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Article Dans Une Revue Physica A: Statistical Mechanics and its Applications Année : 2004

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

Y. Y. Yamaguchi
  • Fonction : Auteur
Julien Barré
Thierry Dauxois
S. Ruffo
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-00008414 , version 1 (04-09-2005)

Identifiants

Citer

Y. Y. Yamaguchi, Julien Barré, Freddy Bouchet, Thierry Dauxois, S. Ruffo. Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model. Physica A: Statistical Mechanics and its Applications, 2004, 337, pp.36-66. ⟨hal-00008414⟩
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