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Problem

$N$ 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: http://en.wikipedia.org/wiki/Chebyshev_polynomials#Roots_and_extrema

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    $\begingroup$ I can do no better than point out dx.doi.org/10.1088/0305-4470/31/3/014 and dx.doi.org/10.1098/rspa.2001.0913 . $\endgroup$
    – J. M. ain't a mathematician
    Sep 17, 2010 at 9:36
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    $\begingroup$ 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 $N<32$. $\endgroup$
    – alext87
    Sep 17, 2010 at 18:45
  • $\begingroup$ Can't you do this with a (kinetic) Monte Carlo algorithm ? $\endgroup$
    – max
    Jul 18, 2011 at 21:08

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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.

http://neilsloane.com/electrons/index.html

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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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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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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