Vlasov stability of the Hamiltonian Mean Field model

TL;DR

This paper analyzes the stability of the Hamiltonian Mean Field model using Vlasov dynamics, identifying conditions under which homogeneous states are stable or unstable, and confirming previous numerical findings about phase transitions.

Contribution

It applies a nonlinear stability test to Vlasov solutions and finite-N distributions, providing new insights into the stability of quasi-stationary states in the model.

Findings

Homogeneous solutions are stable above a critical energy.

Stability decreases as energy approaches the critical point.

Results align with previous numerical evidence of phase transition.

Abstract

We investigate the dynamical stability of a fully-coupled system of $N$ inertial rotators, the so-called Hamiltonian Mean Field model. In the limit $N \to \infty$, and after proper scaling of the interactions, the $\mu$-space dynamics is governed by a Vlasov equation. We apply a nonlinear stability test to (i) a selected set of spatially homogeneous solutions of Vlasov equation, qualitatively similar to those observed in the quasi-stationary states arising from fully magnetized initial conditions, and (ii) numerical coarse-grained distributions of the finite-$N$ dynamics. Our results are consistent with previous numerical evidence of the disappearance of the homogenous quasi-stationary family below a certain energy.