# Distributing n Charges on a Sphere

## Description:

A classic question in physics is the known as Thomson Problem, consisting in distributing a number n of equal charges on a sphere. Charges repel each other, and the equilibrium state is attained when for each particle the total sum of repulsion forces is null. To find this equilibrium state is equivalent to minimize the potential energy function

where xi is the position of the electron i.

The difficult of the problem consist in the complexity of the potential function. If for each charge we have three variables, the function f(N) has 3N variables. In addition to the complexity of the problem, this function presents local minimal that grows exponentially with N [MD96]

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## Instances and best known solutions for those instances:

In order to define an instance of this function we need to provide the number N of charges. In the web [HSS] we can find the potential for some numbers of charges (max 282), also approximations to the potential function [EH91][LL99][MD96], limits and bounds [KS]. We found the best approximation to the potential function in [MD96]

with a=1.10461 and b=0.137.

In the table we have the potential fon N charges (Copyright R. H. Hardin, N. J. A. Sloane & W. D. Smith, Feb 1994). Left click on the number of charges to get the coordinates of the putatively (Copyright R. H. Hardin, N. J. A. Sloane & W. D. Smith, Feb 1994).

Charges Potential Charges Potential Charges Potential Charges Potential
43.674234638593.0385036712190.6499061044822.876523
56.474691539626.389009722255.0011911054919.000638
69.985281440660.6752788732320.6338841065015.984596
714.452977441695.9167443742387.0729821075113.953548
819.675287942732.0781075752454.3696891085212.813508
925.759986543769.1908465762522.6748721095312.73508
1032.716949544807.1742631772591.8501521105413.549294
1140.596450545846.1884011782662.0464751115515.293215
1249.165253146886.1671136792733.2483581125618.044882
1358.853230647927.0592707802805.3558761135721.824978
1469.306363348968.7134553812878.522831145826.521572
1580.6702441491011.557183822952.5696751155932.181286
1692.9116553501055.182315833027.5284891166038.815594
17106.0504048511099.81929843103.4651241176146.342447
18120.0844674521145.418964853180.3614431186254.877028
19135.0894676531191.92229863258.2116061196364.347318
20150.8815683541239.361475873337.000751206474.756325
21167.6416224551287.772721883416.7201971216586.12195
22185.2875361561337.094945893497.4390191226698.374499
23203.9301907571387.383229903579.0912231236811.827228
24223.3470741581438.618251913661.7136991246926.169974
25243.8127603591490.773335923745.2916361257041.473264
26265.1333263601543.830401933829.8443381267157.669225
27287.302615611597.94183943915.309271277274.819505
28310.4915424621652.90941954001.7716761287393.007443
29334.6344399631708.879682964089.154011297512.107319
30359.6039459641765.802578974177.53361307632.167379
31385.5308381651823.66796984266.8224641317753.205167
32412.2612747661882.441525994357.1391631327875.045343
33440.2040574671942.12271004448.35063419216963.33839
34468.9048533682002.8747021014540.59005221220768.05309
35498.5698725692064.5334831024633.73656627234515.19329
36529.1224084702127.1009021034727.83661728237147.29442
37560.6188877

## Related Papers:

[LIH04] C. Luque, P. Isasi, J.C. Hernández, "Distribución de Cargas en una Esfera mediante Estrategias Evolutivas", Revista IEEE América Latina, v2, n2 (2004).

[EH91] T. Erber, G. M. Hockney, "Equilibrium configurations of n equal charges on a sphere", J Phys A: Math, 1991.

[KS97] A. B. J. Kuijlaars, E. B. Saff, "Distributing many points on a sphere", Mathematical Intelligencer, v19 n1 (1997), pp. 5-11.

[KS] A. B. J. Kuijlaars, E. B. Saff, "Asymptotics for minimal discrete energy on the sphere", Trans. Amer. Math. Soc., to appear.

[LL99] A. M. Livshits, Yu E. Lozovik, "Coulomb clusters on a sphere: topological classification" Chemical Physics Letters 314 (1999), pp. 577-583.

[MD96] J. R. Morris, D. M. Deaven and K. M. Ho. "Genetic-algorithm energy minimization for point charges on a sphere". Physical Review B, 53(4): pp. 1740--1743, 1996.

[Wil84] L. T. Wille, "Searching potential energy surfaces by simulated annealing", Nature v 324 n 6 (1984), pp. 46-48.

## Related Webs:

[HSS] R. H. Hardin, N. J. A. Sloane and W. D. Smith http://www.research.att.com/~njas/electrons/

[NSWZ] Neubauer, Schilling, Watkins & Zeitlin, 1998 http://www.csun.edu/~hcmth007/algorithm.html

Click here to get the bibliography in bibtex fotmat.

Last Updated: 21/11/05                                                                               For any question or suggestion, click here to contact with us.