Phase mixing

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

ttn12 Kinetic theory, Plasma Physics , , , ,

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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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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