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

Abstract

We comment on the recent work by Yamaguchi and Barré [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.

Figures (4)

• Figure 1: Momentum distribution functions for the unimodal, flat-top, and bimodal cases, where $f_{\alpha}(p) = \int F_{\alpha}(\theta, p) \, d\theta$. The distribution differ in terms of their concavity at the origin.
• Figure 2: Magnetization before and after the bifurcation point $(K_c=0.95)$ identified by YB for the bimodal distribution with $\alpha = 0.0311643$. In both instances, the time-averaged magnetization is zero -- indicating that the system remains in a paramagnetic state above the instability threshold.
• Figure 3: For $K = 1.070$, corresponding to the bimodal distribution with $\alpha = 0.0311643$, we observe a coexistence of two distinct qSS using particles with exactly the same initial momentum and differing only by a random number exchange in the homogeneous angular distribution. In all cases, the initial magnetization was $|m_0| < 10^{-6}$, and the particles evolve using a fourth-order symplectic integrator Yoshida1990 with a time step of $dt = 0.02$. The energy conservation in the simulation is maintained with a precision accurate to the seventh decimal place.
• Figure 4: Schematic representation of the system's behavior as a function of the parameter $K$ for the bimodal case ($\alpha > 0$). Below the threshold $K_c$, the magnetization exhibits microscopic fluctuations around zero. Beyond the threshold, the system develops oscillations around $\overline{m}=0$. For larger $K$ we reach a coexistence region, in which initial conditions drawn from the same distribution evolve either to a paramagnetic or a ferromagnetic state. For still larger $K$ all the initial conditions evolve to a ferromagnetic state. Notably, as $\alpha \to 0$, the region characterized by the paramagnetic state with oscillating magnetization appears to shrink, causing the lower boundary of the coexistence region to move towards $K_c$. In either cases, we conclude that the system undergoes a discontinuous order-disorder phase transition.