Due to the apparent ease with which they can be numerically simulated, one-dimensional gravitational systems were first introduced by astronomers to explore different modes of gravitational evolution. These include violent relaxation and the approach to thermal equilibrium. Careful work by dynamicists and statistical physicists has shown that several claims made by astronomers regarding these models were incorrect. Unusual features of the evolution include the development of long lasting structures on large scales, which can be thought of as one-dimensional analogs of Jupiter's red spot or a galactic spiral density wave or bar. The existence of these structures demonstrates that in gravitational systems evolution is not entirely dominated by the second law of thermodynamics and also appears to contradict the Arnold diffusion ansatz. Thus it is correct to assert that the one-dimensional planar sheet gravitational system is the nonextensive analog of the Fermi-Pasta-Ulam model of dynamical systems. This paper is an extension of a preliminary study where we conclusively showed mass segregation and equipartition of kinetic energy in a two-mass planar sheet system for the first time. Here we employ both mean-field theory and dynamical simulation to more thoroughly probe the statistical and ergodic properties of these systems. Valuable information is obtained from local and global time averaging, and temporal and spatial correlation functions. Using these tools we show that the system appears to approach the equilibrium distribution on very long time scales, but the relaxation is incomplete.

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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 (NϽ11) 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 (Nϭ64) 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.

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