Relaxation time distributions for an anomalously diffusing particle

TL;DR

This paper investigates the dynamics of anomalously diffusing particles using a generalized Langevin equation with a Mittag-Leffler memory kernel, revealing conditions under which relaxation time distributions exist for different dynamical quantities.

Contribution

It introduces a novel analysis of relaxation time distributions in anomalous diffusion, highlighting their existence for certain quantities depending on the diffusive regime.

Findings

Relaxation time distributions exist for the memory kernel and scattering function in subdiffusion.

In superdiffusion, relaxation time distributions are found for the particle's mean velocity.

The study clarifies the mathematical conditions for defining relaxation time distributions in anomalous diffusion.

Abstract

As well known, the generalized Langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle. Here, we focus attention on some new aspects of the dynamics, successively considering the memory kernel, the particle's mean velocity, and the scattering function. All these quantities are studied from a unique angle, namely, the discussion of the possible existence of a distribution of relaxation times characterizing their time decay. Although a very popular concept, a relaxation time distribution cannot be associated with any time-decreasing quantity (from a mathematical point of view, the decay has to be described by a completely monotonic function). Technically, we use a memory kernel decaying as a Mittag-Leffler function (the Mittag-Leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times). We show that, in the case of a subdiffusive motion, relaxation time distributions can be defined for the memory kernel and for the scattering function, but not for the particle's mean velocity. The situation is opposite in the superdiffusive case.