Occurrence of anomalous diffusion and non-local response in highly-scattering acoustic periodic media

TL;DR

This paper demonstrates that anomalous diffusive transport of acoustic waves can occur in perfectly periodic, highly-scattering media due to anisotropic bandgaps, leading to hybrid propagation characterized by heavy-tailed Lévy distributions.

Contribution

It reveals that anomalous diffusion is possible in periodic media without disorder, using a combination of models to explain this unexpected behavior.

Findings

Anomalous diffusion occurs in perfectly periodic media with anisotropic bandgaps.

Acoustic wave intensity profiles exhibit Lévy alpha-stable heavy tails.

A fractional diffusive model accurately describes the hybrid transport mechanism.

Abstract

We investigate the occurrence of anomalous diffusive transport associated with acoustic wave fields propagating through highly-scattering periodic media. Previous studies had correlated the occurrence of anomalous diffusion to either the random properties of the scattering medium or to the presence of localized disorder. In this study, we show that anomalous diffusive transport can occur also in perfectly periodic media and in the absence of disorder. The analysis of the fundamental physical mechanism leading to this unexpected behavior is performed via a combination of deterministic, stochastic, and fractional-order models in order to capture the different elements contributing to this phenomenon. Results indicate that this anomalous transport can indeed occur in perfectly periodic media when the dispersion behavior is characterized by anisotropic (partial) bandgaps. In selected frequency ranges, the propagation of acoustic waves not only becomes diffusive but its intensity distribution acquires a distinctive L\'evy alpha-stable profile having pronounced heavy-tails. In these ranges, the acoustic transport in the medium occurs according to a hybrid transport mechanism which is simultaneously propagating and anomalously diffusive. We show that such behavior is well captured by a fractional diffusive transport model whose order can be obtained by the analysis of the heavy tails.