### Abstract

We study the dynamical properties of the canonical ordered phase of the Hamiltonian mean-field (HMF) model, in which N particles, globally coupled via pairwise attractive interactions, form a rotating cluster. Using a combination of numerical and analytical arguments, we first show that the largest Lyapunov exponent remains strictly positive in the infinite-size limit, converging to its asymptotic value with 1/ln N corrections. We then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered, clustered phase and the disordered phase present at high-energy densities. We also show that the full spectrum of Lyapunov exponents consists of a bulk component converging to the (zero) value taken by a test oscillator forced by the mean field, plus subextensive bands of O(ln N) exponents taking finite values. We finally investigate the robustness of these results by studying a "2D" extension of the HMF model where each particle is endowed with 4 degrees of freedom, thus allowing the emergence of chaos at the level of a single particle. Altogether, these results illustrate the subtle effects of global (or long-range) coupling and the importance of the order in which the infinite-time and infinite-size limits are taken: For an infinite-size HMF system represented by the Vlasov equation, no chaos is present, while chaos exists and subsists for any finite system size.

Original language | English |
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Article number | 066211 |

Number of pages | 15 |

Journal | Physical Review. E, Statistical, Nonlinear and Soft Matter Physics |

Volume | 84 |

Issue number | 6 |

DOIs | |

Publication status | Published - 28 Dec 2011 |

### Keywords

- largest lyapunov exponent
- phase-transition
- interacting oscillators
- statistical-mechanics
- scaling law
- systems
- dynamics

### Cite this

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*,

*84*(6), [066211]. https://doi.org/10.1103/PhysRevE.84.066211