Controlling number of particles in fragmentation equations

One of the most interesting features of fragmentation models is the possibility of breaching the mass conservation principle through ‘shattering’; that is, the formation of a dust of zero-size particles. A similar phenomenon may occur in the evolution of the number of particles in the system which, to some extent, is intertwined with the evolution of its total mass. To investigate these phenomena, we consider the fragmentation equation in the space of densities yielding both a finite number of particles and a finite mass of the ensemble, and show, in particular, that in a non-shattering fragmentation one can typically control the total number of particles in the system. On the other hand, both mass and particles in the shattering fragmentation can disappear from the system. It is conjectured that in such a case the fragmentation equation alone does not offer the full description of the dynamics of the problem.