Aging properties of an anomalously diffusing particle

Abstract

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 ${\omega }^{\delta -1}$ at low frequencies, with $0<\delta <1$ (subdiffusion) or $1<\delta <2$ (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 $\delta =1$), 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.

15 pages

References in corpus (2)

• Fractional Langevin equation
• Aging effects in the quantum dynamics of a dissipative free particle: non-ohmic case

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