Exponents of the grain size distributions for three different kernels as function of the fragment distribution index. The plot shows the simulation results (solid lines), the analytic solution for the growth cascade (case A, dotted lines), the intermediate regime (case B, dashed lines), and the fragment dominated regime (case C, dash-dotted lines). The upper panel was calculated with a (ν = 5/6,γ = 0)-kernel (i.e. turbulent relative velocities, see Sect. 4.1), the middle panel with a (ν = 1/3,γ = 1/3)-kernel and the lower panel with a (ν = 1/6,γ =  − 1/2)-kernel (i.e. Brownian motion relative velocities). In the grey shaded area, Eq. (20) is not fulfilled, and the size distribution is not a power-law.