Ergodic properties and equilibrium of one-dimensional self-gravitating systems

Abstract

Recent studies of one-dimensional self-gravitating systems have raised new questions about their ergodic properties, what defines equilibrium for these systems, and their ability to reach a state of thermal equilibrium in a finite time. Earlier studies of small-$N$ systems $(N<11)$ using Lyapunov exponents have shown that stable regions exist in the phase space which prevent these systems from thermalizing. Here we investigate several small-$N$ systems with specific initial states in an attempt to answer some of the questions of ergodicity and relaxation toward equilibrium which have been sparked by recent large-$N$ $(N=64)$ simulations. Using time averages of the specific particle energy deviations from equipartition, we see similar peaks occurring in the data for small-$N$ simulations as have been reported for large $N.$ Instead of being an indication of the onset of equilibrium, these peaks may indicate regions of the phase space where the system resides for extremely long periods of time. The existence of sticky regions in the phase space in both small- and large-$N$ systems raises questions about the structure of the phase space, relaxation, and the appropriateness of various tests of equilibrium. Here we show that equipartition is not sufficient to remove fundamental doubts concerning the system's ergodic properties.