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. 
Anoop Varghese, Satyavani Vemparala, R. Rajesh
cond-mat.stat-mech