# Lyapunov Exponent and Criticality in the Hamiltonian Mean Field Model

## TL;DR

This paper studies how the largest Lyapunov exponent behaves in a Hamiltonian mean field model at equilibrium, revealing critical behavior and chaos transition related to phase changes and resonant particle interactions.

## Contribution

It provides numerical confirmation of a critical exponent for the Lyapunov exponent and discusses the limitations of geometrical analytical estimates in the model.

## Key findings

- Existence of a critical exponent for the Lyapunov exponent at phase transition.
- Chaos persists in the magnetized phase even in the thermodynamic limit.
- Transition from weak to strong chaos coincides with the onset of diffusive center of mass motion.

## 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.