Anomalous diffusion of a particle in an aging medium

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 ωδ−1 at small frequencies, with 0<δ<2. The aging properties of the medium are encoded in a frequency-dependent effective temperature Teff.(ω). The latter is modelized by a function proportional to ωα at small frequencies, with α<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 δ as it would be the case in a thermal bath, but also to the exponent α characterizing the density of slow modes.