Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
5  
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 . –  J. M. Sep 17 '10 at 9:36
1  
@J.M. Thank you very much! –  alext87 Sep 17 '10 at 14:39
1  
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

4 Answers 4

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.

share|improve this answer

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".

share|improve this answer
1  
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.

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.