Algebraic damping in the one-dimensional Vlasov equation

We investigate the asymptotic behaviour of a perturbation around a spatially non-homogeneous stable stationary state of a one-dimensional Vlasov equation. Under general hypotheses, after transient exponential Landau damping, a perturbation evolving according to the linearized Vlasov equation decays algebraically with the exponent -2 and a well-defined frequency. The theoretical results are successfully tested against numerical N-body simulations, corresponding to the full Vlasov dynamics in the large N limit, in the case of the Hamiltonian mean-field model. For this purpose, we use a weighted particles code, which allows us to reduce finite size fluctuations and to observe the asymptotic decay in the N-body simulations.