Natural Occurrence of Fractional Derivatives in Physics

TL;DR

This paper explores how fractional derivatives naturally arise in various physical phenomena characterized by power-law behaviors across different fields such as acoustics, viscoelasticity, and dielectrics, highlighting their fundamental role.

Contribution

It demonstrates the intrinsic presence of fractional derivatives in physical models with power-law responses, connecting mathematical concepts with real-world phenomena.

Findings

Power-law frequency dependence in ultrasound and acoustics explained by fractional derivatives.

Viscodynamic operators in boundary problems can be formulated with fractional derivatives.

Power-law and stretched exponential responses in dielectrics relate to fractional derivative models.

Abstract

Power laws in time and frequency appear in fields such as linear viscoelasticity and acoustics, viscous boundary layer problems, and dielectrics. This is consistent with fractional derivatives in the fundamental descriptions, since power laws in time and frequency are related by the Fourier transform, and also associated with fractional derivatives. Examples here include power-law frequency dependent attenuation in ultrasound, elastography and sediment acoustics. In viscous boundary problems there is a viscodynamic operator in the Biot poroviscoelastic theory which may be formulated with a fractional derivative. Power law and stretched exponential temporal responses of non-ideal capacitors can also be shown to relate to the Cole-Cole power-law dielectric model.