Vlasov-HMF videos

Here are some simulations of the full Vlasov-HMF equation. This equation describes the time evolution of a density function $f$ in the phase space of position $x$ (or $q$ depending of the notation used at that time) and velocity $v$ (or $p$). The colors represent the magnitude of $f(x,v,t)$ (or $f(q,p,t)$). I show the evolution of the density $f(x,v,t)$ under a small initial perturbation $\delta f(x,v)$ when the initial distribution is a stationnary solution $f^0(x,v)$ of the Vlasov equation. I show different cases, whether the initial distribution are homogeneous in space or not, stable, or weakly unstable. The homogeneous case is understood theoretically and numerically since the 90s. My PhD work consisted in studying (both theoretically and numerically) the case when the initial distribution is non homegenous $f^0(x,v)$ (depends on $x$).

The simulations were done thanks to a GPU Vlasov-HMF solver provided by T. M. Rocha Filho Comput. Phys. Commun. 184, 34 (2013). More details to come on the context of these simulations. In the meantime you can refer to Physical Review E, 93(4), 042207 or the more recent arXiv:1909.11344 . For an even more complete descriptions of the problems (theory and simulation details), please refer to the Part Two, Chapter V and VI, of PhD thesis. These two Chapters correspond respectively to the homogeneous and non-homogeneous Vlasov equation cases.

If the videos do not work directly on your web browser, try to click on the links below or download the video or contact me.


Unstable inhomogeneous steady state of the Vlasov-HMF equation. It is perturbated with a small "negative" perturbation. The perturbation is not saturated close by: subcritical bifurcation.

Video: Fermi_eps_-.webm, Fermi_eps_-.mp4



Unstable inhomogeneous steady state of the Vlasov-HMF equation. It is perturbated with a small "positive" perturbation. We observe a saturation close by: supercritical bifurcation (which means we do not see much happening on the movie).

Video: Fermi_eps_+.webm, Fermi_eps_+.mp4



Same video but with the intial state subtracted to highlight the perturbative dynamics.

Video: Fermi_eps_+_diff.webm, Fermi_eps_+_diff.mp4



Unstable inhomogeneous steady state of the Vlasov-HMF equation. This time the initial condition is no more a Fermi distribution but a non monotonic distribution of energy. It is perturbated with a small "negative" perturbation. The perturbation is not saturated close by: subcritical bifurcation.

Video: Torr_G_eps_-.webm, Torr_G_eps_-.mp4



Stable homogeneous steady state of the Vlasov-HMF equation. It is perturbated with a small perturbation. We observe Landau damping (mixing in the velocity variable).

Video: Landau_Damping.webm, Landau_Damping.mp4



Unstable homogeneous steady state of the Vlasov-HMF equation. It is perturbated with a small perturbation. This time the perturbation grows and we observe the "trapping phenomenon" (with a characteristic cat-eye pattern).

Video: Cat_eye.webm, Cat_eye.mp4