Scaled Brownian motion as a mean-field model for continuous-time random walks

Abstract

We consider scaled Brownian motion (sBm), a random process described by a diffusion equation with explicitly time-dependent diffusion coefficient D(t)=αD0tα-1 (Batchelor's equation) which, for α<1, is often used for fitting experimental data for subdiffusion of unclear genesis. We show that this process is a close relative of subdiffusive continuous-time random walks and describes the motion of the rescaled mean position of a cloud of independent walkers. It shares with subdiffusive continuous-time random walks its nonstationary and nonergodic properties. The nonergodicity of sBm does not however go hand in hand with strong difference between its different realizations: its heterogeneity ("ergodicity breaking") parameter tends to zero for long trajectories.