Errors, chaos, and the collisionless limit

We simultaneously study the growth of errors and the question of the faithfulness of simulations of $N$-body systems. The errors are quantified through the numerical reversibility of trajectories of small-$N$ spherical systems integrated to high accuracy. Initially, the errors add randomly, before exponential divergence sets in. Though the exponentiation rate is virtually independent of $N$, the instability saturates at scales $1/\sqrt{N}$. This is interpreted by adopting a model due to Goodman, Heggie \& Hut (1993). In the third phase, the (diminished) growth is initially driven by multiplicative enhancement of errors as in the exponential stage. It is then qualitatively different for the errors in the phase space variables and the mean field conserved quantities (energy and momentum); the former grow systematically through phase mixing while the latter grow diffusively. For energy, the $N$-variation of the `relaxation time' of error growth follows expectations of two-body relaxation theory. This is not the case for angular momentum, at least up to the particle numbers and timescales considered, and even less so for the velocities. Due to increasingly smaller saturation scales, the information loss associated with the exponential instability decreases with $N$, especially when viewed in terms of the mean-field conserved quantities. Indeed, the dynamical entropy vanishes at any finite resolution as $N \rightarrow \infty$. In this sense there is convergence to the collisionless limit and confidence that numerical simulations may faithfully represent it, despite the exponential instability and loss of information on phase space trajectories. Nevertheless, the rapid initial growth of errors. and the relatively slow $N$-variation in its saturation, point to the slowness of the convergence.