Ensemble Equivalence for Counterion Condensation on a Two Dimensional Charged Disc

TL;DR

This paper investigates counterion condensation on a 2D charged disc, demonstrating ensemble equivalence in energy-temperature relations despite long-range interactions, through analysis of canonical and microcanonical ensembles.

Contribution

It shows that for a 2D charged disc, the counterion condensation transition points are identical in both ensembles, even with long-range interactions.

Findings

Microcanonical entropy is piecewise linear in energy.

Entropy remains concave for all energies.

Energy-temperature relations are identical in both ensembles.

Abstract

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.