Gravity in one dimension: the evolution of large and small N systems

Abstract

The search for relaxation to equilibrium (thermalization) of one-dimensional, self-gravitating systems has been investigated via theory and computer simulation for the past three decades. The casual reader of these investigations may be easily confused by the abundance of contradictory conjectures that have appeared in the recent literature. These conjectures have been examined in this research and new statistical tools and procedures to test for equilibrium have been developed. This research examines the problem by simulating the evolution of large and small N systems, where N is the system population. For large N systems (N = 100) procedures including correlations in time, chi-square and central limit theorem statistics applied to Rybicki's (1971) equilibrium density functions, and ensemble averaging have been employed. For small N systems (3 $\leq$ N $\leq$ 20) the stability of selected periodic modes has been examined by calculating their Lyapunov characteristic numbers. For large N systems it may be concluded from this research that thermalization has never really been observed. A system may enter a state that mimics equilibrium but slowly drifts away from it. This state is referred to as "quasi-equilibrium" and has been shown to have long-term, weak correlations in positions and velocities. Systems with N $\leq$ 10 are definitely not ergodic as shown by the stability of selected periodic orbits. Stability appeared to decrease logarithmically with increasing N $\leq$ 10. For N $\geq$ 11 any perturbation caused instability in the periodic orbits. However, the failure of the Lyapunov characteristic numbers to converge to a single positive value from different modes for a given N suggests that the energy surface in phase space may be segmented. Thus, one-dimensional, self-gravitating systems may only be merely ergodic which would require a time t $\to$ $\infty$ for a system to explore its entire energy surface on the phase space. This would account for the fact that relaxation to equilibrium has never been seen.