Change of Scaling and Appearance of Scale-Free Size Distribution in Aggregation Kinetics by Additive Rules
The idealized general model of aggregate growth is considered on the basis of the simple additive rules that correspond to one-step aggregation process. The two idealized cases were analytically investigated and simulated by 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 "zero cluster size" boundary condition were determined by analysis of evolving cumulative distribution functions. The "pile-up" case with a \textit{minimum} active surface (singularity) could imitate piling up aggregations of dislocations, and the case with a \textit{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" \textit{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 \textit{asymmetric} distribution of pile-ups. In contrast, the initial "singular" \textit{symmetric} distributions of walls initially evolve by the diffusive scaling law and later it is replaced by the other ballistic (linear) scaling law with \textit{scale-free} exponential distributions without distinctive peaks. The conclusion was made as to possible applications of such approach for scaling, fitting, moment, and bootstrapping analyses of distributions in simulated and experimental data.