Landau damping

Remarks

The mechanism is phase mixing: under free transport $\partial_t f + v \partial_x f = 0$, the Fourier mode at time $t$ is $\hat f(t, k, \xi) = \hat f_{\mathrm{in}}(k, \xi + k t)$. So the density mode $\hat f(t, k, 0) = \hat f_{\mathrm{in}}(k, k t)$ decays as fast as the data are smooth in $v$. The proof treats the field term as a perturbation, which gives a Volterra equation for $\hat E(t, k)$. The Maxwellian satisfies the Penrose stability condition, so this equation has no growing modes. The decay is not dissipation: the Vlasov equation is time-reversible, and the information moves to ever finer oscillations in $v$. The smoothness hypothesis is needed: for data with only finite regularity in $v$, the field decays only at a polynomial rate. Lev Landau predicted the effect in 1946, and Clément Mouhot and Cédric Villani proved in 2011 that it persists for the nonlinear Vlasov–Poisson system with small analytic or Gevrey perturbations.

Going deeper

For a student

This is the linear theorem, in units where the plasma frequency and the Debye length are $1$, with electrons moving in a fixed neutralising ion background. The damping rate is the imaginary part of the root of Landau's dispersion relation, obtained by continuing the Laplace transform of the kernel analytically. For long waves the rate is exponentially small in $1/k^2$, so damping is weak at large scales and strong at the Debye scale. The nonlinear result of Mouhot and Villani needs new estimates to control plasma echoes, in which nonlinear interactions bring old oscillations back.