Scaling with system size of the Lyapunov exponents for the Hamiltonian Mean Field model

Abstract

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 $U_c=3/4$ and its dynamics is exactly described by the Vlasov equation in the $N\to \infty$ 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 $U>U_c$, as suggested by theoretical approaches based on a random matrix approximation, but also below a threshold energy $U_t\approx 0.2$. Using a recently proposed method (GALI) devised to rapidly check the chaotic or regular nature of an orbit, we find that $U_t$ 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.

18 pages, 7 figures, (revised version, several minor typos fixed - accepted for publication in Transport Theory and Statistical Physics)

References in corpus (3)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Abundance of regular orbits and out-of-equilibrium phase transitions in the thermodynamic limit for long-range systems
• Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method

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• Many-body synchronisation in a classical Hamiltonian system
• Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model
• Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method
• $N$-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited
• Lyapunov Exponent and Criticality in the Hamiltonian Mean Field Model
• Importance-sampling computation of statistical properties of coupled oscillators
• Quantum fluctuations inhibit symmetry breaking in the HMF model
• Classical Goldstone modes in Long-Range Interacting Systems