Consider $n$ particles in a unit interval. The potential energy between two particles at $x_i$ and $x_j$ is $f(|x_i-x_j|)$. $f$ is a positive-valued, monotonically decreasing convex function. There is hard-wall repulsion between particles and walls at $x=0$ and $x=1$. How are the particles distributed so that the total potential energy is minimized?
From numerical simulation I found that for the same $f$, the particles are distributed on the same curve regardless of the total number.
For $f(r)=r^{-1}$, the positions of particles can be found through Chebyshev polynomials.
Are there more general results on this?
One generalization is this: http://emis.ams.org/journals/ETNA/vol.26.2007/pp439-452.dir/pp439-452.pdf
Which generalized to the case where apart from free charges on the [-1,1] interval, there are fixed charges around it in the complex plane. The equilibrium positions correspond the points for near-best rational interpolation with poles at the locations of fixed charges. When these charges recede to infinity, you get the zeros of Chebyshev polynomials, which are optimal for polynomial interpolation.