Extensive analytical and numerical investigation of the kinetic and stochastic Cantor set

TL;DR

This paper analytically and numerically explores the kinetic and stochastic variants of the Cantor set, revealing their fractal dimensions and sum properties, and providing exact algorithms for their dynamics.

Contribution

It introduces exact algorithms for kinetic and stochastic Cantor sets and demonstrates their properties, including fractal dimensions and sum rules, with analytical and numerical validation.

Findings

Kinetic Cantor set has the same fractal dimension as the classical set.

Stochastic Cantor set has a lower fractal dimension of approximately 0.56155.

Sum of the $d_f$th power of all intervals equals the initial size, regardless of the variant.

Abstract

We investigate, both analytically and numerically, the kinetic and stochastic counterpart of the triadic Cantor set. The generator that divides an interval either into three equal pieces or into three pieces randomly and remove the middle third is applied to only one interval, picked with probability proportional to its size, at each generation step in the kinetic and stochastic Cantor set respectively. We show that the fractal dimension of the kinetic Cantor set coincides with that of its classical counterpart despite the apparent differences in the spatial distribution of the intervals. For the stochastic Cantor set, however, we find that the resulting set has fractal dimension $d_f=0.56155$ which is less than its classical value $d_f={{\ln 2}\over{\ln 3}}$. Nonetheless, in all three cases we show that the sum of the $d_f$th power, $d_f$ being the fractal dimension of the respective set, of all the intervals at all time is equal to one or the size of the initiator $[0,1]$ regardless of whether it is recursive, kinetic or stochastic Cantor set. Besides, we propose exact algorithms for both the variants which can capture the complete dynamics described by the rate equation used to solve the respective model analytically. The perfect agreement between our analytical and numerical simulation is a clear testament to that.