Self-similar size distribution functions

The resulting steady-state size distribution functions were compared with experiments, and it was demonstrated that size distributions were indeed self-similar, and the functionality deduced in the development of the model was observed. 

For instance, in the SLSP model, a HSR may be obtained by taking into account both the self-similarity of the percolating cluster and the scale invariance of the cluster size distribution function (71). By utilizing the renormalization procedure [213], in which the size L of the lattice changes to a new size Lc with a scaling coefficient b = Lc/L, the relationship between the distribution function of the original lattice and the lattice with the adjusted size can be presented as w(s, sm) = bd nDw(s, sm), where s = s D and sm = smD. 

Substituting the self-similar form for the size distribution function, (5.23), we obtain... 

The median diameter of limestone pellets in the ball growth region as a function of the granulation time on log-log scales is shown in Fig. 20. The corresponding size distributions are self-similar as illustrated in Fig. 21. 

However, there are essential differences between the Lorentz gas model and IS structures in many-dimensional molecular systems. The most obvious difference is the size distribution of basins. In the Lorentz gas model, the size of unit cells is identical but the basin size, as well as the depth of potentials, of molecular systems is believed to range quite broadly, and possibly distributed in a self-similar way, reflecting that local potential minima increase exponentially as a function of the system size. As a result, one expects that the diffusion among multibasin structures bears different characters. The situation of the latter is... 

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