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.
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Video: Fermi_eps_-.webm, Fermi_eps_-.mp4
Video: Fermi_eps_+.webm, Fermi_eps_+.mp4
Video: Fermi_eps_+_diff.webm, Fermi_eps_+_diff.mp4
Video: Torr_G_eps_-.webm, Torr_G_eps_-.mp4
Video: Landau_Damping.webm, Landau_Damping.mp4
Video: Cat_eye.webm, Cat_eye.mp4