Anomalous diffusion: A dynamic perspective

Abstract

This paper investigates whether spontaneous, stationary velocity fluctuations can lead to deviations from the regular Fickian diffusion. A kinematic analysis reveals that anomalous diffusion, both fast and slow, arises from long-tailed velocity auto-correlation functions (VACF). This infinite span of interdependence of the random velocity leads to the breakdown of the central limit theorem for particle displacements. A generalized Langevin equation, which features a retarded friction, has been used to describe the particle dynamics in the long-time limit. The analysis reveals that simple power-law decay models for the friction kernel are adequate to yield the pathological VACFs which imply anomalous diffusion. The fluctuation dissipation theorem is invoked to infer that a fractional noise gives rise to anomalous diffusion. Such a Langevin equation represents a mean-field description of disorder effects and the friction kernel then becomes a constitutive property of the medium.