Ensemble Equivalence for Counterion Condensation on a Two Dimensional Charged Disc

We study the counterion condensation on a two dimensional charged disc in the limit of infinite dilution, and compare the energy-temperature relation obtained from the canonical free energy and microcanonical entropy. The microcanonical entropy is piecewise linear in energy, and is shown to be concave for all energies. As a result, even though the interactions are long-ranged, the energy-temperature relation and hence the counterion condensation transition points are identical in both the ensembles.