# Distributions of point charges

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

• 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 . Sep 17, 2010 at 9:36
• 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 $N<32$. Sep 17, 2010 at 18:45
• Can't you do this with a (kinetic) Monte Carlo algorithm ?
– max
Jul 18, 2011 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.

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.

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".

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.

• Some months after Wok's answer, I abandoned the domain ogre.nu and moved that page to bendwavy.org/sphere.htm Nov 3, 2015 at 9:25

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.