Emergence of skew distributions in controlled growth processes

TL;DR

This paper derives a master equation-based model for evolving systems, showing how different growth, division, and production mechanisms lead to skew distributions like power-law, log-normal, and Weibull, supported by numerical simulations.

Contribution

It introduces a unified framework to derive various skew distributions from controlled growth processes starting from a master equation.

Findings

Power-law distributions emerge from uniform size production.

Log-normal distributions result from size-dependent production without new element creation.

Weibull distributions arise from binary fission processes.

Abstract

Starting from a master equation, we derive the evolution equation for the size distribution of elements in an evolving system, where each element can grow, divide into two, and produce new elements. We then probe general solutions of the evolution quation, to obtain such skew distributions as power-law, log-normal, and Weibull distributions, depending on the growth or division and production. Specifically, repeated production of elements of uniform size leads to power-law distributions, whereas production of elements with the size distributed according to the current distribution as well as no production of new elements results in log-normal distributions. Finally, division into two, or binary fission, bears Weibull distributions. Numerical simulations are also carried out, confirming the validity of the obtained solutions.