Chaos in violent relaxation dynamics. Disentangling micro- and macro-chaos in numerical experiments of dissipationless collapse

Violent relaxation (VR) is often regarded as the mechanism leading stellar systems to collisionless meta equilibrium via rapid changes in the collective potential. We investigate the role of chaotic instabilities on single particle orbits in leading to nearly-invariant phase-space distributions, aiming at disentangling it from the chaos induced by collective oscillations in the self-consistent potential. We explore as function of the systems size (i.e. number of particles $N$) the chaoticity in terms of the largest Lyapunov exponent of test trajectories in a simplified model of gravitational cold collapse, mimicking a $N-$body calculation via a time dependent smooth potential and a noise-friction process accounting for the discreteness effects. A new numerical method to evaluate effective Lyapunov exponents for stochastic models is presented and tested. We find that the evolution of the phase-space of independent trajectories reproduces rather well what observed in self-consistent $N-$body simulations of dissipationless collapses. The chaoticity of test orbits rapidly decreases with $N$ for particles that remain weakly bounded in the model potential, while it decreases with different power laws for more bound orbits, consistently with what observed in previous self-consistent $N$-body simulations. The largest Lyapunov exponents of ensembles of orbits starting from initial conditions uniformly sampling the accessible phase-space are somewhat constant for $N\lesssim 10^9$, while decreases towards the continuum limit with a power-law trend. Moreover, our numerical results appear to confirm the trend of a specific formulation of dynamical entropy and its relation with Lyapunov time scales.