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.