Harmonically Confined Particles with Long-Range Repulsive Interactions

We study an interacting system of N classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repel each other via pairwise interaction potential that behaves as a power law proportional to Sigma i not equal j(N) vertical bar x(i) - x(j)vertical bar(-k) (with k > -2) of their mutual distance. This is a generalization of the well-known cases of the one-component plasma (k = -1), Dyson's log gas (k -> 0(+)), and the Calogero-Moser model (k = 2). Because of the competition between harmonic confinement and pairwise repulsion, the particles spread over a finite region of space for all k > -2. We compute exactly the average density profile for large N for all k > -2 and show that while it is independent of temperature for sufficiently low temperature, it has a rich and nontrivial dependence on k with distinct behavior for -2 < k < 1, k > 1 and k = 1.