Transition from random to ordered fractals in fragmentation of particles in an open system

We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space, focusing on dimensional analysis. This exhibits a spectrum of mass exponents theta, whose exact numerical values are given for which x(-theta) or t(theta z) has the dimension of particle size distribution function psi(x,t), where z is the kinetic exponent. We obtained conditions for which the scaling and fragmentation process altogether breaks down, and we give an explicit scaling solution for a special case. Finally, we identify a new class of fractals ranging from random to nonrandom and show that the fractal dimension increases with increasing order and a transition to a strictly self-similar pattern occurs when randomness completely ceases.