Kirkwood superposition approximation - Revision history

Carl McBride: Added a recent publication

2013-11-06T16:27:39Z

Added a recent publication

← Older revision		Revision as of 18:27, 6 November 2013
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	the [[Pressure equation \|pressure]] and the [[compressibility equation]] if this superposition approximation is used to generate <math>g(r)</math>.		the [[Pressure equation \|pressure]] and the [[compressibility equation]] if this superposition approximation is used to generate <math>g(r)</math>.
	This approximation is rigorously correct for one-dimensional systems, and is only true in three-dimensions in the limit of zero density.		This approximation is rigorously correct for one-dimensional systems, and is only true in three-dimensions in the limit of zero density.
			It has recently been shown that the Kirkwood superposition approximation precludes the existence of a critical point (Ref. 6).
	==References==		==References==
	#[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood, "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]		#[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood, "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]
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	#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]		#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]
	#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216''' pp. 203-218 (1953)]		#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216''' pp. 203-218 (1953)]
	#[http://dx.doi.org/10.1143/PTP.21.421 Ryuzo Abe "On the Kirkwood Superposition Approximation", Progress of Theoretical Physics '''21''' pp. 421-430 (1959)]		#[http://dx.doi.org/10.1063/1.4824388 Jarosław Piasecki , Piotr Szymczak and John J. Kozak "Communication: Nonexistence of a critical point within the Kirkwood superposition approximation", Journal of Chemical Physics '''139''' 141101 (2013)]
	#[http://dx.doi.org/10.1063/1.1725757 Russell V. Cochran and L. H. Lund "On the Kirkwood Superposition Approximation", Journal of Chemical Physics '''41''' pp. 3499-3504 (1964)]		;Related reading
	#[http://dx.doi.org/10.1088/0034-4885/31/2/301 G. H. A. Cole "Classical fluids and the superposition approximation", Reports on Progress in Physics '''31''' pp. 419-470 (1968)]		*[http://dx.doi.org/10.1143/PTP.21.421 Ryuzo Abe "On the Kirkwood Superposition Approximation", Progress of Theoretical Physics '''21''' pp. 421-430 (1959)]
			*[http://dx.doi.org/10.1063/1.1725757 Russell V. Cochran and L. H. Lund "On the Kirkwood Superposition Approximation", Journal of Chemical Physics '''41''' pp. 3499-3504 (1964)]
			*[http://dx.doi.org/10.1088/0034-4885/31/2/301 G. H. A. Cole "Classical fluids and the superposition approximation", Reports on Progress in Physics '''31''' pp. 419-470 (1968)]
	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Nice and Tidy: /* References */ Added a reference

2009-06-30T10:13:24Z

References: Added a reference

← Older revision		Revision as of 12:13, 30 June 2009
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	#[http://dx.doi.org/10.1143/PTP.21.421 Ryuzo Abe "On the Kirkwood Superposition Approximation", Progress of Theoretical Physics '''21''' pp. 421-430 (1959)]		#[http://dx.doi.org/10.1143/PTP.21.421 Ryuzo Abe "On the Kirkwood Superposition Approximation", Progress of Theoretical Physics '''21''' pp. 421-430 (1959)]
	#[http://dx.doi.org/10.1063/1.1725757 Russell V. Cochran and L. H. Lund "On the Kirkwood Superposition Approximation", Journal of Chemical Physics '''41''' pp. 3499-3504 (1964)]		#[http://dx.doi.org/10.1063/1.1725757 Russell V. Cochran and L. H. Lund "On the Kirkwood Superposition Approximation", Journal of Chemical Physics '''41''' pp. 3499-3504 (1964)]
			#[http://dx.doi.org/10.1088/0034-4885/31/2/301 G. H. A. Cole "Classical fluids and the superposition approximation", Reports on Progress in Physics '''31''' pp. 419-470 (1968)]
	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Carl McBride at 13:27, 25 October 2007

2007-10-25T13:27:16Z

← Older revision		Revision as of 15:27, 25 October 2007
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	[[John G. Kirkwood]] (Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)		The '''Kirkwood superposition approximation''' takes its name from [[John G. Kirkwood]] (see Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)


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	It appears that this was used as a basis of a closure for the		It appears that this was used as a basis of a closure for the
	Kirkwood integral equation (Ref. 1) and the Yvon, and Born-Green		Kirkwood [[integral equations \|integral equation]] (Ref. 1) and the Yvon, and Born-Green
	(Ref. 2) until the work of Morita and Hiroike (Ref. 3).		(Ref. 2) until the work of Morita and Hiroike (Ref. 3).
	It was pointed out in Ref.s 4 and 5, that there is an inconsistency between		It was pointed out in Ref.s 4 and 5, that there is an inconsistency between
	the pressure and the compressibility equation if this superposition approximation is used to generate <math>g(r)</math>.		the [[Pressure equation \|pressure]] and the [[compressibility equation]] if this superposition approximation is used to generate <math>g(r)</math>.
	This approximation is rigorously correct for one-dimensional systems, and is only true in three-dimensions in the limit of zero density.		This approximation is rigorously correct for one-dimensional systems, and is only true in three-dimensions in the limit of zero density.
	==References==		==References==

Carl McBride: /* References */

2007-09-06T13:09:15Z

References

← Older revision		Revision as of 15:09, 6 September 2007
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	#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216''' pp. 203-218 (1953)]		#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216''' pp. 203-218 (1953)]
	#[http://dx.doi.org/10.1143/PTP.21.421 Ryuzo Abe "On the Kirkwood Superposition Approximation", Progress of Theoretical Physics '''21''' pp. 421-430 (1959)]		#[http://dx.doi.org/10.1143/PTP.21.421 Ryuzo Abe "On the Kirkwood Superposition Approximation", Progress of Theoretical Physics '''21''' pp. 421-430 (1959)]
			#[http://dx.doi.org/10.1063/1.1725757 Russell V. Cochran and L. H. Lund "On the Kirkwood Superposition Approximation", Journal of Chemical Physics '''41''' pp. 3499-3504 (1964)]
	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Carl McBride at 13:50, 10 July 2007

2007-07-10T13:50:26Z

← Older revision		Revision as of 15:50, 10 July 2007
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	:<math>~~g_N~~^{(3)}(~~r_1~~,~~r_2~~,~~r_3~~)=~~g_N~~^{(2)}(~~r_1~~,~~r_2~~)~~g_N~~^{(2)}(~~r_2~~,~~r_3~~)~~g_N~~^{(2)}(~~r_3~~,~~r_1~~)</math>		:<math>{\rm g}_N^{(3)}({\mathbf r}_1,{\mathbf r}_2,{\mathbf r}_3)={\rm g}_N^{(2)}({\mathbf r}_1,{\mathbf r}_2){\rm g}_N^{(2)}({\mathbf r}_2,{\mathbf r}_3){\rm g}_N^{(2)}({\mathbf r}_3,{\mathbf r}_1)</math>

Carl McBride at 12:48, 28 June 2007

2007-06-28T12:48:24Z

← Older revision		Revision as of 14:48, 28 June 2007
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	It was pointed out in Ref.s 4 and 5, that there is an inconsistency between		It was pointed out in Ref.s 4 and 5, that there is an inconsistency between
	the pressure and the compressibility equation if this superposition approximation is used to generate <math>g(r)</math>.		the pressure and the compressibility equation if this superposition approximation is used to generate <math>g(r)</math>.
			This approximation is rigorously correct for one-dimensional systems, and is only true in three-dimensions in the limit of zero density.
	==References==		==References==
	#[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood, "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]		#[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood, "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]

Carl McBride at 10:26, 11 June 2007

2007-06-11T10:26:23Z

← Older revision		Revision as of 12:26, 11 June 2007
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	#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]		#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]
	#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216''' pp. 203-218 (1953)]		#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216''' pp. 203-218 (1953)]
			#[http://dx.doi.org/10.1143/PTP.21.421 Ryuzo Abe "On the Kirkwood Superposition Approximation", Progress of Theoretical Physics '''21''' pp. 421-430 (1959)]
	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Carl McBride at 13:34, 27 February 2007

2007-02-27T13:34:00Z

← Older revision		Revision as of 15:34, 27 February 2007
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	[[John G. Kirkwood]] ~~1935~~ (Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)		[[John G. Kirkwood]] (Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)


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	#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]		#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]
	#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]		#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]
	#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216'''pp. 203-218 (1953)]		#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216''' pp. 203-218 (1953)]

	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Carl McBride at 13:31, 27 February 2007

2007-02-27T13:31:53Z

← Older revision		Revision as of 15:31, 27 February 2007
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	#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216'''pp. 203-218 (1953)]		#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216'''pp. 203-218 (1953)]

	[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Carl McBride: /* References */

2007-02-27T13:31:42Z

References

← Older revision		Revision as of 15:31, 27 February 2007
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	#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]		#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]
	#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]		#[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]
	#[~~PRSA_1953_216_0203~~]		#[http://links.jstor.org/sici?sici=0080-4630%2819530122%29216%3A1125%3C203%3AOTTOF%3E2.0.CO%3B2-5 G. S. Rushbrooke and H. I. Scoins "On the Theory of Fluids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, '''216'''pp. 203-218 (1953)]

			[Category: Statistical mechanics]]