Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model

TL;DR

This paper uses the Nyquist method to analyze the linear dynamical stability of the HMF model with various distributions, revealing critical temperatures, re-entrant phases, and stability criteria relevant to long-range interacting systems.

Contribution

It applies the Nyquist method to the HMF model for the first time, exploring stability of different distributions and extending results to arbitrary potentials, linking to plasmas and gravitational systems.

Findings

System stable above critical temperature T_c

Re-entrant phases depend on distribution asymmetry

Single-humped distributions are always stable in repulsive case

Abstract

We apply the Nyquist method to the Hamiltonian Mean Field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Delta>1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1<Delta<25.6, a double re-entrant phase for 25.6<Delta<43.9 and no re-entrant phase for Delta>43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for inhomogeneous systems.