Landau damping

Survival threshold for plasma oscillations

ttn12 Kinetic theory, Plasma Physics , , , ,

In plasma physics, plasma oscillations, also known as Langmuir’s oscillatory waves, refer to the oscillatory behavior of excited electrons in a non-trivial non-equilibrium state of a plasma. In this  article, I shall describe how plasma oscillations arise via the Vlasov’s collisionless kinetic theory with long-range Coulomb’s pair interaction between charged particles, namely through the Vlasov-Poisson system. I shall then introduce the so-called survival threshold of spatial frequencies (namely, the inverse of wavelengths) that characterizes the dynamics of excited electrons near spatially homogenous steady states:

  • Phase mixing above survival threshold

  • Plasma oscillations below survival threshold

in which above / below refers to elementary waves with wavenumbers larger / smaller than the threshold. Phase mixing is a pure transport damping mechanism which yields rapid decay for the electric field, while plasma oscillations are not damped, but disperse like a Klein-Gordon dispersive wave in the whole space (i.e. the survival of oscillations below threshold, noting these oscillatory modes may occur on a large torus as well). The classical Landau damping then occurs due to resonant interaction between the two regimes at survival threshold, which we shall detail below. Eventually, the following figure captures the dynamics of the electric field whose dispersion relation, say \lambda_\pm(k), obeys (1) Klein-Gordon’s pure oscillations below survival threshold (i.e. no damping \Re \lambda_\pm(k)=0),  (2) Landau damping at the threshold (i.e. onset of damping \Re \lambda_\pm(k)<0), and (3) phase mixing above the threshold (i.e. exponential damping \Re \lambda_\pm(k)\lesssim-|k|):

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Generator functions and their applications

ttn12 fluid dynamics, Kinetic theory, New papers , , , ,

The Cauchy-Kovalevskaya theorem is a classical convenient tool to construct analytic solutions to partial differential equations, which allows one to view and treat them as if they are ordinary differential equations:

\displaystyle u_t = F(t,x,u,u_x)

for unknown functions {u(t,x)} in {t\ge 0} and {x\in \mathbb{R}^d} (or some spatial domain). Roughly, if {F(t,x,u,w)} is locally analytic near a point {(0,x_0,u_0,w_0)} then the PDE has a unique solution {u(t,x)} which is analytic near {(0,x_0)}, as established by Cauchy (1842) and generalized by Kovalevskaya as part of her dissertation (1875). The theorem has found many applications such as in fluid dynamics and kinetic theory where it is used to provide existence of analytic solutions including those that are obtained at certain asymptotic limits (e.g., inviscid limit, Boltzmann-to-fluid limit, Landau damping, inviscid damping,…). There are modern formulations of the abstract theorem: see, for instance, Asano, Baouendi and Goulaouic, Caflisch, Nirenberg, and Safonov. In this blog post, I present generator functions, as an alternative approach to the use of the Cauchy-Kovalevskaya theorem, recently introduced in my joint work with E. Grenier (ENS Lyon), and discuss the versatility and simplicity of their use to applications. In addition to providing existence of analytic solutions, I will also mention another use of generator functions to capture some physics that would be otherwise missed for analytic data (namely, an analyticity framework to capture physical phenomena that are not seen for analytic data!).

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Landau damping and extra dissipation for plasmas in the weakly collisional regime

ttn12 Kinetic theory, Plasma Physics , , , ,

Sanchit Chaturvedi (Stanford), Jonathan Luk (Stanford), and I just submitted the paper “The Vlasov–Poisson–Landau system in the weakly collisional regime”, where we prove Landau damping and extra dissipation for plasmas modeled by the physical Vlasov-Poisson-Landau system in the weakly collisional regime {\nu\ll1}, where {\nu} is the collisional parameter. The results are obtained for Sobolev data that are {\nu^{1/3}}-close to global Maxwellians on the torus {\mathbb{T}_x^3\times \mathbb{R}_v^3}. While Landau damping is a classical subject in plasma physics that predicts mixing and relaxation without dissipation of the electric field in a plasma, extra dissipation arises due to the interplay between phase mixing and entropic relaxation, or between transport and diffusion, which enhances decay to a faster rate than the usual diffusion rate. In this blog post, I give a flavor of the proof of our results, where we develop a purely energy method which combines Guo’s weighted energy method with the hypocoercive energy method and the Klainerman’s vector field method.

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Plasma echoes near stable Penrose data

ttn12 Kinetic theory, Plasma Physics , ,

Echoes in a plasma are the excitement of new waves due to nonlinear interaction. The excitement may happen at an arbitrarily large time, which is the main source of difficulties in understanding Landau damping. For analytic data, the echoes are suppressed as the electric field is exponentially localized in time, and the nonlinear Landau damping holds for such data, as was first obtained by Mouhot and Villani in their celebrated work (Acta Math 2011; see also the extension to include Gevrey data). The nonlinear Landau damping remains largely elusive for less regular data (e.g., data with Sobolev regularity).

Recently, in a collaboration with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), we give an elementary proof of the known Landau damping results, which I also blogged it here, that were seen as a perturbation of the free transport dynamics, whose damping is direct (that is, the phase mixing). In the companion paper with E. Grenier and I. Rodnianski, we construct a class of echo solutions, which are arbitrarily large in any Sobolev spaces (in particular, they do not belong to the analytic or Gevrey classes studied by Mouhot and Villani), but nonetheless, the nonlinear Landau damping holds. In this blog post, I shall briefly discuss the plasma echo mechanism and our new results.

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Landau damping for analytic and Gevrey data

ttn12 Kinetic theory, Plasma Physics ,

Landau damping is a classical subject in Plasma Physics, which studies decay of the electric field in a collisionless plasma in the large time. The damping was discovered and fully understood by Landau in the 40s for the linearized evolution near Maxwellians, and later extended by O. Penrose in the 60s for general spatially homogenous equilibria. The first mathematical proof of the nonlinear Landau damping was given by Mouhot and Villani for analytic data in their celebrated work (Acta Math, 2011). Their proof was then simplified, and the result was extended by Bedrossian, Masmoudi, and Mouhot to include data in certain Gevrey classes (Annals of PDEs, 2016).

Recently, in a collaboration with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), we give an elementary proof of these same results, which I shall give a sketch of it in this blog post. To avoid some tedious algebra, I mainly focus on the analytic case, which is precisely the case originally studied by Mouhot and Villani, leaving some remarks to the Gevrey cases at the very end of the post, where you’ll also find the slides of my recent lectures over Zoom on this topics.

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Landau damping for screened Vlasov-Poisson on the whole space

ttn12 Kinetic theory ,

In a recent joint work with Daniel Han-Kwan (CMLS, Ecole polytechnique) and Frédéric Rousset (Paris-Sud University), we give an alternative proof of the Landau damping for screened Vlasov-Poisson system near stable homogenous equilibria on the whole space, a result that was first established by Bedrossian, Masmoudi and Mouhot, for data with finite Sobolev regularity (they remarked that 36 derivatives were sufficient).

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