Ergodicity in a two-dimensional self gravitating many body system

TL;DR

This study investigates the ergodic behavior of a two-dimensional self-gravitating system through molecular dynamics simulations, revealing that ergodicity takes a very long time to establish, especially without a Kac factor, and depends on system size.

Contribution

It provides a comprehensive analysis of ergodicity in 2D self-gravitating systems using multiple tests and compares results with short-range and mean-field models, highlighting the impact of the Kac factor.

Findings

Ergodicity time is very long in 2D self-gravitating systems.

With a Kac factor, ergodicity time becomes independent of particle number.

Without a Kac factor, ergodicity time diverges as √N.

Abstract

We study the ergodic properties of a two-dimensional self-gravitating system using molecular dynamics simulations. We apply three different tests for ergodicity: a direct method comparing the time average of a particle momentum and position to the respective ensemble average, sojourn times statistics and the dynamical functional method. For comparison purposes they are also applied to a short-range interacting system and to the Hamiltonian mean-field model. Our results show that a two-dimensional self-gravitating system takes a very long time to establish ergodicity. If a Kac factor is used in the potential energy, such that the total energy is extensive, then this time is independent of particle number, and diverges with $\sqrt{N}$ without a Kac factor.