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 ∝∑[under i≠j][over N]|x_{i}-x_{j}|^{-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 -21 and k=1.