Age representation of Levy walks: partial density waves, relaxation and first passage time statistics

TL;DR

This paper investigates how aging affects Levy walks using a hyperbolic partial density wave framework, revealing anomalous relaxation and first passage time behaviors, and introduces the concept of Levy walks with wearing to model mobility loss.

Contribution

It develops a hyperbolic age formulation for Levy walks, enabling simple integral representations and analysis of aging effects on relaxation and first passage times.

Findings

Normal diffusive Levy walks can show stretched exponential relaxation.

Aging significantly impacts first passage time statistics.

The concept of Levy walks with wearing models mobility losses.

Abstract

Levy walks (LWs) define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks. Alternatively, there is a hyperbolic representation of them in terms of partial probability density waves. Using the latter framework we explore the impact of aging on LWs, which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution. We show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear Volterra equations for any initial preparation. On this basis relaxation properties and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics. We find that even normal diffusive LWs may display anomalous relaxation properties such as stretched exponential decay. We then discuss the impact of aging on the first passage time statistics of LWs by developing the corresponding Volterra integral representation. As a further natural generalization the concept of LWs with wearing is introduced to account for mobility losses.