Emergence of fractal behavior in condensation-driven aggregation.

We investigate the condensation-driven aggregation model that we recently proposed whereby an initial ensemble of chemically identical Brownian particles are continuously growing by condensation and at the same time undergo aggregation upon collision. We solved the model exactly by using scaling theory for the case when a particle, say of size x , grows by an amount alphax over the time it takes to collide with another particle of any size. It is shown that the particle size spectra exhibit transition to scaling c(x,t) approximately t;{-beta}varphi(xt{z}) accompanied by the emergence of a fractal of dimension d {f}=1/(1+2alpha) . A remarkable feature of this model is that it is governed by a nontrivial conservation law, namely, the d {f}th moment of c(x,t) is time invariant. The reason why it remains conserved is explained. Exact values for the exponents beta , z , and d{f} are obtained and it is shown that they obey a generalized scaling relation beta=(1+d{f})z .