Out-of-equilibrium phase transitions in the HMF model: a closer look

TL;DR

This paper analyzes out-of-equilibrium phase transitions in the Hamiltonian Mean Field model using Lynden-Bell's theory, revealing complex transition behaviors, phase reentrance, and ensemble inequivalence depending on initial conditions.

Contribution

It provides a detailed characterization of phase transition types and their dependence on initial conditions in the HMF model, including the discovery of new reentrant phenomena and ensemble inequivalence.

Findings

Different regions exhibit second or first order phase transitions.

Identification of a tricritical point where transition order changes.

Discovery of phase reentrance and ensemble inequivalence phenomena.

Abstract

We provide a detailed discussion of out-of-equilibrium phase transitions in the Hamiltonian Mean Field (HMF) model in the framework of Lynden-Bell's statistical theory of the Vlasov equation. For two-levels initial conditions, the caloric curve $\beta(E)$ only depends on the initial value $f_0$ of the distribution function. We evidence different regions in the parameter space where the nature of phase transitions between magnetized and non-magnetized states changes: (i) for $f_0>0.10965$, the system displays a second order phase transition; (ii) for $0.109497<f_0<0.10965$, the system displays a second order phase transition and a first order phase transition; (iii) for $0.10947<f_0<0.109497$, the system displays two second order phase transitions; (iv) for $f_0<0.10947$, there is no phase transition. The passage from a first order to a second order phase transition corresponds to a tricritical point. The sudden appearance of two second order phase transitions from nothing corresponds to a second order azeotropy. This is associated with a phenomenon of phase reentrance. When metastable states are taken into account, the problem becomes even richer. In particular, we find a new situation of phase reentrance. We consider both microcanonical and canonical ensembles and report the existence of a tiny region of ensembles inequivalence. We also explain why the use of the initial magnetization $M_0$ as an external parameter, instead of the phase level $f_0$, may lead to inconsistencies in the thermodynamical analysis.