Emergence of fractal behavior in condensation-driven aggregation

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 $c(x,t)\sim t^{-\beta }\varphi (x/t^z)$ accompanied by the emergence of fractal of dimension $d_f=\frac{1}{1+2\alpha }$. 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 $\beta =(1+d_f)z$.

8 pages, 6 figures, to appear in Phys. Rev. E

References in corpus (6)

• Droplets nucleation, and Smoluchowski's equation with growth and injection of particles
• Fractal dimension and degree of order in sequential deposition of mixture
• Multifractality and multiscaling in two dimensional fragmentation
• Smoluchowski's equation for cluster exogenous growth
• Condensation-Driven Aggregation in One Dimension
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Cited by in corpus (3)

• Emergence of fractal in aggregation with stochastic self-replication
• Dyadic Cantor set and its kinetic and stochastic counterpart
• Dynamic scaling and stochastic fractal in nucleation and growth processes