Is subdiffusional transport slower than normal?

TL;DR

This paper demonstrates that subdiffusive transport in viscoelastic media, while appearing slower transiently, actually results in larger instantaneous displacements than normal diffusion, challenging the notion of 'ultra-slowness' in biological systems.

Contribution

It shows that subdiffusive transport can be faster than normal diffusion at each instant, despite its slower scaling over time, due to viscoelastic memory effects.

Findings

Transient subdiffusion occurs for t<τ.

Mean displacement is always larger than in normal diffusion.

Implications for biological transport and reactions.

Abstract

We consider anomalous non-Markovian transport of Brownian particles in viscoelastic fluid-like media with very large but finite macroscopic viscosity under the influence of a constant force field F. The viscoelastic properties of the medium are characterized by a power-law viscoelastic memory kernel which ultra slow decays in time on the time scale \tau of strong viscoelastic correlations. The subdiffusive transport regime emerges transiently for t<\tau. However, the transport becomes asymptotically normal for t>>\tau. It is shown that even though transiently the mean displacement and the variance both scale sublinearly, i.e. anomalously slow, in time, <\delta x(t)> ~ F t^\alpha, <\delta x^2(t)> ~ t^\alpha, 0<\alpha<1, the mean displacement at each instant of time is nevertheless always larger than one obtained for normal transport in a purely viscous medium with the same macroscopic viscosity obtained in the Markovian approximation. This can have profound implications for the subdiffusive transport in biological cells as the notion of "ultra-slowness" can be misleading in the context of anomalous diffusion-limited transport and reaction processes occurring on nano- and mesoscales.