Generalized Model of Migration-Driven Aggregate Growth - Asymptotic Distributions, Power Laws and Apparent Fractality

TL;DR

This paper develops a generalized model for migration-driven aggregation, deriving exact solutions and asymptotic distributions that connect Weibull, normal, exponential, and power-law behaviors, with implications for experimental analysis.

Contribution

It introduces a unified framework linking various statistical distributions in aggregation kinetics and confirms findings through simulations without simplifying assumptions.

Findings

Exact solutions for unbiased aggregation match asymptotic distributions.

Distribution transitions depend on the exponent nd include Weibull, normal, exponential, and power laws.

Simulation results validate the theoretical predictions and enable parameter estimation from experimental data.

Abstract

The rate equation for exchange-driven aggregation of monomers between clusters of size $n$ by power-law exchange rate ($\sim{n}^\alpha$), where detaching and attaching processes were considered separately, is reduced to Fokker-Planck equation. Its exact solution was found for unbiased aggregation and agreed with asymptotic conclusions of other models. Asymptotic transitions were found from exact solution to Weibull/normal/exponential distribution, and then to power law distribution. Intermediate asymptotic size distributions were found to be functions of exponent $\alpha$ and vary from normal ($\alpha=0$) through Weibull ($0<\alpha<1$) to exponential ($\alpha=1$) ones, that gives the new system for linking these basic statistical distributions. Simulations were performed for the unbiased aggregation model on the basis of the initial rate equation without simplifications used for reduction to Fokker-Planck equation. The exact solution was confirmed, shape and scale parameters of Weibull distribution (for $0<\alpha<1$) were determined by analysis of cumulative distribution functions and mean cluster sizes, which are of great interest, because they can be measured in experiments and allow to identify details of aggregation kinetics (like $\alpha$). In practical sense, scaling analysis of \emph{evolving series} of aggregating cluster distributions can give much more reliable estimations of their parameters than analysis of \emph{solitary} distributions. It is assumed that some apparent power and fractal laws observed experimentally may be manifestations of such simple migration-driven aggregation kinetics even.