Aging properties of an anomalously diffusing particule

We report new results about the two-time dynamics of an anomalously diffusing classical particle, as described by the generalized Langevin equation with a frequency-dependent noise and the associated friction. The noise is defined by its spectral density proportional to ωδ−1 at low frequencies, with 0<δ<1 (subdiffusion) or 1<δ<2 (superdiffusion). Using Laplace analysis, we derive analytic expressions in terms of Mittag–Leffler functions for the correlation functions of the velocity and of the displacement. While the velocity thermalizes at large times (slowly, in contrast to the standard Brownian motion case δ=1), the displacement never attains equilibrium: it ages. We thus show that this feature of normal diffusion is shared by a subdiffusive or superdiffusive motion. We provide a closed form analytic expression for the fluctuation–dissipation ratio characterizing aging.