We consider a birth and growth model for crystallization processes in d space dimensions, where growth is driven by the gradient of the concentration. A nonlinear condition for the concentration is given on the boundary and a multi-front moving boundary problem arises.We propose a new formulation based on the Schwartz distributions by coupling the growth of the crystals and the diffusion of the concentration. We complete the deterministic growth model by considering stochastic nucleations in space and time. The coupling of the growth dynamics with the evolution of the underlying field of the concentration of matter finally causes the stochastic geometry of the crystals.