Difference between revisions of "Kirkwood superposition approximation"

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[[John G. Kirkwood]]  (Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)
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The '''Kirkwood superposition approximation''' takes its name from [[John G. Kirkwood]]  (see Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)
  
  
:<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>
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:<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>
  
  
 
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
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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>.
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the [[Pressure equation |pressure]] and the [[compressibility equation]] if this superposition approximation is used to generate <math>g(r)</math>.
 
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This approximation is rigorously correct for one-dimensional systems, and is only true in three-dimensions in the limit of zero density.
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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)]
 
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#[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)]
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;Related reading
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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)]
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*[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)]
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*[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]]

Latest revision as of 18:27, 6 November 2013

The Kirkwood superposition approximation takes its name from John G. Kirkwood (see Eq. 40 Ref. 1, Eq. 5.6 Ref. 2)


{\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)


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 (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 the pressure and the compressibility equation if this superposition approximation is used to generate g(r). 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[edit]

  1. John G. Kirkwood, "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics 3 pp. 300-313 (1935)
  2. M. Born and H. S. Green "A General Kinetic Theory of Liquids. I. The Molecular Distribution Functions" Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 188 pp. 10-18 (1946)
  3. Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics 23 pp. 1003-1027 (1960)
  4. 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)
  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)
  6. 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)
Related reading