Emergence of fractal in aggregation with stochastic self-replication

TL;DR

This paper introduces a model for Brownian particle aggregation with stochastic self-replication, revealing conditions for fractal formation, deriving the fractal dimension, and confirming findings through exact solutions, scaling theory, and numerical simulations.

Contribution

It presents an exact solution and scaling theory for aggregation with self-replication, establishing conditions for fractal emergence and linking fractal dimension to conserved quantities.

Findings

Particle size distribution exhibits dynamic scaling

Fractal dimension of the system is analytically derived

Numerical simulations confirm theoretical predictions

Abstract

We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings. In particular, we show analytically that the particle size distribution function exhibits dynamic scaling and we verified it numerically using the idea of data-collapse. Besides, the conditions under which the resulting system emerges as a fractal are found, the fractal dimension of the system is given and the relationship between this fractal dimension and a conserved quantity is pointed out.