Editing Hard sphere: virial coefficients

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>
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>[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>
<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>.
The calculation of <math>B_5</math> had to wait for the Rosenbluths
<ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881- (1954)]</ref> in 1954. Thus far no analytical expressions for <math>B_5</math> and beyond have been derived. One has:

:<math>\frac{B_2}{V(\mathbb{R}^3)}=4</math>

:<math>\frac{B_3}{V(\mathbb{R}^3)^2}=10</math>

:<math>\frac{B_4}{V(\mathbb{R}^3)^3}= \frac{2707\pi+[438\sqrt{2}-4131 \arccos(1/3)]}{70\pi}= 18.3647684</math>

where <math>V(\mathbb{R}^3)</math> is the volume of a sphere in three dimensions. For [[hard disks]] (ie. 2-dimensional hard spheres) one has<ref>[http://dx.doi.org/10.1103/PhysRevE.71.021105 Stanislav Labík, Jirí Kolafa, and Anatol Malijevský, "Virial coefficients of hard spheres and hard disks up to the ninth", Physical  Review E '''71''' pp. 021105 (2005)]</ref>

:<math>\frac{B_2}{V(\mathbb{R}^2)}=2</math>

:<math>\frac{B_3}{V(\mathbb{R}^2)^2}=\frac{16}{3}- \frac{4 \sqrt{3}}{\pi}</math>

:<math>\frac{B_4}{V(\mathbb{R}^2)^3}= 16-\frac{36\sqrt{3}}{\pi}+\frac{80}{\pi^2}</math>

where <math>V(\mathbb{R}^2)</math> is the area of a circle.
{| style="width:100%; height:250px; text-align:center" border="1"
|- 
| Virial / Dimension || 2 || 3 || 4 || 5 || 6 || 7 || 8
|- 
| <math>B_3/B_2^2</math> || 0.782004...  || 0.625        || 0.506340...   || 0.414063... || 0.340941... || 0.282227... || 0.234614...
|-
| <math>B_4/B_2^3</math> || 0.53223180...|| 0.2869495... || 0.15184606... || 0.0759724807... || 0.03336314... || 0.00986494662... || -0.00255768...
|-
| <math>B_5/B_2^4</math> || 0.33355604(1) || 0.110252(1) ||  0.0357041(17)|| 0.0129551(13) || 0.0075231(11) || 0.0070724(10) || 0.00743092(93)
|-
| <math>B_6/B_2^5</math> || 0.1988425(42)|| 0.03888198(91)|| 0.0077359(16) || 0.0009815(14) ||  -0.0017385(13)|| -0.0035121(11) || -0.0045164(11)
|-
| <math>B_7/B_2^6</math> || 0.1148728(43)||0.01302354(91) || 0.0014303(19) ||  0.0004162(19)||  0.0013066(18)|| 0.0025386(16) || 0.0034149(15)
|-
| <math>B_8/B_2^7</math> || 0.0649930(34)|| 0.0041832(11)|| 0.0002888(18) || -0.0001120(20) || -0.0008950(30) ||  -0.0019937(28)|| -0.0028624(26)
|-
| <math>B_9/B_2^8</math> ||0.0362193(35) || 0.0013094(13)|| 0.0000441(22) || 0.0000747(26) || 0.0006673(45) ||  0.0016869(41)|| 0.0025969(38)
|-
| <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)|| ||  ||  ||  || 
|}
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==
*[[Equations of state for hard disks]]
*[[Equations of state for hard spheres]]
== References ==
<references/>
'''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)]
*[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/10.1063/1.2951456 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/10.1063/1.2958914 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/10.1080/00268976.2014.904945 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)]



[[category:virial coefficients]]
[[category: hard sphere]]
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