The Vlasov equation and the Hamiltonian Mean-Field model

TL;DR

This paper demonstrates that quasi-stationary states in the Hamiltonian Mean-Field model are Vlasov stable homogeneous states, which eventually relax to Boltzmann-Gibbs equilibrium due to finite N effects, with a focus on the stability and variety of these states.

Contribution

It establishes the connection between quasi-stationary states and Vlasov stability in the HMF model, highlighting the diversity of stable states and their eventual relaxation.

Findings

Quasi-stationary states are Vlasov stable homogeneous states.

Infinite variety of stable states corresponding to different initial conditions.

Finite N effects cause eventual relaxation to Boltzmann-Gibbs equilibrium.

Abstract

We show that the quasi-stationary states observed in the $N$-particle dynamics of the Hamiltonian Mean-Field (HMF) model are nothing but Vlasov stable homogeneous (zero magnetization) states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis $q$-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite $N$ effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann-Gibbs equilibrium.