Vlasov stability of the Hamiltonian mean field model

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→∞, and after proper scaling of the interactions, the μ-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 homogeneous quasi-stationary family below a certain energy.