Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems

TL;DR

This paper establishes a comprehensive criterion for linear dynamical stability of inhomogeneous stationary states in the Hamiltonian Mean Field model, extending previous results for homogeneous states and providing practical stability conditions.

Contribution

It derives explicit necessary and sufficient conditions for the linear stability of inhomogeneous Vlasov stationary states, generalizing known criteria for homogeneous states.

Findings

Derived a necessary and sufficient stability disequality for inhomogeneous states.

Established that for homogeneous states, linear dynamical stability and formal stability are equivalent.

Provided simpler stability conditions that are useful in practical analysis.

Abstract

We derive a necessary and sufficient condition of linear dynamical stability for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF) model. The condition is expressed by an explicit disequality that has to be satisfied by the stationary state, and it generalizes the known disequality for homogeneous stationary states. In addition, we derive analogous disequalities that express necessary and sufficient conditions of formal stability for the stationary states. Their usefulness, from the point of view of linear dynamical stability, is that they are simpler, although they provide only sufficient criteria of linear stability. We show that for homogeneous stationary states the relations become equal, and therefore linear dynamical stability and formal stability become equivalent.