omputed via N-body simulation (the solid line is a Gaussian fit). Panel (b): Entropy $S$ at the stationary points, as function of energy, $e$: magnetized solution (solid line) and non–magnetized one (dashed line).[]{data-label="fig1"}](fig1.eps){width="7cm"}

![Phase space snapshots for $(M_{0},e)=(0.5,0.69)$.[]{data-label="fig2"}](fig2.ps){width="7cm"}

Qualitatively, one can track the evolution of the system in phase space, both for the homogeneous and non homogeneous cases. Results of the Vlasov integration are displayed in figure \[fig2\] for $(M_{0},e)=(0.5,0.69)$, where the system is shown to evolve towards a non magnetized QSS. The initial water-bag distribution splits into two large resonances, which persist asymptotically: the latter acquire constant opposite velocities which are maintained during the subsequent time evolution, in agreement with the findings of [@antoniazziPRL]. The two bumps are therefore an emergent property of the model, which is correctly reproduced by the Vlasov dynamics. For larger values of the initial magnetization ($M_{0}>0.89$), while keeping $e=0.69$, the system evolves towards an asymptotic magnetized state, in agreement with the theory. In this case several resonances are rapidly developed which eventually coaelesce giving rise to complex patterns in phase space. More quantitatively, one can compare the velocity distributions resulting from, respectively, Vlasov and N-body simulations. The curves are diplayed in figure \[fig3\] (a),(b),(c) for various choices of the initial conditions in the magnetized region. The agreement is excellent, thus reinforcing our former conclusion about the validity of the Vlasov model. Finally, let us stress that, when $e=0.69$, the two solutions (resp. magnetized and non magnetized) [@antoniazziPRL] are associated to a practically indistinguishible entropy level (see figure \[fig3\] (d)). As previously discussed, the system explores an almost flat entropy landscape and can be therefore be stuck in local traps, because of finite size effects. 