Vlasov stability of the Hamiltonian Mean Field model

Abstract

We investigate the dynamical stability of a fully-coupled system of $N$ inertial rotators, the so-called Hamiltonian Mean Field model. In the limit $N\to \infty$, and after proper scaling of the interactions, the $\mu$-space dynamics is governed by a Vlasov equation. We apply a nonlinear stability test to (i) a selected set of spatially homogeneous solutions of Vlasov equation, qualitatively similar to those observed in the quasi-stationary states arising from fully magnetized initial conditions, and (ii) numerical coarse-grained distributions of the finite-$N$ dynamics. Our results are consistent with previous numerical evidence of the disappearance of the homogenous quasi-stationary family below a certain energy.

11 pages, 5 figures. Submitted as a contribution to the proceedings of the International Workshop on Trends and Perspectives on Extensive and Non-Extensive Statistical Mechanics, November, 19-21, 2003, Angra dos Reis, Brazil

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• Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model
• Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system
• Stability and ensemble inequivalence in a globally coupled system
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Cited by in corpus (9)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index
• Lynden-Bell and Tsallis distributions for the HMF model
• The Vlasov equation and the Hamiltonian Mean-Field model
• On the diffusive anomalies in a long-range Hamiltonian system
• Dynamics and thermodynamics of rotators interacting with both long and short range couplings
• Quasi-stationary trajectories of the HMF model: a topological perspective
• Effective spin-glass Hamiltonian for the anomalous dynamics of the HMF model
• A Monte Carlo Investigation of the Hamiltonian Mean Field Model