### 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 |
---|---|

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

**Chaos in the Hamiltonian mean-field model.** / Ginelli, Francesco; Takeuchi, Kazumasa A.; Chate, Hugues; Politi, Antonio; Torcini, Alessandro.

Research output: Contribution to journal › Article

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*, vol. 84, no. 6, 066211. https://doi.org/10.1103/PhysRevE.84.066211

}

TY - JOUR

T1 - Chaos in the Hamiltonian mean-field model

AU - Ginelli, Francesco

AU - Takeuchi, Kazumasa A.

AU - Chate, Hugues

AU - Politi, Antonio

AU - Torcini, Alessandro

PY - 2011/12/28

Y1 - 2011/12/28

N2 - 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.

AB - 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.

KW - largest lyapunov exponent

KW - phase-transition

KW - interacting oscillators

KW - statistical-mechanics

KW - scaling law

KW - systems

KW - dynamics

U2 - 10.1103/PhysRevE.84.066211

DO - 10.1103/PhysRevE.84.066211

M3 - Article

VL - 84

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 6

M1 - 066211

ER -