Ageing in Mortal Superdiffusive L\'evy Walkers

TL;DR

This paper investigates how the finite lifetime of superdiffusive Lévy walkers affects their observed motion, revealing that stopping rates temper the mean squared displacement and impact the interpretation of diffusive behaviors.

Contribution

It introduces a framework to distinguish natural death from experimental artefacts in mortal Lévy walkers and links stopping rates to tempered diffusion processes.

Findings

Stopping rate $ heta$ tempers MSD in mortal walkers.

Invariant front velocity under time lags.

Tempering can lead to misinterpretation of superdiffusive motion.

Abstract

A growing body of literature examines the effects of superdiffusive subballistic movement pre-measurement (ageing or time lag) on observations arising from single-particle tracking. A neglected aspect is the finite lifetime of these L\'{e}vy walkers, be they proteins, cells or larger structures. We examine the effects of ageing on the motility of mortal walkers, and discuss the means by which permanent stopping of walkers may be categorised as arising from `natural' death or experimental artefacts such as low photostability or radiation damage. This is done by comparison of the walkers' mean squared displacement (MSD) with the front velocity of propagation of a group of walkers, which is found to be invariant under time lags. For any running time distribution of a mortal random walker, the MSD is tempered by the stopping rate $\theta$. This provides a physical interpretation for truncated heavy-tailed diffusion processes and serves as a tool by which to better classify the underlying running time distributions of random walkers. Tempering of aged MSDs raises the issue of misinterpreting superdiffusive motion which appears Brownian or subdiffusive over certain time scales.