Natural Occurrence of Fractional Derivatives in Physics

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.