Origin of scaling structure and non-Gaussian velocity distribution in a self-gravitating ring model

Fractal structures and non-Gaussian velocity distributions are characteristic properties commonly observed in virialized self-gravitating systems, such as galaxies and interstellar molecular clouds. We study the origin of these properties using a one-dimensional ring model that we propose in this paper. In this simple model, N particles are moving, on a circular ring fixed in three-dimensional space, with mutual interaction of gravity. This model is suitable for the accurate symplectic integration method by which we argue the phase transition in this system. Especially, in between the extended phase and the collapsed phase, we find an interesting phase (halo phase) that has negative specific heat at the intermediate energy scale. Moreover, in this phase, there appear scaling properties and nonthermal and non-Gaussian velocity distributions. In contrast, these peculiar properties are never observed in other gas and core phases. Particles in each phase have a typical time scale of motions determined by the cutoff length $\xi ,$ the ring radius R, and the total energy E. Thus all relaxation patterns of the system are determined by these three time scales.