# The discreteness-driven relaxation of collisionless gravitating systems: entropy evolution and the Nyquist-Shannon theorem

## TL;DR

This paper demonstrates that the rapid relaxation of collisionless gravitating systems can be explained by their discrete nature and the Nyquist-Shannon sampling theorem, eliminating the need for subjective coarse-graining assumptions.

## Contribution

It establishes a connection between entropy evolution in discrete systems and the Nyquist-Shannon theorem, providing a fundamental explanation for the fast relaxation in astrophysical systems.

## Key findings

- Finite N systems tend to a uniform distribution after a relaxation time scaling as N^{1/d}
- The Nyquist-Shannon criterion constrains the smallest resolvable phase-space structures in discrete data
- Explains rapid relaxation in galaxies and star clusters without subjective coarse-graining

## Abstract

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.