Localization and universal fluctuations in ultraslow diffusion processes

TL;DR

This paper compares ultraslow diffusion models, revealing universal fluctuation behaviors and non-ergodic properties, and connects Golosov localization with non-ergodicity in complex systems.

Contribution

It demonstrates the universality of fluctuations in ultraslow diffusion processes and links Golosov localization to non-ergodic behavior in these systems.

Findings

Logarithmic mean squared displacement in ultraslow diffusion.

Universal exponential suppression of mobile trajectories.

Connection between Golosov localization and non-ergodicity.

Abstract

We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) $\langle x^2(t)\rangle\simeq\log^{\gamma}t$. Comparison of annealed continuous time random walks (CTRWs) with logarithmic waiting time distribution $\psi(\tau)\simeq1/(\tau\log^{1+\gamma}\tau)$ and Sinai diffusion in quenched random landscapes shows striking similarities, despite their very different physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time and ensemble averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble and time averaged MSD.