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. We point out in particular the role played by the infinity of stationary states of the associated $N o infty$ 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. The system first shows a rapid convergence towards a stationary state of the Vlasov equation. We numerically characterize 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. When starting from a Vlasov stable homogenous initial state, the finite $N$ system 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 cannot be fitted by the single particle Tsallis distributions used in Latora, Rapisarda, Tsallis, Phys. Rev. E extbf{64}, 056134 (2001).

Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model / Yamaguchi, Y. Y.; Barre', J.; Bouchet, F.; Dauxois, T.; Ruffo, Stefano. - In: PHYSICA. A. - ISSN 0378-4371. - 337:1-2(2004), pp. 36-66. [10.1016/j.physa.2004.01.041]

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

Ruffo, Stefano
2004-01-01

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. We point out in particular the role played by the infinity of stationary states of the associated $N o infty$ 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. The system first shows a rapid convergence towards a stationary state of the Vlasov equation. We numerically characterize 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. When starting from a Vlasov stable homogenous initial state, the finite $N$ system 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 cannot be fitted by the single particle Tsallis distributions used in Latora, Rapisarda, Tsallis, Phys. Rev. E extbf{64}, 056134 (2001).
2004
337
1-2
36
66
https://arxiv.org/abs/cond-mat/0312480
Yamaguchi, Y. Y.; Barre', J.; Bouchet, F.; Dauxois, T.; Ruffo, Stefano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/11904
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