Anomalous diffusion of a particle in an aging medium

TL;DR

This paper investigates how a particle diffuses anomalously in an aging medium, revealing that the diffusion behavior depends on both the medium's frequency-dependent friction and its density of slow modes, characterized by new exponents.

Contribution

It introduces a generalized Langevin model incorporating aging effects through frequency-dependent parameters, linking anomalous diffusion to the medium's slow mode density.

Findings

Velocity correlation function exhibits specific asymptotic behavior.

Mean square displacement shows anomalous diffusion with exponents depending on medium properties.

Diffusion exponent is influenced by both the friction and slow mode density exponents.

Abstract

We report new results about the anomalous diffusion of a particle in an aging medium. For each given age, the quasi-stationary particle velocity is governed by a generalized Langevin equation with a frequency-dependent friction coefficient proportional to $|\omega|^{\delta-1}$ at small frequencies, with $0<\delta<2$. The aging properties of the medium are encoded in a frequency dependent effective temperature $T_{\rm eff.}(\omega)$. The latter is modelized by a function proportional to $|\omega|^\alpha$ at small frequencies, with $\alpha<0$, thus allowing for the medium to have a density of slow modes proportionally larger than in a thermal bath. Using asymptotic Fourier analysis, we obtain the behaviour at large times of the velocity correlation function and of the mean square displacement. As a result, the anomalous diffusion exponent in the aging medium appears to be linked, not only to $\delta$ as it would be the case in a thermal bath, but also to the exponent $\alpha$ characterizing the density of slow modes.