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
- 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 L(F_e = 0) =\frac{V}{k_BT} \int_0^\infty \left\langle {J(0)J(s)} \right\rangle _{0} ~{\mathrm{d}} s}
where is the flux.
Shear viscosity
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
The Green-Kubo relations can be derived from the Evans-Searles transient fluctuation theorem[3]
References
- ↑ 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)