On Linear Damping Around Inhomogeneous Stationary States of the Vlasov-HMF Model

TL;DR

This paper demonstrates algebraic Landau damping in the Vlasov-HMF model for inhomogeneous stationary states, showing solutions decay over time and scatter to a modified state, confirming linear damping effects.

Contribution

It provides a rigorous proof of algebraic decay and scattering behavior for perturbations around inhomogeneous states in the Vlasov-HMF model, extending understanding of Landau damping.

Findings

Algebraic decay of Fourier modes of density perturbations

Solutions scatter to a modified state, indicating damping

Linear Landau damping occurs with algebraic rate

Abstract

We study the dynamics of perturbations around an inhomogeneous stationary state of the Vlasov-HMF (Hamiltonian Mean-Field) model, satisfying a linearized stability criterion (Penrose criterion). We consider solutions of the linearized equation around the steady state, and prove the algebraic decay in time of the Fourier modes of their density. We prove moreover that these solutions exhibit a scattering behavior to a modified state, implying a linear Landau damping effect with an algebraic rate of damping.