Change of Scaling and Appearance of Scale-Free Size Distribution in Aggregation Kinetics by Additive Rules

TL;DR

This paper investigates how aggregation kinetics change under additive rules, revealing the emergence of scale-free size distributions and different scaling laws through analytical and simulation methods.

Contribution

It introduces a model analyzing aggregation processes with additive rules, demonstrating the transition of size distributions and scaling laws, supported by analytical and Monte Carlo simulation results.

Findings

Change of scaling law for pile-ups and walls.

Emergence of scale-free distributions in wall aggregation.

Different evolution patterns for pile-up and wall distributions.

Abstract

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.