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We investigate the old problem of the fast relaxation of collisionless$N$-body systems which are collapsing or perturbed, emphasizing the importanceof (non-collisional) discreteness effects. We integrate orbit ensembles infixed external potentials, estimating the entropy of the ensemble to analyzethe time evolution of the distribution function. We show that these estimatescapture the correct physical behavior expected from the 2nd Law ofThermodynamics, without any spurious entropy production. For self-consistent(i.e. stationary) samples, the entropy is conserved, while fornon-self-consistent samples, it increases within a few dynamical times up to amaximum where it stabilizes (even in integrable potentials). Our results shedlight on the main ingredients for this fast collisionless relaxation. Thefundamental ingredient is the discreteness (finite $N$) of gravitationalsystems in any potential. Additionally, in non-integrable potentials, thepresence of chaotic orbits accelerates the entropy production. Contrary to thetraditional violent relaxation scenario, our results indicate that atime-dependent potential is not necessary for this fast relaxation. For thefirst time, in connection with the Nyquist-Shannon theorem we determine the$N$-dependence of this discreteness-driven relaxation, deriving a typicalrelaxation time $T/\tau_{cr}\approx 0.1 N^{1/6}$, with slightly weaker$N$-dependencies for non-integrable potentials with substantial fractions ofchaotic orbits. This timescale is much smaller than the collisional timescaleeven for small-$N$ systems such as open clusters and represents an upper limitfor real collisionless $N$-body systems. Additionally, our results reinforcethe conclusion draw in Beraldo e Silva et al. (2017) that the Vlasov equationdoes not provide an adequate kinetic description of the fast relaxation ofcollapsing collisionless $N$-body systems.

The discreteness-driven relaxation of collisionless gravitating systems: entropy evolution in external potentials, N-dependence and the role of chaos