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   

      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]

     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
4	3.6742346	38	593.0385036	71	2190.649906	104	4822.876523
5	6.4746915	39	626.389009	72	2255.001191	105	4919.000638
6	9.9852814	40	660.6752788	73	2320.633884	106	5015.984596
7	14.4529774	41	695.9167443	74	2387.072982	107	5113.953548
8	19.6752879	42	732.0781075	75	2454.369689	108	5212.813508
9	25.7599865	43	769.1908465	76	2522.674872	109	5312.73508
10	32.7169495	44	807.1742631	77	2591.850152	110	5413.549294
11	40.5964505	45	846.1884011	78	2662.046475	111	5515.293215
12	49.1652531	46	886.1671136	79	2733.248358	112	5618.044882
13	58.8532306	47	927.0592707	80	2805.355876	113	5721.824978
14	69.3063633	48	968.7134553	81	2878.52283	114	5826.521572
15	80.6702441	49	1011.557183	82	2952.569675	115	5932.181286
16	92.9116553	50	1055.182315	83	3027.528489	116	6038.815594
17	106.0504048	51	1099.81929	84	3103.465124	117	6146.342447
18	120.0844674	52	1145.418964	85	3180.361443	118	6254.877028
19	135.0894676	53	1191.92229	86	3258.211606	119	6364.347318
20	150.8815683	54	1239.361475	87	3337.00075	120	6474.756325
21	167.6416224	55	1287.772721	88	3416.720197	121	6586.12195
22	185.2875361	56	1337.094945	89	3497.439019	122	6698.374499
23	203.9301907	57	1387.383229	90	3579.091223	123	6811.827228
24	223.3470741	58	1438.618251	91	3661.713699	124	6926.169974
25	243.8127603	59	1490.773335	92	3745.291636	125	7041.473264
26	265.1333263	60	1543.830401	93	3829.844338	126	7157.669225
27	287.302615	61	1597.94183	94	3915.30927	127	7274.819505
28	310.4915424	62	1652.90941	95	4001.771676	128	7393.007443
29	334.6344399	63	1708.879682	96	4089.15401	129	7512.107319
30	359.6039459	64	1765.802578	97	4177.5336	130	7632.167379
31	385.5308381	65	1823.66796	98	4266.822464	131	7753.205167
32	412.2612747	66	1882.441525	99	4357.139163	132	7875.045343
33	440.2040574	67	1942.1227	100	4448.350634	192	16963.33839
34	468.9048533	68	2002.874702	101	4540.590052	212	20768.05309
35	498.5698725	69	2064.533483	102	4633.736566	272	34515.19329
36	529.1224084	70	2127.100902	103	4727.836617	282	37147.29442
37	560.6188877

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.