Short note on conditional collapse of self-gravitating system with positive total energy

In this study, we first show a finite time collapse of a three-dimensional Smoluchowski-Poisson system in the micro-canonical ensemble (i.e., μSP system) with positive total energy. Using a precise estimation of the bound of the gravitational contribution to total energy, we provide blow-up conditions on the initial state of the μSP system. The conditions indicate that highly clustered self-gravitating particles in the μSP system induce a finite time collapse, even for positive total energy.