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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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    I can do no better than point out http://dx.doi.org/10.1088/0305-4470/31/3/014 and http://dx.doi.org/10.1098/rspa.2001.0913 .2010-09-17
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    @J.M. Thank you very much!2010-09-17
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    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 $N<32$.2010-09-17
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    Can't you do this with a (kinetic) Monte Carlo algorithm ?2011-07-18

4 Answers 4