Green-Kubo relations - Revision history

Carl McBride at 14:33, 22 December 2009

2009-12-22T14:33:49Z

← Older revision		Revision as of 16:33, 22 December 2009
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	:<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>		:<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.		where <math>L</math> is the linear transport coefficient and <math>J</math> is the flux.
	==Shear viscosity==		==Shear viscosity==
	The [[Viscosity \|shear viscosity]] is related to the [[Pressure \|pressure tensor]] via		The [[Viscosity \|shear viscosity]] is related to the [[Pressure \|pressure tensor]] via

Carl McBride: /* References */ Added a reference to a book chapter.

2009-12-22T14:12:37Z

References: Added a reference to a book chapter.

← Older revision		Revision as of 16:12, 22 December 2009
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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]]

Carl McBride: Added a little content.

2009-12-22T13:55:02Z

Added a little content.

← Older revision		Revision as of 15:55, 22 December 2009
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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>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>

Carl McBride: Changed references to Cite format.

2009-12-22T12:44:37Z

Changed references to Cite format.

← Older revision		Revision as of 14:44, 22 December 2009
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	{{Stub-general}}		{{Stub-general}}
	The '''Green-Kubo relations''' ~~provide exact mathematical~~ expressions ~~for the~~ [[transport coefficients]] ~~in terms of~~ integrals of		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>
			<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]].
	The Green-Kubo relations can be derived from the [[Evans-Searles transient 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>
	==References==		==References==
	~~#[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)]~~		<references/>
	~~#[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)]~~		'''Related reading'''
	#[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~~)]		*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])
	[[Category: Non-equilibrium thermodynamics]]		[[Category: Non-equilibrium thermodynamics]]

Carl McBride at 10:59, 27 September 2007

2007-09-27T10:59:52Z

← Older revision		Revision as of 12:59, 27 September 2007
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			{{Stub-general}}
	The '''Green-Kubo relations''' provide exact mathematical expressions for the [[transport coefficients]] in terms of integrals of		The '''Green-Kubo relations''' provide exact mathematical expressions for the [[transport coefficients]] in terms of integrals of
	[[time correlation functions]].		[[time correlation functions]].

Carl McBride at 13:22, 13 June 2007

2007-06-13T13:22:42Z

← Older revision		Revision as of 15:22, 13 June 2007
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	The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]].		The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]].
	==References==		==References==
			#[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)]
			#[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)]
	#[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)]		#[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)]
	[[Category: Non-equilibrium thermodynamics]]		[[Category: Non-equilibrium thermodynamics]]

Carl McBride at 13:10, 13 June 2007

2007-06-13T13:10:26Z

← Older revision		Revision as of 15:10, 13 June 2007
Line 1:		Line 1:
			The '''Green-Kubo relations''' provide exact mathematical expressions for the [[transport coefficients]] in terms of integrals of
			[[time correlation functions]].
	The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]].		The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]].
	==References==		==References==
	#[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)]		#[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)]
	[[Category: Non-equilibrium thermodynamics]]		[[Category: Non-equilibrium thermodynamics]]

Carl McBride: New page: The Green-Kubo relations can be derived from the Evans-Searles transient fluctuation theorem. ==References== #[http://dx.doi.org/10.1063/1.481610 Debra J. Searles and Denis J. Evans "T...

2007-06-13T13:06:04Z

New page: The Green-Kubo relations can be derived from the Evans-Searles transient fluctuation theorem. ==References== #[http://dx.doi.org/10.1063/1.481610 Debra J. Searles and Denis J. Evans "T...

New page

The Green-Kubo relations can be derived from the [[Evans-Searles transient fluctuation theorem]].
==References==
#[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)]
[[Category: Non-equilibrium thermodynamics]]