  Fractal structures and non-Gaussian velocity distributions are characteristic
properties commonly observed in virialized self-gravitating systems such as
galaxies or interstellar molecular clouds. We study the origin of these
properties using the one-dimensional ring model which we newly propose in this
paper. In this simple model, $N$ particles are moving, on a circular ring fixed
in the three dimensional space, with mutual interaction of gravity. This model
is suitable for accurate symplectic integration method by which we find the
phase transition in this system from extended-phase to collapsed-phase through
an interesting phase (\halo-phase) which has negative specific heat. In this
intermediate energy scale, there appear scaling properties, non-thermal and
non-Gaussian velocity distributions. In contrast, these peculiar properties are
never observed in other \gas and \core phases. Particles in each phase have
typical time scales of motion 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.
