Out-of-Equilibrium Phase Transitions and Time-Asymptotic One-Particle Dynamics in the Vlasov Limit of The Hamiltonian Mean Field Model

When starting from specific initial conditions, the ferromagnetic-like XY Hamiltonian mean field (HMF) model evolves toward quasistationary states, with lifetimes diverging with the number N of degrees of freedom that violate equilibrium statistical mechanics predictions. Phase transitions have been reported between low-energy magnetized quasistationary states and large energy unexpected, antiferromagnetic-like ones with low, but nonvanishing, magnetization. This issue is addressed here in the Vlasov N→∞ limit. It is argued that the time asymptotic states emerging in the Vlasov limit can be related to simple generic time asymptotic forms for the force field. The proposed picture unveils the nature of the out-of-equilibrium phase transitions reported for the ferromagnetic HMF in the second order regime: This is a bifurcation point connecting an effective integrable Vlasov one-particle time-asymptotic dynamic to a partly ergodic one, which means an abrupt open-up of the Vlasov one-particle phase space. This is proposed as a mechanism for second-order phase transitions compatible with nonvanishing time-asymptotic values of the order parameter in mean-field long-range systems.