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Diffusion in a Class of Fractal Sets

Dhurjati Prasad Datta, Santanu Raut, Anuja Ray Chaudhuri

Abstract


The anomalous mean square fluctuations are shown to arise naturally from the ordinary diffusion equation interpreted scale invariantly in a formalism endowing real numbers with a nonarchimedean multiplicative structure. A variable t approaching 0 linearly in the ordinary analysis is shown to enjoy a sublinear t log t^−1 flow in the presence of this scale invariant structure. Diffusion on an ultrametric Cantor set is also generically subdiffusive with the above sublinear mean square deviation. The present study is likely to offer a new explanation of the origin of anomalous diffusive motion in Cantor set like fractal sets. Further, it also offers a novel mechanism of realizing a complex structure from a relatively simple initial
system.

Keywords


Anomalous diffusion, Cantor set, Scale invariant, Nonarchimedean

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