Ornstein-Zernike relation - Revision history

Carl McBride: /* References */ Added a recent publication

2016-12-21T13:17:17Z

References: Added a recent publication

← Older revision		Revision as of 15:17, 21 December 2016
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	'''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 § 3.5		*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5
			*[http://doi.org/10.1063/1.4972020 Yan He, Stuart A. Rice, and Xinliang Xu "Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids", Journal of Chemical Physics '''145''' 234508 (2016)]


	[[Category: Integral equations]]		[[Category: Integral equations]]

128.174.229.94: /* Ornstein-Zernike relation in Fourier space */

2012-06-05T18:51:43Z

Ornstein-Zernike relation in Fourier space

← Older revision		Revision as of 20:51, 5 June 2012
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	:<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}(r) dk</math>		:<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}(k) dk</math>

	Note:		Note:

155.185.147.143: /* Ornstein-Zernike relation in Fourier space */ latex fix

2012-05-14T11:49:06Z

Ornstein-Zernike relation in Fourier space: latex fix

← Older revision		Revision as of 13:49, 14 May 2012
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	The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):		The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):

	:<math>\hat{\gamma} = ({\mathbf I} - \rho {\mathbf \hat{c}})^{-1} {\mathbf \hat{c}} \rho {\mathbf \hat{c}}</math>		:<math>\hat{\gamma} = (\mathbf{I} - \rho \mathbf{\hat{c}})^{-1} \mathbf{\hat{c}} \rho \mathbf{\hat{c}}</math>

	The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly		The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly

Carl McBride: Added an internal link

2010-09-02T12:50:11Z

Added an internal link

← Older revision		Revision as of 14:50, 2 September 2010
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	particle labels.		particle labels.
	The Ornstein-Zernike relation can be derived by performing a functional differentiation		The Ornstein-Zernike relation can be derived by performing a functional differentiation
	of the grand canonical distribution function ~~(HM check this)~~.		of the [[Grand canonical ensemble \|grand canonical]] distribution function.
	==Ornstein-Zernike relation in Fourier space==		==Ornstein-Zernike relation in Fourier space==
	The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):		The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):

Carl McBride: Added an internal link

2010-09-02T12:48:21Z

Added an internal link

Carl McBride: Created a "notation" box.

2010-08-18T14:15:01Z

Created a "notation" box.

← Older revision		Revision as of 16:15, 18 August 2010
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	Notation:		<div style="border:1px solid #f3f3ff; padding-left: 0.5em !important; background-color: #f3f3ff; border-width: 0 0 0 1.4em; clear:right; float:right;">
			Notation used:
	*<math>g(r)</math> is the [[Pair distribution function \| pair distribution function]].		*<math>g(r)</math> is the [[Pair distribution function \| pair distribution function]].
	*<math>\Phi(r)</math> is the [[Intermolecular pair potential \| pair potential]] acting between pairs.		*<math>\Phi(r)</math> is the [[Intermolecular pair potential \| pair potential]] acting between pairs.
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	*<math>\omega(r)</math> is the [[Thermal potential \| thermal potential]].		*<math>\omega(r)</math> is the [[Thermal potential \| thermal potential]].
	*<math>f(r)</math> is the [[Mayer f-function]].		*<math>f(r)</math> is the [[Mayer f-function]].
			</div>

			The '''Ornstein-Zernike relation''' (OZ) integral equation <ref>L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)</ref> is given by:
	The '''Ornstein-Zernike relation''' (OZ) integral equation is
	:<math>h=h\left[c\right]</math>		:<math>h=h\left[c\right]</math>
	where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact.		where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact.
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	(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)		(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)
	:<math>h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math>		:<math>h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math>
	If the system is both homogeneous and isotropic, the OZ relation becomes (Ref. ~~1Eq. 6~~)		If the system is both homogeneous and isotropic, the OZ relation becomes (Eq. 6 of Ref. 1)

	:<math>\gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(\|{\mathbf r} - {\mathbf r'}\|) {\rm d}{\mathbf r'}</math>		:<math>\gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(\|{\mathbf r} - {\mathbf r'}\|) {\rm d}{\mathbf r'}</math>

	In words, this equation (Hansen and McDonald, section 5.2 p. 107)		In words, this equation (Hansen and McDonald, section 5.2 p. 107)
	``...describes the fact that the ''total'' correlation between particles 1 and 2, represented by <math>h(1,2)</math>,		:"...describes the fact that the ''total'' correlation between particles 1 and 2, represented by <math>h(1,2)</math>, is due in part to the ''direct'' correlation between 1 and 2, represented by <math>c(1,2)</math>, but also to the ''indirect'' correlation, <math>\gamma (r)</math>, propagated via increasingly large numbers of intermediate particles."
	is due in part to the ''direct'' correlation between 1 and 2, represented by <math>c(1,2)</math>, but also to the ''indirect'' correlation,
	<math>\gamma (r)</math>, propagated via increasingly large numbers of intermediate particles."

	Notice that this equation is basically a convolution, ''i.e.''		Notice that this equation is basically a convolution, ''i.e.''
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	:::::''etc.''		:::::''etc.''

	Diagrammatically this expression can be written as (~~Ref. 2~~):		Diagrammatically this expression can be written as <ref>[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)]</ref>:

	:[[Image:oz_diag.png]]		:[[Image:oz_diag.png]]
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	of the grand canonical distribution function (HM check this).		of the grand canonical distribution function (HM check this).
	==OZ equation in Fourier space==		==OZ equation in Fourier space==
	The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (Eq. 5 ~~in Ref. 3~~):		The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):

	:<math>\hat{\gamma} = ({\mathbf I} - \rho {\mathbf \hat{c}})^{-1} {\mathbf \hat{c}} \rho {\mathbf \hat{c}}</math>		:<math>\hat{\gamma} = ({\mathbf I} - \rho {\mathbf \hat{c}})^{-1} {\mathbf \hat{c}} \rho {\mathbf \hat{c}}</math>
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	==References==		==References==
	#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)		<references/>
	~~#[http://dx.doi.org/10.1103~~/~~PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)]~~
	#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]
	'''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 § 3.5		*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5

Carl McBride: Reverted edits by Marina D. Kung (Talk) to last version by Carl McBride

2010-07-26T11:11:38Z

Reverted edits by Marina D. Kung (Talk) to last version by Carl McBride

← Older revision		Revision as of 13:11, 26 July 2010
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	#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)		#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)
	#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)]		#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)]
	#[http://~~www~~.~~essaywriters~~.~~net~~/ ~~freelance writer~~]		#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]
	'''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 § 3.5		*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5

Marina D. Kung at 11:10, 26 July 2010

2010-07-26T11:10:53Z

← Older revision		Revision as of 13:10, 26 July 2010
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	#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)		#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)
	#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)]		#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)]
	#[http://dx.~~doi~~.~~org~~/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]		#[http://www.essaywriters.net/ freelance writer]
	'''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 § 3.5		*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5

Carl McBride: /* References */ Corrected reference.

2010-03-16T15:57:00Z

References: Corrected reference.

← Older revision		Revision as of 17:57, 16 March 2010
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	==References==		==References==
	#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)		#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)
	#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816 - 824 (1992)]		#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)]
	#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]		#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]
	#Hansen and ~~MacDonald~~ "Theory of Simple Liquids"		'''Related reading'''
			*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5


	[[Category: Integral equations]]		[[Category: Integral equations]]

Carl McBride at 15:08, 12 February 2008

2008-02-12T15:08:27Z

← Older revision		Revision as of 17:08, 12 February 2008
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	:<math>c=c\left[h\right]</math>		:<math>c=c\left[h\right]</math>
	Note that <math>h</math> depends on <math>c</math>, and <math>c</math> depends on <math>h</math>.		Note that <math>h</math> depends on <math>c</math>, and <math>c</math> depends on <math>h</math>.
	Because of this <math>h</math> must be determined [[self-consistently]].		Because of this <math>h</math> must be determined self-consistently.
	This need for self-consistency is characteristic of all many-body problems.		This need for self-consistency is characteristic of all many-body problems.
	(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)		(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)

← Older revision		Revision as of 14:48, 2 September 2010
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	</div>		</div>

	The '''Ornstein-Zernike relation''' ~~(OZ)~~ integral equation <ref>L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)</ref> is given by:		The '''Ornstein-Zernike relation''' integral equation <ref>L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)</ref> is given by:
	:<math>h=h\left[c\right]</math>		:<math>h=h\left[c\right]</math>
	where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact.		where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact.
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	Because of this <math>h</math> must be determined self-consistently.		Because of this <math>h</math> must be determined self-consistently.
	This need for self-consistency is characteristic of all many-body problems.		This need for self-consistency is characteristic of all many-body problems.
	(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)		(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the Ornstein-Zernike relation has the form (5.2.7)
	:<math>h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math>		:<math>h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math>
	If the system is both homogeneous and isotropic, the OZ relation becomes (Eq. 6 of Ref. 1)		If the system is both homogeneous and isotropic, the Ornstein-Zernike relation becomes (Eq. 6 of Ref. 1)

	:<math>\gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(\|{\mathbf r} - {\mathbf r'}\|) {\rm d}{\mathbf r'}</math>		:<math>\gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(\|{\mathbf r} - {\mathbf r'}\|) {\rm d}{\mathbf r'}</math>
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	where the bold lines connecting root points denote <math>c</math> functions, the blobs denote <math>h</math> functions.		where the bold lines connecting root points denote <math>c</math> functions, the blobs denote <math>h</math> functions.
	An arrow pointing from left to right indicates an uphill path from one root		An arrow pointing from left to right indicates an uphill path from one root
	point to another. An `uphill path' is a sequence of Mayer bonds passing through increasing		point to another. An `uphill path' is a sequence of [[Mayer f-function \|Mayer bonds]] passing through increasing
	particle labels.		particle labels.
	The OZ relation can be derived by performing a functional differentiation		The Ornstein-Zernike relation can be derived by performing a functional differentiation
	of the grand canonical distribution function (HM check this).		of the grand canonical distribution function (HM check this).
	==~~OZ equation~~ in Fourier space==		==Ornstein-Zernike relation in Fourier space==
	The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):		The Ornstein-Zernike equation may be written in [[Fourier analysis \|Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):