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$N$ point charges are distributed in the unit ball in $\mathbb{R}^k$, $k=2,3$. Given locations of the particles $x_1,\ldots,x_N$ the potential energy is

$E=\sum_{j=1}^{N-1}\sum_{k=j+1}^N |x_j-x_k|^{-1}$

where $|x_j-x_k|$ is Euclidean distance between $x_j$ and $x_k$. I'm interested in both the minimal value of $E$ over all possible locations of the particles in the unit ball and what this configuration looks like.

On the Unit Interval For $k=1$ the $N$ charges are distributed on the interval $[-1,1]$ according to the roots of the $N+1$th Chebshev polynomial. See:

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I can do no better than point out and . – J. M. Sep 17 '10 at 9:36
@J.M. Thank you very much! – alext87 Sep 17 '10 at 14:39
This problem was much harder than I indeed. As J.M. kindly pointed out in his comment it appears that for $k=2$ the problem has been approximately solved for $N<80$ and for $k=3$ only for $N<32$. – alext87 Sep 17 '10 at 18:45
Can't you do this with a (kinetic) Monte Carlo algorithm ? – max Jul 18 '11 at 21:08

The canonical thing to do for a question like this is to look at Neal Sloane's home page. Sure enough, there is a table giving some good arrangements.

This was indeed one of the links on the page in wok's answer, but it may be the most complete resource.

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You can look at one particular configuration at "Animated (Java) Illustrations {of} 24 Electrons on a Sphere" and a few more at "Min-Energy Configurations of Electrons On A Sphere".

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Your first link is broken; it ends with ".ht" where it should be ".htm". – Rahul Sep 24 '10 at 22:07
@Rahul : Thank you. fixed. – David Cary Sep 27 '10 at 22:03

The most comprehensive webpage I have found is about evenly distributing N points on a sphere. To be more general, this is known as the seventh of Stephen Smale's problems: the "optimal" distribution of points on the 2-sphere. It is still unsolved.

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Some months after Wok's answer, I abandoned the domain and moved that page to – Anton Sherwood Nov 3 '15 at 9:25
Thanks. I updated the link. Great webpage! – Wok Nov 5 '15 at 16:19

This problem seems related to evenly distributing points across the surface of a sphere (specifically, that is very similar to your k=2 case). That problem is addressed in a programming competition challenge named PSPHERE at SPOJ. The solutions there are not public, but perhaps approaching the leading contestants on that particular challenge could prove helpful.

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