Comment on "Discontinuous codimension-two bifurcation in a Vlasov equation"

We comment on the recent work by Yamaguchi and Barré [Phys. Rev. E 107, 054203 (2023)2470-004510.1103/PhysRevE.107.054203], which uses a linear stability analysis of the Vlasov equation to characterize phase transitions in a generalized Hamiltonian mean field 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. These results indicate that the linear stability analysis, in general, is not sufficient to predict symmetry-breaking phase transitions in long-range interacting systems.