On the microcanonical solution of a system of globally coupled rotators

TL;DR

This paper analyzes the microcanonical ensemble of the Hamiltonian Mean Field model, deriving entropy from canonical free energy, and investigates metastable states and their relaxation dynamics near the phase transition.

Contribution

It provides a microcanonical solution for the HMF model, connecting canonical and microcanonical descriptions, and studies the relaxation of metastable states.

Findings

Metastable homogeneous states exist below the critical energy.

Water bag states relax to equilibrium with a time diverging linearly with system size.

The temperature-energy relation matches the canonical ensemble results.

Abstract

We study the Hamiltonian Mean Field (HMF) model, a system of $N$ fully coupled particles, in the microcanonical ensemble. We use the previously obtained free energy in the canonical ensemble to derive entropy as a function of energy, using Legendre transform techniques. The temperature-energy relation is found to coincide with the one obtained in the canonical ensemble and includes a {\it metastable} branch which represents spatially homogeneous states below the critical energy. "Water bag" states, with removed tails momentum distribution, lying on this branch, are shown to relax to equilibrium on a time which diverges linearly with $N$ in an energy region just below the phase transition.