Time-space fabric underlying anomalous diffusion

TL;DR

This paper explores the fundamental time-space structure behind anomalous diffusion, proposing new fractional quantum relationships and a Hausdorff derivative approach, leading to novel models and solutions for anomalous transport phenomena.

Contribution

It introduces a new conceptual framework linking fractal spacetime to anomalous diffusion, deriving fractional quantum relations and a Hausdorff derivative-based diffusion equation.

Findings

Derived fractional quantum relationships between energy, frequency, momentum, and wavenumber.

Proposed a Hausdorff derivative-based anomalous diffusion equation.

Obtained a new stretched Gaussian distribution as the Green's function solution.

Abstract

This study unveils the time-space transforms underlying anomalous diffusion process. Based on this finding, we present the two hypotheses concerning the effect of fractal time-space fabric on physical behaviors and accordingly derive fractional quantum relationships between energy and frequency, momentum and wavenumber which further give rise to fractional Schrodinger equation. As an alternative modeling approach to the standard fractional derivatives, we introduce the concept of the Hausdorff derivative underlying the Hausdorff dimensions of metric spacetime. And in terms of the proposed hypotheses, the Hausdorff derivative is used to derive a linear anomalous transport-diffusion equation underlying anomalous diffusion process. Its Green's function solution turns out to be a new type of stretched Gaussian distribution and is compared with that from the Richardson's diffusion equation.