The Vlasov-HMF equation: simulations

The Vlasov-HMF equation

The full Vlasov-HMF equation reads

tF+pqFqϕ[F]pF=0\begin{aligned} \qquad\partial_t F+p\partial_q F-\partial_q \phi[F]\partial_p F=0 \end{aligned} ϕ[F](q)=ππVHMF(qq)F(q,p,t)dqdp=VHMFqFdp\begin{aligned} \phi[F](q)=\int_{-\infty}^\infty\int_{-\pi}^\pi V_{\rm{HMF}}(q-q')F(q',p',t)\,\mathrm{d}q'\mathrm{d}p'=V_{\rm{HMF}}\star_q \int_{-\infty}^\infty F\,\mathrm{d}p \end{aligned} ππFdqdp=1\begin{aligned} \int_{-\infty}^\infty\int_{-\pi}^\pi F \,\mathrm{d}q\mathrm{d}p =1 \end{aligned}

where q\star_q is the convolution in space (this formulation makes easy the Fourier transform). From now on we will omit the integration bounds. For the HMF potential the mean field potential ϕ[F]\phi[F] becomes

ϕ[F](q)=Mc[F](t)cosqMs[F](t)sinq=Mcos(qφM)\begin{aligned} \phi[F](q)&=-M_c[F](t)\cos q-M_s[F](t)\sin q=-\|M\|\cos(q-\varphi_M) \end{aligned} M[F]=Mc[F]+iMs[F]=Feiqdqdp,\begin{aligned} M[F]&=M_c[F]+iM_s[F]=\iint Fe^{i q}\,\mathrm{d}q\mathrm{d}p, \end{aligned}

where M[F](t)M[F](t) is the order parameter (magnetization). In all simulation shown below by symmetry, it will always be real. This equation describes the time evolution of a density of articles FF in the phase space of position qq (or xx or θ\theta depending on the context) and velocity pp (or vv). The colors represent the magnitude of F(q,p,t)F(q,p,t) (or F(x,v,t)F(x,v,t)). I show the evolution of the density F(q,p,t)=F0(q,p)+P(q,p,t)F(q,p,t) = F^0(q,p) + P(q,p,t) under a small initial perturbation P(q,p)P(q,p) when the initial distribution is a stationary solution F0(q,p)F^0(q,p) of the Vlasov equation. I show different cases, whether the initial distribution is homogeneous in space or not, stable, or weakly unstable. The homogeneous case is already well documented theoretically and numerically. My PhD work consisted in studying (both theoretically and numerically) the case when the initial distribution is non-homogeneous F0(q,p)F^0(q,p) (depends on qq).

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 most recent Phys. Rev. E 102, 052208. For an longer description 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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I want to illustrate the fundamental differences between the dynamics around homogeneous steady states F0(p)F^0(p) with associate magnetization M0=0M_0=0 and the non-homogeneous states F0(q,p)F^0(q,p) with associate magnetization M00M_0\neq 0. In the first case we get a "singular supercritical" bifurcation, while in the latter we get a "transcritical" bifurcation. The quotes mean that the situation is more subtle than that.

Homogeneous situation | Non Homogeneous situation

Simulations movies

Homogeneous steady states F0(p)F^0(p)

Stable homogeneous steady state of the Vlasov-HMF equation. It is perturbed with a small perturbation. We observe Landau damping i.e., phase space mixing. The perturbation looks as if it is damped but it actually disappear into the velocity high Fourier modes.

Unstable homogeneous steady state of the Vlasov-HMF equation. It is perturbed with a small perturbation. This time the perturbation grows and we observe the \"trapping phenomenon\" with a characteristic cat-eye pattern. There are strong resonances between the particles with a velocity close to the frequency of the perturbation, here close to p=0p=0 that leads to the trapping.

Non-Homogeneous steady states F0(q,p)F^0(q,p)

Unstable inhomogeneous steady state F0(q,p)F^0(q,p) of the Vlasov-HMF equation.

  • Small "positive" perturbation P(q,p)P(q,p) perturbation, meaning that the associate magnetization is positive. The perturbation stops is quickly saturated by the nonlinear terms at O(λ2)O(\lambda^2). We observe small nonlinear oscillations.

  • To highlight this perturbative dynamic, we remove subtract the reference state and show FF0F-F^{0}.

  • Small "negative" perturbation P(q,p)P(q,p) perturbation (sponge scenario), meaning that the associate magnetization is negative. The perturbation grows outside the perturbative regime. Outside the perturbative regime, we observe a limit cycle type behavior.

  • We show the same simulations for another initial non-homogeneous state F0F^0.

CC BY-SA 4.0 David Métivier. Last modified: June 13, 2024. Website built with Franklin.jl and the Julia programming language.