The discreteness-driven relaxation of collisionless gravitating systems:
  entropy evolution and the Nyquist-Shannon theorem
  The time irreversibility and fast relaxation of collapsing $N$-body
gravitating systems (as opposed to the time reversibility of the equations of
motion for individual stars or particles) are traditionally attributed to
information loss due to coarse-graining in the observation. We show that this
subjective element is not necessary once one takes into consideration the
fundamental fact that these systems are discrete, i.e. composed of a finite
number $N$ of stars or particles. We show that a connection can be made between
entropy estimates for discrete systems and the Nyquist-Shannon sampling
criterion. Specifically, given a sample with $N$ points in a space of $d$
dimensions, the Nyquist-Shannon criterion constrains the size of the smallest
structures defined by a function in the continuum that can be uniquely
associated with the discrete sample. When applied to an $N$-body system, this
theorem sets a lower limit to the size of phase-space structures (in the
continuum) that can be resolved in the discrete data. As a consequence, the
finite $N$ system tends to a uniform distribution after a relaxation time that
typically scales as $N^{1/d}$. This provides an explanation for the fast
achievement of a stationary state in collapsing $N$-body gravitating systems
such as galaxies and star clusters, without the need to advocate for the
subjective effect of coarse-graining.
