Algebraic damping in the one-dimensional Vlasov equation

TL;DR

This paper studies the long-term decay of perturbations in a one-dimensional Vlasov system, revealing algebraic damping after initial exponential Landau damping, supported by numerical simulations.

Contribution

It provides a theoretical analysis of algebraic damping in the Vlasov equation and validates it with large-scale N-body simulations using a weighted particles code.

Findings

Perturbations decay algebraically with exponent -2 after Landau damping.

Numerical simulations confirm theoretical predictions in the Hamiltonian mean-field model.

Weighted particles code effectively reduces finite size effects in N-body simulations.

Abstract

We investigate the asymptotic behavior 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.