2010/06/30 by Thanos Manos, Stefano Ruffo · 1 citation
Physics and Astronomy · #nlin.CD #cond-mat.stat-mech
paper · pdf · doi:10.1080/00411450.2011.651035
18 pages, 7 figures, (revised version, several minor typos fixed - accepted for publication in Transport Theory and Statistical Physics)
arxiv created 2010/09/07 · arxiv updated 2013/03/26
The Hamiltonian Mean Field (HMF) model is a prototype for systems with long-range interactions. It describes the motion of N particles moving on a ring, coupled through an infinite-range potential. The model has a second order phase transition at the energy Uc=3/4 and its dynamics is exactly described by the Vlasov equation in the N → ∞ limit. Its chaotic properties have been investigated in the past, but the determination of the scaling with N of the Lyapunov Spectrum (LS) of the model remains a challenging open problem. We here show that the N-1/3 scaling of the Maximal Lyapunov Exponent (MLE), found in previous numerical and analytical studies, extends to the full LS; not only, scaling is "precocious" for the LS, meaning that it becomes manifest for a much smaller number of particles than the one needed to check the scaling for the MLE. Besides that, the N-1/3 scaling appears to be valid not only for U>Uc, as suggested by theoretical approaches based on a random matrix approximation, but also below a threshold energy Ut ≈ 0.2. Using a recently proposed method (GALI) devised to rapidly check the chaotic or regular nature of an orbit, we find that Ut is also the energy at which a sharp transition from \it weak to \it strong chaos is present in the phase-space of the model. Around this energy the phase of the vector order parameter of the model becomes strongly time dependent, inducing a significant untrapping of particles from a nonlinear resonance.