  We investigate the old problem of the fast relaxation of collisionless
$N$-body systems which are collapsing or perturbed, emphasizing the importance
of (non-collisional) discreteness effects. We integrate orbit ensembles in
fixed external potentials, estimating the entropy of the ensemble to analyze
the time evolution of the distribution function. We show that these estimates
capture the correct physical behavior expected from the 2nd Law of
Thermodynamics, without any spurious entropy production. For self-consistent
(i.e. stationary) samples, the entropy is conserved, while for
non-self-consistent samples, it increases within a few dynamical times up to a
maximum where it stabilizes (even in integrable potentials). Our results shed
light on the main ingredients for this fast collisionless relaxation. The
fundamental ingredient is the discreteness (finite $N$) of gravitational
systems in any potential. Additionally, in non-integrable potentials, the
presence of chaotic orbits accelerates the entropy production. Contrary to the
traditional violent relaxation scenario, our results indicate that a
time-dependent potential is not necessary for this fast relaxation. For the
first time, in connection with the Nyquist-Shannon theorem we determine the
$N$-dependence of this discreteness-driven relaxation, deriving a typical
relaxation time $T/\tau_{cr}\approx 0.1 N^{1/6}$, with slightly weaker
$N$-dependencies for non-integrable potentials with substantial fractions of
chaotic orbits. This timescale is much smaller than the collisional timescale
even for small-$N$ systems such as open clusters and represents an upper limit
for real collisionless $N$-body systems. Additionally, our results reinforce
the conclusion draw in Beraldo e Silva et al. (2017) that the Vlasov equation
does not provide an adequate kinetic description of the fast relaxation of
collapsing collisionless $N$-body systems.
