Viscoelastic subdiffusion: from anomalous to normal

TL;DR

This paper investigates viscoelastic subdiffusion in various potentials, revealing slow transition kinetics, the limits of non-Markovian rate theory, and the ergodic nature of potential-free subdiffusion, with implications for understanding anomalous diffusion behaviors.

Contribution

It provides a detailed analysis of viscoelastic subdiffusion dynamics, highlighting the conditions under which non-Markovian rate theory applies and contrasting it with trap-based subdiffusion mechanisms.

Findings

Transition kinetics are stretched-exponential for moderate barriers.

Non-Markovian rate theory approximates well at high barriers.

Potential-free subdiffusion is ergodic and insensitive to barrier height asymptotically.

Abstract

We study viscoelastic subdiffusion in bistable and periodic potentials within the Generalized Langevin Equation approach. Our results justify the (ultra)slow fluctuating rate view of the corresponding bistable non-Markovian dynamics which displays bursting and anti-correlation of the residence times in two potential wells. The transition kinetics is asymptotically stretched-exponential when the potential barrier $V_0$ several times exceeds thermal energy $k_BT$ ($V_0\sim 2 - 10 k_BT$) and it cannot be described by the non-Markovian rate theory (NMRT). The well-known NMRT result approximates, however, ever better with the increasing barrier height, the most probable logarithm of the residence times. Moreover, the rate description is gradually restored when the barrier height exceeds a fuzzy borderline which depends on the power law exponent of free subdiffusion $\alpha$. Such a potential-free subdiffusion is ergodic. Surprisingly, in periodic potentials it is not sensitive to the barrier height in the long time asymptotic limit. However, the transient to this asymptotic regime is extremally slow and it does profoundly depend on the barrier height. The time-scale of such subdiffusion can exceed the mean residence time in a potential well, or in a finite spatial domain by many orders of magnitude. All these features are in sharp contrast with an alternative subdiffusion mechanism involving jumps among traps with the divergent mean residence time in these traps.