Ultraslow diffusion revisited: Logarithmic scaling in single-term fractional diffusion models for anomalous transport of complex systems

Abstract

This study establishes rigorous ties between ultraslow diffusion dynamics and single-term Caputo–Hadamard time-fractional diffusion equations via mean square displacement (MSD) analysis. We derive the explicit logarithmic scaling law $\mathrm{MSD}\left(t\right)\sim {(logt)}^{\alpha }$ for constant-order Hadamard diffusion equations and verify this functional relationship through systematic numerical simulations. The revealed ${(logt)}^{\alpha }$ scaling definitively departs from the classical $t^{\alpha }$ behavior characteristic of Caputo models, establishing a streamlined framework for modeling anomalous transport in mesoscopic complex systems.