki. If you add sufficient material to this article then please remove the {{Stub-general}} template from this page. |
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is the flux.
Shear viscosity[edit]
The shear viscosity is related to the pressure tensor via
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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]
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)