Lyapunov stability of Vlasov Equilibria using Fourier-Hermite modes

TL;DR

This paper introduces a Fourier-Hermite mode expansion method to efficiently compute Lyapunov exponents and eigenvectors for Vlasov systems, enabling stability analysis of long-range interacting particles with fewer computational resources.

Contribution

It presents a novel Fourier-Hermite expansion approach for analyzing the linear stability of Vlasov equilibria, improving computational efficiency and accuracy.

Findings

Fast convergence of the expansion with an appropriate scaling parameter.

Ability to predict stability of states in both near and far-from equilibrium regimes.

Application to the Hamiltonian mean-field model demonstrating the method's effectiveness.

Abstract

We propose an efficient method to compute Lyapunov exponents and Lyapunov eigenvectors of long-range interacting many-particle systems, whose dynamics is described by the Vlasov equation. We show that an expansion of a distribution function using Hermite modes (in momentum variable) and Fourier modes (in configuration variable) converges fast if an appropriate scaling parameter is introduced and identified with the inverse of the system temperature. As a consequence, dynamics and linear stability properties of many-particle states, both in the close-to and in the far-from equilibrium regimes can be predicted using a small number of expansion coefficients. As an example of a long-range interacting system we investigate stability properties of stationary states of the Hamiltonian mean-field model.