Hard sphere: virial coefficients - Revision history

ManfredS: Added the given Name of G. Jäger

2025-06-07T05:48:00Z

Added the given Name of G. Jäger

← Older revision		Revision as of 07:48, 7 June 2025
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	The [[virial equation of state]] of the [[hard sphere model]], in various dimensions, has long been of interest.		The [[virial equation of state]] of the [[hard sphere model]], in various dimensions, has long been of interest.
	In 3-dimensions analytical results were derived for <math>B_2</math> by [[Johannes Diderik van der Waals]]<ref>[http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publDetail&pId=PU00014537 J. D. van der Waals "Simple deduction of the characteristic equation for substances with extended and composite molecules", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 138-143 (1899)]</ref>, <math>B_3</math> by Jäger <ref>G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 15- (1896)</ref>		In 3-dimensions analytical results were derived for <math>B_2</math> by [[Johannes Diderik van der Waals]]<ref>[http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publDetail&pId=PU00014537 J. D. van der Waals "Simple deduction of the characteristic equation for substances with extended and composite molecules", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 138-143 (1899)]</ref>, <math>B_3</math> by Gustav Jäger <ref>G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 15- (1896)</ref>
	and [[Ludwig Eduard Boltzmann]] <ref>L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 695- (1896)</ref>		and [[Ludwig Eduard Boltzmann]] <ref>L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 695- (1896)</ref>
	<ref>L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. '''7''' pp. 484- (1899)</ref>, and <math>B_4</math> by [[Johannis Jacobus van Laar]]		<ref>L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. '''7''' pp. 484- (1899)</ref>, and <math>B_4</math> by [[Johannis Jacobus van Laar]]

Iurrutia at 17:14, 22 January 2024

2024-01-22T17:14:27Z

← Older revision		Revision as of 19:14, 22 January 2024
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	*[https://doi.org/10.1007/s10955-005-3020-6 I. Lyberg "The fourth virial coefficient of a fluid of hard spheres in odd dimensions", Journal of Statistical Physics '''119''' pp. 747-764 (2005)]		*[https://doi.org/10.1007/s10955-005-3020-6 I. Lyberg "The fourth virial coefficient of a fluid of hard spheres in odd dimensions", Journal of Statistical Physics '''119''' pp. 747-764 (2005)]
	*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]		*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]
	*[https://doi.org/10.1007/s10955-022-02913-7 I. Urrutia "The ~~Fourth Virial Coefficient~~ for ~~Hard Spheres~~ in ~~Even Dimension~~", Journal of Statistical Physics '''187''' pp. 29-50 (2022)]		*[https://doi.org/10.1007/s10955-022-02913-7 I. Urrutia "The fourth virial coefficient for hard spheres in even dimension", Journal of Statistical Physics '''187''' pp. 29-50 (2022)]
	*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]		*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]
	*[http://dx.doi.org/10.1063/1.2951456 René D. Rohrmann, Miguel Robles, Mariano López de Haro, and Andrés Santos "Virial series for fluids of hard hyperspheres in odd dimensions", Journal of Chemical Physics '''129''' 014510 (2008)]		*[http://dx.doi.org/10.1063/1.2951456 René D. Rohrmann, Miguel Robles, Mariano López de Haro, and Andrés Santos "Virial series for fluids of hard hyperspheres in odd dimensions", Journal of Chemical Physics '''129''' 014510 (2008)]

Iurrutia: I added a new reference in Related Reading.

2024-01-22T17:09:26Z

I added a new reference in Related Reading.

← Older revision		Revision as of 19:09, 22 January 2024
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	*[https://doi.org/10.1007/s10955-005-3020-6 I. Lyberg "The fourth virial coefficient of a fluid of hard spheres in odd dimensions", Journal of Statistical Physics '''119''' pp. 747-764 (2005)]		*[https://doi.org/10.1007/s10955-005-3020-6 I. Lyberg "The fourth virial coefficient of a fluid of hard spheres in odd dimensions", Journal of Statistical Physics '''119''' pp. 747-764 (2005)]
	*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]		*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]
			*[https://doi.org/10.1007/s10955-022-02913-7 I. Urrutia "The Fourth Virial Coefficient for Hard Spheres in Even Dimension", Journal of Statistical Physics '''187''' pp. 29-50 (2022)]
	*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]		*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]
	*[http://dx.doi.org/10.1063/1.2951456 René D. Rohrmann, Miguel Robles, Mariano López de Haro, and Andrés Santos "Virial series for fluids of hard hyperspheres in odd dimensions", Journal of Chemical Physics '''129''' 014510 (2008)]		*[http://dx.doi.org/10.1063/1.2951456 René D. Rohrmann, Miguel Robles, Mariano López de Haro, and Andrés Santos "Virial series for fluids of hard hyperspheres in odd dimensions", Journal of Chemical Physics '''129''' 014510 (2008)]

Urrutia: New related reading by Lyberg

2020-12-26T16:34:55Z

New related reading by Lyberg

← Older revision		Revision as of 18:34, 26 December 2020
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	<references/>		<references/>
	'''Related reading'''		'''Related reading'''
			*[https://doi.org/10.1007/s10955-005-3020-6 I. Lyberg "The fourth virial coefficient of a fluid of hard spheres in odd dimensions", Journal of Statistical Physics '''119''' pp. 747-764 (2005)]
	*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]		*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]
	*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]		*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]

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

2014-05-20T12:14:10Z

References: Added a recent publication

← Older revision		Revision as of 14:14, 20 May 2014
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	*[http://dx.doi.org/10.1063/1.2958914 André O. Guerrero and Adalberto B. M. S. Bassi "On Padé approximants to virial series", Journal of Chemical Physics '''129''' 044509 (2008)]		*[http://dx.doi.org/10.1063/1.2958914 André O. Guerrero and Adalberto B. M. S. Bassi "On Padé approximants to virial series", Journal of Chemical Physics '''129''' 044509 (2008)]
	*[http://dx.doi.org/10.1063/1.3558779 Miguel Ángel G. Maestre, Andrés Santos, Miguel Robles, and Mariano López de Haro "On the relation between virial coefficients and the close-packing of hard disks and hard spheres", Journal of Chemical Physics '''134''' 084502 (2011)]		*[http://dx.doi.org/10.1063/1.3558779 Miguel Ángel G. Maestre, Andrés Santos, Miguel Robles, and Mariano López de Haro "On the relation between virial coefficients and the close-packing of hard disks and hard spheres", Journal of Chemical Physics '''134''' 084502 (2011)]
			*[http://dx.doi.org/10.1080/00268976.2014.904945 Cheng Zhang and B. Montgomery Pettitt "Computation of high-order virial coefficients in high-dimensional hard-sphere fluids by Mayer sampling", Molecular Physics '''112''' pp. 1427-1447 (2014)]

85.50.104.134: fix some broken reference URLs

2014-02-01T22:00:48Z

fix some broken reference URLs

← Older revision		Revision as of 00:00, 2 February 2014
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	The [[virial equation of state]] of the [[hard sphere model]], in various dimensions, has long been of interest.		The [[virial equation of state]] of the [[hard sphere model]], in various dimensions, has long been of interest.
	In 3-dimensions analytical results were derived for <math>B_2</math> by [[Johannes Diderik van der Waals]]<ref>[http://www.~~digitallibrary~~.nl/~~proceedings~~/~~search~~/~~detail.cfm~~?~~pubid=220&view~~=~~image~~&~~startrow~~=1 J. D. van der Waals "Simple deduction of the characteristic equation for substances with extended and composite molecules", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 138-143 (1899)]</ref>, <math>B_3</math> by Jäger <ref>G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 15- (1896)</ref>		In 3-dimensions analytical results were derived for <math>B_2</math> by [[Johannes Diderik van der Waals]]<ref>[http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publDetail&pId=PU00014537 J. D. van der Waals "Simple deduction of the characteristic equation for substances with extended and composite molecules", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 138-143 (1899)]</ref>, <math>B_3</math> by Jäger <ref>G. Jäger "", Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 15- (1896)</ref>
	and [[Ludwig Eduard Boltzmann]] <ref>L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 695- (1896)</ref>		and [[Ludwig Eduard Boltzmann]] <ref>L. Boltzmann "",Sitzber. Akad. Wiss. Wien. Ber. Math. Natur-w. Kl. (Part 2a) '''105''' pp. 695- (1896)</ref>
	<ref>L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. '''7''' pp. 484- (1899)</ref>, and <math>B_4</math> by [[Johannis Jacobus van Laar]]		<ref>L. Boltzmann "On the characteristic equation of v.d.Waals", Versl. Gewone Vergad. Afd. Natuurkd., K. Ned. Akad. Wet. '''7''' pp. 484- (1899)</ref>, and <math>B_4</math> by [[Johannis Jacobus van Laar]]
	<ref>[http://www.~~digitallibrary~~.nl/~~proceedings~~/~~search~~/~~detail.cfm~~?~~pubid~~=~~203~~&~~view~~=~~image~~&~~startrow~~=1 J. J. Van Laar "Calculation of the second correction to the quantity b of the equation of condition of Van der Waals", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 273-287 (1899)]</ref>		<ref>[http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?page_id=&pagetype=publDetail&pId=PU00014563 J. J. Van Laar "Calculation of the second correction to the quantity b of the equation of condition of Van der Waals", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''1''' pp. 273-287 (1899)]</ref>
	as well as Boltzmann <ref>[http://dx.doi.org/10.1119/1.1986605 John H. Nairn and John E. Kilpatrick "van der Waals, Boltzmann, and the Fourth Virial Coefficient of Hard Spheres", American Journal of Physics '''40''' pp. 503-515 (1972)]</ref>		as well as Boltzmann <ref>[http://dx.doi.org/10.1119/1.1986605 John H. Nairn and John E. Kilpatrick "van der Waals, Boltzmann, and the Fourth Virial Coefficient of Hard Spheres", American Journal of Physics '''40''' pp. 503-515 (1972)]</ref>
	<ref>[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)]</ref>.		<ref>[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)]</ref>.

Aidan t: /* References */

2013-10-08T14:38:49Z

References

← Older revision		Revision as of 16:38, 8 October 2013
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	<references/>		<references/>
	'''Related reading'''		'''Related reading'''
	*[http://link.aps.org/doi/10.1103/PhysRevLett.110.200601 Richard J. Wheatley "Calculation of High-Order Virial Coefficients with Applications to Hard and Soft Spheres", Physical review Letters, '''110''' 200601 (2013)]]
	*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]		*[http://dx.doi.org/10.1023/B:JOSS.0000013959.30878.d2 N. Clisby and B.M. McCoy "Analytic Calculation of B4 for Hard Spheres in Even Dimensions", Journal of Statistical Physics '''114''' pp. 1343-1361 (2004)]
	*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]		*[http://dx.doi.org/10.1063/1.2821962 Marvin Bishop, Nathan Clisby and Paula A. Whitlock "The equation of state of hard hyperspheres in nine dimensions for low to moderate densities", Journal of Chemical Physics '''128''' 034506 (2008)]

198.102.153.1 at 23:18, 7 October 2013

2013-10-07T23:18:32Z

← Older revision		Revision as of 01:18, 8 October 2013
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	\| <math>B_{12}/B_2^{11}</math> \|\| \|\| 0.000027 (7)\|\| \|\| \|\| \|\| \|\|		\| <math>B_{12}/B_2^{11}</math> \|\| \|\| 0.000027 (7)\|\| \|\| \|\| \|\| \|\|
	\|}		\|}
	This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref>		This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref>. The values of <math>B_{11}</math> and <math>B_{12}</math> for three dimensional hard spheres are taken from <ref>[http://link.aps.org/doi/10.1103/PhysRevLett.110.200601 Richard J. Wheatley "Calculation of High-Order Virial Coefficients with Applications to Hard and Soft Spheres", Physical review Letters, '''110''' 200601 (2013)]]</ref>.
	==See also==		==See also==
	*[[Equations of state for hard disks]]		*[[Equations of state for hard disks]]

Aidan t at 21:57, 7 October 2013

2013-10-07T21:57:09Z

← Older revision		Revision as of 23:57, 7 October 2013
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	\|-		\|-
	\| <math>B_{11}/B_2^{10}</math> \|\| \|\| 0.000122 (4)\|\| \|\| \|\| \|\| \|\|		\| <math>B_{11}/B_2^{10}</math> \|\| \|\| 0.000122 (4)\|\| \|\| \|\| \|\| \|\|
			\|-
			\| <math>B_{12}/B_2^{11}</math> \|\| \|\| 0.000027 (7)\|\| \|\| \|\| \|\| \|\|
	\|}		\|}
	This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref>		This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref>

Aidan t at 21:56, 7 October 2013

2013-10-07T21:56:11Z

← Older revision		Revision as of 23:56, 7 October 2013
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	\| <math>B_{10}/B_2^9</math> \|\| 0.0199537(80)\|\| 0.0004035(15)\|\| 0.0000113(31)\|\| -0.0000492(48) \|\| -0.000525(16) \|\| -0.001514(14) \|\| -0.002511(13)		\| <math>B_{10}/B_2^9</math> \|\| 0.0199537(80)\|\| 0.0004035(15)\|\| 0.0000113(31)\|\| -0.0000492(48) \|\| -0.000525(16) \|\| -0.001514(14) \|\| -0.002511(13)
	\|-		\|-
	\| <math>B_{11}/B_2^10</math> \|\| \|\| 0.000122 (4)\|\| \|\| \|\| \|\| \|\|		\| <math>B_{11}/B_2^{10}</math> \|\| \|\| 0.000122 (4)\|\| \|\| \|\| \|\| \|\|
	\|}		\|}
	This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref>		This table is taken directly from Table 1 in Ref.<ref>[http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)]</ref>