Dispersion of scale-invariant size-distribution functions

It is shown that the size-distribution function of nanostructures, which obeys scale invariance (scaling), always displays a dispersion proportional to the square of the average size. Theoretical analysis is carried out for the shape and dispersion of a biparametric scale-invariant distribution describing the growth of homogeneous or heterogeneous nanoobjects with monomer capture rates proportional to their size. It is found that the Gaussian distribution is replaced by a broader and asymmetric distribution upon a decrease in the nucleation probability.