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We investigate, both analytically and numerically, the kinetic and stochasticcounterpart of the triadic Cantor set. The generator that divides an intervaleither into three equal pieces or into three pieces randomly and remove themiddle third is applied to only one interval, picked with probabilityproportional to its size, at each generation step in the kinetic and stochasticCantor set respectively. We show that the fractal dimension of the kineticCantor set coincides with that of its classical counterpart despite theapparent differences in the spatial distribution of the intervals. For thestochastic Cantor set, however, we find that the resulting set has fractaldimension $d_f=0.56155$ which is less than its classical value $d_f={{\ln2}\over{\ln 3}}$. 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 allthe 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 capturethe complete dynamics described by the rate equation used to solve therespective model analytically. The perfect agreement between our analytical andnumerical simulation is a clear testament to that.

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