Perturbation theory in a Vlasovian frame

Theoretical treatment starts with the Vlasov equation in the non-relativistic zero-magnetic field limit, the Vlasov–Poisson set of equations. Explicit solutions are obtained in the limit of a small ${\displaystyle E}$-field. The distribution function ${\displaystyle f}$ and field ${\displaystyle E}$ are expanded in a series: ${\displaystyle f=f_{0}(v)+f_{1}(x,v,t)+\cdots }$, ${\displaystyle E=E_{1}(x,t)+E_{2}(x,t)+\cdots }$ and terms of equal order are collected.

To first order the Vlasov–Poisson equations read

$${\displaystyle (\partial _{t}+v\partial _{x})f_{1}+{e \over m}E_{1}f'_{0}=0,\quad \partial _{x}E_{1}={e \over \epsilon _{0}}\int f_{1}\mathrm {d} v.}$$

Landau calculated^[1] the wave caused by an initial disturbance ${\displaystyle f_{1}(x,v,0)=g(v)\exp(ikx)}$ and found by aid of Laplace transform and contour integration a damped travelling wave of the form ${\displaystyle \exp[ik(x-v_{\text{ph}}t)-\gamma t]}$ with wave number ${\displaystyle k}$ and damping decrement

$${\displaystyle \gamma \approx -{\pi \omega _{p}^{3} \over 2k^{2}N}f'_{0}(v_{\text{ph}}),\quad N=\int f_{0}\mathrm {d} v.}$$

Here ${\displaystyle \omega _{p}}$ is the plasma oscillation frequency and ${\displaystyle N}$ is the electron density. Later Nico van Kampen proved^[11] that the same result can be obtained with Fourier transform. He showed that the linearized Vlasov–Poisson equations have a continuous spectrum of singular normal modes, now known as van Kampen modes

$${\displaystyle {\frac {\omega _{p}^{2}}{kN}}f'_{0}{\frac {\mathcal {P}}{kv-\omega }}+\epsilon \delta \left(v-{\frac {\omega }{k}}\right)}$$

in which ${\displaystyle {\mathcal {P}}}$ signifies principal value, ${\displaystyle \delta }$ is the delta function (see generalized function) and

$${\displaystyle \epsilon =1+{\frac {\omega _{p}^{2}}{kN}}\int f'_{0}{\frac {\mathcal {P}}{\omega -kv}}\mathrm {d} v}$$

is the plasma permittivity. Decomposing the initial disturbance in these modes he obtained the Fourier spectrum of the resulting wave. Damping is explained by phase-mixing of these Fourier modes with slightly different frequencies near ${\displaystyle \omega _{p}}$.

It was not clear how damping could occur in a collisionless plasma: where does the wave energy go? In fluid theory, in which the plasma is modeled as a dispersive dielectric medium,^[12] the energy of Langmuir waves is known: field energy multiplied by the Brillouin factor ${\displaystyle \partial _{\omega }(\omega \epsilon )}$. But damping cannot be derived in this model. To calculate energy exchange of the wave with resonant electrons, Vlasov plasma theory has to be expanded to second order and problems about suitable initial conditions and secular terms arise.

In Ref.^[13] these problems are studied. Because calculations for an infinite wave are deficient in second order, a wave packet is analysed. Second-order initial conditions are found that suppress secular behavior and excite a wave packet of which the energy agrees with fluid theory. The figure shows the energy density of a wave packet traveling at the group velocity, its energy being carried away by electrons moving at the phase velocity. Total energy, the area under the curves, is conserved.

The Cauchy problem for perturbative solutions

The rigorous mathematical theory is based on solving the Cauchy problem for the evolution equation (here the partial differential Vlasov–Poisson equation) and proving estimates on the solution.

First a rather complete linearized mathematical theory has been developed since Landau.^[14]

Going beyond the linearized equation and dealing with the nonlinearity has been a longstanding problem in the mathematical theory of Landau damping. Previously one mathematical result at the non-linear level was the existence of a class of exponentially damped solutions of the Vlasov–Poisson equation in a circle which had been proved in^[15] by means of a scattering technique (this result has been recently extended in^[16]). However these existence results do not say anything about which initial data could lead to such damped solutions.

In a paper published by French mathematicians Cédric Villani and Clément Mouhot,^[17] the initial data issue is solved and Landau damping is mathematically established for the first time for the non-linear Vlasov equation. It is proved that solutions starting in some neighborhood (for the analytic or Gevrey topology) of a linearly stable homogeneous stationary solution are (orbitally) stable for all times and are damped globally in time. The damping phenomenon is reinterpreted in terms of transfer of regularity of ${\displaystyle f}$ as a function of ${\displaystyle x}$ and ${\displaystyle v}$, respectively, rather than exchanges of energy. Large scale variations pass into variations of smaller and smaller scale in velocity space, corresponding to a shift of the Fourier spectrum of ${\displaystyle f}$ as a function of ${\displaystyle v}$. This shift, well known in linear theory, proves to hold in the non-linear case.

Perturbation theory in an N-body frame

The mechanical N-body description, originally deemed impossible, enables a rigorous calculation of Landau damping using Newton’s second law of motion and Fourier series.^[10] Neither the Vlasov equation nor Laplace transforms are required for this derivation. The calculation of the energy (more precisely momentum) exchange of the wave with electrons is done similarly. This calculation makes intuitive the interpretation of Landau damping as the synchronization of almost resonant passing particles.