1104.5381 (Yu. G. Gordienko)

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

Yu. G. Gordienko

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.

View original: http://arxiv.org/abs/1104.5381