Chaos and relaxation to equilibrium in systems with long-range interactions

In the thermodynamic limit, systems with long-range (LR) interactions do not relax to equilibrium, but become trapped in
non-equilibrium stationary states. Once a system is trapped in a non-equilibrium state, two outcomes are possible: if the system has a finite number of particles $N$, residual correlations will eventually drive it to thermodynamic equilibrium (if such equilibrium exists, which is not the case for 3d gravitational systems) after a time $t_\times$ which scales with $N$ as $t_\times \sim N^\delta$, where $\delta$ is a system specific exponent.
On the other hand, in the thermodynamic limit, $N\rightarrow \infty$, the system will remain trapped in a stationary state forever. In this collisionless limit, the relaxation to stationarity is a result of Landau damping, which transfers the energy of collective oscillations to the individual particles. Once the oscillations of the mean-field potential die out, the particles will move in a static mean-field potential.
If a system has sufficient symmetry, the motion of particles in a static potential will be integrable, and the ergodicity will be irrevocably broken. In this paper we will explore the role of chaotic dynamics on the time that a system with LR interactions remains trapped in a QSS. We discover that a small degree of chaos, measured by the Lyapunov exponents, favors a faster relaxation to equilibrium. Surprisingly, a larger degree of chaos
hinders the relaxation to equilibrium.