Lyapunov Exponent and Criticality in the Hamiltonian Mean Field Model

Abstract

We investigate the dependence of the largest Lyapunov exponent of a $N$-particle self-gravitating ring model at equilibrium with respect to the number of particles and its dependence on energy. This model has a continuous phase-transition from a ferromagnetic to homogeneous phase, and we numerically confirm with large scale simulations the existence of a critical exponent associated to the largest Lyapunov exponent, although at variance with the theoretical estimate. The existence of chaos in the magnetized state evidenced by a positive Lyapunov exponent, even in the thermodynamic limit, is explained by the resonant coupling of individual particle oscillations to the diffusive motion of the center of mass of the system due to the thermal excitation of a classical Goldstone mode. The transition from "weak" to "strong" chaos occurs at the onset of the diffusive motion of the center of mass of the non-homogeneous equilibrium state, as expected. We also discuss thoroughly for the model the validity and limits of a geometrical approach for their analytical estimate.

21 pages, 14 figures

References in corpus (7)

• 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
• Statistical Mechanics of Unbound Two Dimensional Self-Gravitating Systems
• Incomplete equilibrium in long-range interacting systems
• Quasi-stationary states and the range of pair interactions
• Topology of Collisionless Relaxation
• Violent relaxation in two-dimensional flows with varying interaction range

Cited by in corpus (6)

• Many-body synchronisation in a classical Hamiltonian system
• $N$-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited
• Critical Exponent for the Lyapunov Exponent and Phase Transitions -- The Generalized Hamiltonian Mean-Field Model
• Quantum fluctuations inhibit symmetry breaking in the HMF model
• Classical Goldstone modes in Long-Range Interacting Systems
• Contribution of individual degrees of freedom to Lyapunov vectors in many-body systems