Comment on: Discontinuous codimension-two bifurcation in a Vlasov equation (arXiv:2212.01250)

TL;DR

This paper critiques previous linear stability analyses of the Vlasov equation in predicting phase transitions, showing through large-scale simulations that actual transitions are discontinuous and occur at different parameters, highlighting the limitations of bifurcation analysis.

Contribution

The paper demonstrates that bifurcation analysis alone cannot predict the true nature or location of phase transitions in long-range systems, emphasizing the need for extensive simulations.

Findings

Bifurcation analysis does not predict the actual phase transition point.

The true transition is discontinuous and occurs at higher coupling strengths.

System exhibits coexistence of states near the transition.

Abstract

We comment on the recent work by Yamaguchi and Barr\'e [Phys. Rev. E 107, 054203 (2023)], which uses linear stability analysis of the Vlasov equation to characterize phase transitions in a generalized Hamiltonian Mean Field (gHMF) model. By performing extensive molecular dynamics simulations with $N=10^8$ particles, we demonstrate that the bifurcation analysis of the initial stationary distribution is insufficient to predict either the location or the nature of the phase transition to a quasi-stationary state (qSS). Specifically, we show that for bimodal momentum distributions, the instability threshold identified by the authors does not correspond to a ferromagnetic transition; instead, the system remains in a paramagnetic state characterized by magnetization oscillations with a zero time-average. We find that the true paramagnetic-ferromagnetic transition is discontinuous (first-order) and occurs at significantly larger coupling strengths, characterized by a clear coexistence of states. These results indicate that linear bifurcation and symmetry-breaking phase transitions are distinct phenomena in long-range interacting systems, and that the former lacks the predictive power to describe the long-time fate of the system.