Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model

TL;DR

This paper investigates the relaxation dynamics of the Hamiltonian Mean-Field model, deriving stability criteria for stationary states, and characterizes the quasi-stationary states and their dependence on initial conditions and system size.

Contribution

It introduces a new stability criterion for homogeneous distributions in the Vlasov framework and numerically verifies the relaxation scenario in the HMF model.

Findings

Stable states depend on initial conditions and system size.

Quasi-stationary states last longer with increasing N, scaling as N^{1.7}.

Momentum distributions in quasi-stationary states lack power-law tails.

Abstract

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in $N$-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated $N ~$ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite $N$, dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via non stationary states: we characterize numerically this dynamical instability in the finite $N$ system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite $N$ system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law $N^{1.7}$. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the $q$-exponential distributions derived from Tsallis statistics.