{
  "id": "cond-mat/0312480",
  "version": "v2",
  "published": "2003-12-18T14:53:10.000Z",
  "updated": "2004-01-15T16:04:16.000Z",
  "title": "Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model",
  "authors": [
    "Y. Y. Yamaguchi",
    "J. Barr'e",
    "F. Bouchet",
    "T. Dauxois",
    "S. Ruffo"
  ],
  "comment": "To appear in Physica A",
  "journal": "Physica A 337, 36-66 (2004)",
  "doi": "10.1016/j.physa.2004.01.041",
  "categories": [
    "cond-mat.stat-mech",
    "nlin.PS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-01-15T16:04:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vlasov equation",
      "quasi-stationary state",
      "hmf model",
      "stability criteria",
      "stable stationary state"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}