Ultrametric Cantor sets and Origin of Anomalous Diffusion

TL;DR

This paper demonstrates that anomalous diffusion and complex patterns can naturally emerge from ordinary diffusion equations when interpreted within a scale-invariant, nonarchimedean framework, particularly on ultrametric Cantor sets.

Contribution

It introduces a novel formalism that endows real numbers with a nonarchimedean structure, providing a new perspective on the origin of anomalous diffusion and complex patterns.

Findings

Anomalous mean square fluctuations arise from a scale-invariant interpretation of the diffusion equation.

Diffusion on ultrametric Cantor sets is inherently subdiffusive with a sublinear mean square deviation.

A new interpretation of complex pattern emergence from simple systems is proposed.

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 instead 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 seems to offer a new interpretation of a possible emergence of complex patterns from an apparently simple system.