Change of scaling and appearance of scale-free size distribution in aggregation kinetics by additive rules

Abstract

The idealized general model of aggregate growth is considered on the basis of the simple additive rules that correspond to a one-step aggregation process. The two idealized cases were analytically investigated and simulated by the Monte Carlo method in the Desktop Grid distributed computing environment to analyze “pile-up” and “wall” cluster distributions in different aggregation scenarios. Several aspects of aggregation kinetics (change of scaling, change of size distribution type, and appearance of scale-free size distribution) driven by the “zero cluster size” boundary condition were determined by the analysis of evolving cumulative distribution functions. The “pile-up” case with a minimum active surface (singularity) could imitate piling up aggregations of dislocations, and the case with a maximum active surface could imitate arrangements of dislocations in walls. The change of scaling law (for pile-ups and walls) and availability of scale-free distributions (for walls) were analytically shown and confirmed by scaling, fitting, moment, and bootstrapping analyses of simulated probability density and cumulative distribution functions. The initial “singular” symmetric distribution of pile-ups evolves by the “infinite” diffusive scaling law and later it is replaced by the other “semi-infinite” diffusive scaling law with asymmetric distribution of pile-ups. In contrast, the initial “singular” symmetric distributions of walls initially evolve by the diffusive scaling law and later it is replaced by the other ballistic (linear) scaling law with scale-free exponential distributions without distinctive peaks. The conclusion was made as to possible applications of such an approach for scaling, fitting, moment, and bootstrapping analyses of distributions in simulated and experimental data.