Time-Dependent Fractional Diffusion and Friction Functions for Anomalous Diffusion

Abstract

The precise determination of diffusive properties is presented for a system described by the generalized Langevin equation. The time-dependent fractional diffusion function and the Green-Kubo relation as well as the generalized Stokes-Einstein formula, in the spirit of ensemble averages, are reconfigured. The effective friction function is introduced as a measure of the influence of frequency-dependent friction on the evolution of the system. This is applied to the generalized Debye model, from which self-oscillation emerges as indicative of ergodicity that breaks due to high finite-frequency cutoff. Moreover, several inconsistent conclusions that have appeared in the literature are revised.