Gravity in one dimension: The critical population

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Abstract
The failure of a one-dimensional gravitational system to relax to equilibrium on predicted time scales has raised questions concerning the ergodic properties of the dynamics. A failure to approach equilibrium could be caused by the segmentation of the phase space into isolated regions from which the system cannot escape. In general, each region may have distinct ergodic properties. By numerically investigating the stability of two classes of periodic orbits for the N-body system in a previous work [Phys. Rev. A 46, 837 (1992)], we demonstrated that phase-space segmentation occurred when N≤10. Tentative results suggested that segmentation also occurred for 11≤N≤20. Here this work has been refined. Based on calculations of Lyapunov characteristic numbers, we argue that segmentation disappears and the system is both ergodic and mixing for N≥11, the critical population.

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