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

2003/12/31 by Yoshiyuki Y. Yamaguchi, Y. Y. Yamaguchi, Julien Barré +7 · 4 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Classical mechanics #Complex Systems and Time Series Analysis #Hamiltonian (control theory) #Hamiltonian system #Instability #Markov chain #Mathematical optimization #Mathematics #Physics #Plasma #Power law #Quantum mechanics #Stationary distribution #Stationary state #Statistical Mechanics and Entropy #Statistical physics #Tsallis statistics #Vlasov equation #cond-mat.stat-mech #nlin.PS

paper · pdf · doi:10.1016/j.physa.2004.01.041

published as Physica A 337, 36-66 (2004) · To appear in Physica A

arxiv created 2004/01/15 · openalex publication_date 2004/03/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two-dimensional turbulence
• Influence of expansion on hierarchical structure
• Aging in an infinite-range Hamiltonian system of coupled rotators

Cited by

• Attractor nonequilibrium stationary states in perturbed long-range interacting systems
• Microcanonical quasistationarity of long-range interacting systems in contact with a heat bath
• Onset of synchronization in the disordered Hamiltonian mean-field model
• Microcanonical solution of the mean-field model: Comparison with time averages at finite size