Emergence of fractal behavior in condensation-driven aggregation

TL;DR

This paper presents an exactly solved model of condensation-driven aggregation, revealing fractal structures with a conserved moment and establishing scaling relations between growth dynamics and fractal dimension.

Contribution

The study introduces a novel exactly solvable model demonstrating fractal formation and a non-trivial conserved moment in condensation-driven aggregation.

Findings

Particle size spectra exhibit dynamic scaling with fractal dimension d_f.

The fractal dimension d_f is given by 1/(1+2α).

A conserved moment of order d_f remains invariant over time.

Abstract

We investigate a model in which an ensemble of chemically identical Brownian particles are continuously growing by condensation and at the same time undergo irreversible aggregation whenever two particles come into contact upon collision. We solved the model exactly by using scaling theory for the case whereby a particle, say of size $x$, grows by an amount $\alpha x$ over the time it takes to collide with another particle of any size. It is shown that the particle size spectra of such system exhibit transition to dynamic scaling $c(x,t)\sim t^{-\beta}\phi(x/t^z)$ accompanied by the emergence of fractal of dimension $d_f={{1}\over{1+2\alpha}}$. One of the remarkable feature of this model is that it is governed by a non-trivial conservation law, namely, the $d_f^{th}$ moment of $c(x,t)$ is time invariant regardless of the choice of the initial conditions. The reason why it remains conserved is explained by using a simple dimensional analysis. We show that the scaling exponents $\beta$ and $z$ are locked with the fractal dimension $d_f$ via a generalized scaling relation $\beta=(1+d_f)z$.