The Origin of Fractal Distribution in Self-Gravitating Virialized System and Self-Organized Criticality

Fractal structure and non-Gaussian velocity distribution are characteristic and universal properties in the self-gravitating virialized system such as galaxies or interstellar molecular clouds. Using one-dimensional ring model in which $N$ particles constrained on a circular ring are interacting by three-dimensional (3-D) gravitating force, we examine the origin of these characteristic properties from the viewpoint of the gravitational phase transition. We find that a virialized state with negative specific heat appears at the intermediate energy scale, where a phase transition occurs and a cluster is formed. Classifying the particles in each state into three phases (core, halo and gas phases), we can characterize this virialized state by non-thermal property of the particles in \halo phase. Although the relaxation time of the particles in the \core phase is determined by the free-fall time, no typical time scale appears for the particles in the \halo phase. As a result, the relaxation time of \halo particles becomes very long comparing to that of \core particles, then non-thermal properties are expected in a quasi-equilibrium state. In fact, non-Gaussian velocity distribution and fractal structure are found for \halo particles. The fractal dimension is independent of the total energy. These results suggest that a scale free criticality in the gravitational interaction could be the main origin to cause the universal properties such as fractal structure and non-Gaussian velocity distribution in the 3-D self-gravitating system.