Errors, chaos and the collisionless limit
  We simultaneously study the dynamics of 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 small-$N$ spherical systems,
and by comparing fixed-timestep runs with different stepsizes. The errors add
randomly, before exponential divergence sets in, with exponentiation rate
virtually independent of $N$, but scale saturating as $\sim 1/\sqrt{N}$, in
line with theoretical estimates presented. In a third phase, the growth rate is
initially driven by multiplicative enhancement of errors, as in the exponential
stage. It is then qualitatively different for the phase space variables and
mean field conserved quantities (energy and momentum); for the former, the
errors grow systematically through phase mixing, for the latter they grow
diffusively. For energy, the $N$-variation of the `relaxation time' of error
growth follows the $N$-scaling of two-body relaxation. This is also true for
angular momentum in the fixed stepsize runs, although the associated error
threshold is higher and the relaxation time smaller. Due to shrinking
saturation scales, the information loss associated with the exponential
instability decreases with $N$ and the dynamical entropy vanishes at any finite
resolution as $N \rightarrow \infty$. A distribution function depending on the
integrals of motion in the smooth potential is decreasingly affected. In this
sense there is convergence to the collisionless limit, despite the persistence
of exponential instability on infinitesimal scales. Nevertheless, the slow
$N$-variation in its saturation points to the slowness of the convergence.
