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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 $|ω|^{δ-1}$ at small frequencies, with $0<δ<2$. The aging properties of the medium are encoded in a frequency dependent effective temperature $T_{\rm eff.}(ω)$. The latter is modelized by a function proportional to $|ω|^α$ 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.