Plasma Physics

Landau damping for the borderline Vlasov-Dirac-Benney system

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The Vlasov-Dirac-Benney system refers to the Vlasov theory for charged particles when the interaction potential is a Dirac delta function, namely

\displaystyle \partial_t f + v\cdot \nabla_x f + E \cdot \nabla_v f = 0 \ \ \ \ \ (1)

where the self-consistent electric field is computed through

\displaystyle E = -\nabla_x \rho[f] \ \ \ \ \ (2)

with density \rho[f] =\int f(t,x,v)\; dv - n_{0} (compared with the classical Vlasov-Poisson system E = -\nabla_x (-\Delta_x)^{-1}\rho[f]). The name was coined by C. Bardos due to a close link to a model for water waves derived by Benney. The system is in fact ill-posed for general initial data in any Sobolev and Gevrey spaces, and the question of Landau damping for the borderline analytic data has always been open and of great interest. I shall briefly discuss how such a Vlasov theory arises, and mention my recent work that resolves this very question. This work is dedicated to my teacher and friend Dang Duc Trong in occasion of his 60th birthday.     

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Remarks on Landau damping

ttn12 Kinetic theory, New papers, Plasma Physics , , , ,

I’ve just posted on arxiv this short paper “Remarks on Landau damping”, which gives a quick overview of phase mixing, Landau damping, plasma echoes, and in particular, provides a self-contained proof, for non specialists, of the celebrated result by Mouhot-Villani in the analytic case. This was written in honor of Dang Duc Trong, a kind mentor to many Vietnamese mathematicians, on the occasion of his 60th birthday.  Specifically, consider the following classical Vlasov-Poisson system

\displaystyle \partial_t f + v\cdot \nabla_x f + E \cdot \nabla_v f = 0 \ \ \ \ \ (1)

\displaystyle E = -\nabla_x \phi, \qquad -\Delta_x \phi = \rho \ \ \ \ \ (2)

modeling the dynamics of excited electrons confined on a torus {\mathbb{T}_x^3\times \mathbb{R}^3} or in the whole space {\mathbb{R}^3_x\times \mathbb{R}^3_v}, in which {\rho(t,x) = \int_{\mathbb{R}^3} f(t,x,v)\; dv-n_{\mathrm{ion}}} denotes the charged density, and { n_{\mathrm{ion}}} is a non-negative constant representing the uniform ions background. The Cauchy problem is rather classical, going back to the works by Lions-Perthame, Pfaffelmoser, and Schaeffer in the early 90s, which assert that smooth initial data {f(0,x,v)} with finite moments give rise to global-in-time smooth solutions. However, their large time behavior is largely open due to the presence of plasma echoes and rich underlying physics, which we shall now discuss.

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The Vlasov theory for relativistic plasmas near vacuum

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In plasma physics, the dynamics of collisionless charged particles can often be modeled by the relativistic Vlasov-Maxwell system, namely the transport equation

\displaystyle \partial_t f + \hat v \cdot \nabla_x f + \mathcal K\cdot \nabla_v f = 0 \ \ \ \ \ (1)

for the unknown one-particle density distribution {f(t,x,v)} in the phase space {\mathbb{R}_x^3 \times \mathbb{R}^3_v}, coupled with the Lorentz force {\mathcal K = E + \hat v \times B}, whose electromagnetic fields are computed though the classical Maxwell equations

\displaystyle \begin{aligned} \partial_t B + \nabla_x \times E = 0, \qquad \nabla_x \cdot E &= \rho[f], \\ - \partial_t E + \nabla_x \times B = {\bf j}[f], \qquad \nabla_x \cdot B & =0. \end{aligned} \ \ \ \ \ (2)

The particles travel with relativistic velocities {\hat v = v/\langle v\rangle} with {\langle v\rangle =\sqrt{1+|v|^2}} (normalizing the speed of light to be one), and the charge and current densities are defined by {\rho[f] = \int f(t,x,v) dv - n_{\mathrm{ion}}} and {{\bf j}[f] = \int \hat v f(t,x,v)\; dv - {\bf j}_{\mathrm{ion}}} for some fixed constant background densities {n_{\mathrm{ion}}} and {{\bf j}_{\mathrm{ion}}} (i.e. for sake of presentation, we simply focus on the dynamics of excited electrons, fixing a uniform ion background).

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Landau damping for relativistic plasmas

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Of great interest is to resolve the Final State Conjecture of charged particles in the kinetic theory of plasma physics (borrowing words from general relativity, though amusingly lesser mathematical progress is made for the case of plasmas!), asserting the scattering to neutrality in the far field plus a superposition of coherent states or trapped particles, and identifying relaxation mechanism plays a fundamental role in resolving the conjecture. Despite an extensive list of works in the mathematical literature towards the understanding of the large-time dynamics of non-relativistic plasmas (already encoding extremely rich underlying physics and challenging mathematics – see Survival Threshold), very little is known in the case of relativistic plasmas, including the basic question of whether stable equilibria exist, not to mention the apparent lack of study on relaxation mechanism. In this blog article, I shall highlight my recent joint work with D. Han-Kwan (CNRS, Nantes) and F. Rousset (Paris-Sud, Orsay) which resolved the linear stability theory of relativistic plasmas near radial spatially homogenous equilibria, the first such a result, paving the waves for many possible future advances on the subject.

Figure: Depicted is a description of phase mixing on torus (in the whole space, particles simply scatter away), one of the key relaxation mechanisms in plasma physics.

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Survival threshold for plasma oscillations

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

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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

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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

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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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On the non-relativistic limit of Vlasov-Maxwell

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In this note, I briefly explain my recent joint work with D. Han-Kwan (CNRS, Ecole polytechnique) and F. Rousset (Paris-Sud) on the non-relativistic limit of Vlassov-Maxwell. Precisely, we consider the relativistic Vlasov-Maxwell system, modeling the dynamics of electrons with electron density distribution {f(t,x,v)}, which reads

\displaystyle \partial_t f + \hat v \cdot \nabla_x f + (E + \epsilon \hat v \times B)\cdot \nabla_v f = 0

on {\mathbb{T}^3\times \mathbb{R}^3}, with the relativistic velocity {\hat v = v/\sqrt{1+ \epsilon^2 |v|^2}}.

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Stability of a collisionless plasma

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What is a plasma? A plasma is an ionized gas that consists of charged particles: positive ions and negative electrons. To describe the dynamics of a plasma, let {f^\pm(t,x,v)} be the (nonnegative) density distribution of ions and electrons, respectively, at time {t\ge 0}, position {x\in \Omega \subset \mathbb{R}^3}, and particle velocity (or momentum) {v\in \mathbb{R}^3}. The dynamics of a plasma is commonly modeled by the Vlasov equations

\displaystyle \frac{d}{dt} f^\pm (t,X(t), V(t)) = 0 \ \ \ \ \ (1)

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Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits

ttn12 Instabilities, New papers, Plasma Physics

Daniel Han-Kwan and I have just submitted a paper entitled: “Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits”, which is also available on arxiv: arXiv:1506.08537. In this paper, we study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit {\varepsilon \rightarrow 0}, with {\varepsilon} being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution {\mu} of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from {\mu} in arbitrary negative Sobolev norms within time of order {|\log \varepsilon|}. Second, we deduce the invalidity of the quasineutral limit in {L^2} in arbitrarily short time.

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