Fig. 2. Numerical simulation of the CTRW, i.e., Brownian motion with power-law waiting times.

The simulations were performed for α = 0.7 and 10,000 realizations were obtained. a The MSD shows subdiffusive behavior 〈x²(t)〉 ~ t^α, while a linear regression of $\log \left(\mathrm{MSD}\right)$ vs. $\log \left(t\right)$ indicates 〈x²(t)〉 ~ t^0.69. The times and displacements are unitless, i.e., the simulation sampling time is 1. b PSD at five different measurement times exhibits aging. The power law asymptotic S ~ ω⁻² is indicated with a dashed line. The arrow shows the decay in the PSD with measurement time t_m. c The amplitude A(t_m) of the PSD, where 〈S〉 = A(t_m)/ω², shows $A\left(t_{\mathrm{m}}\right)~t_{\mathrm{m}}^{-0.31}$, highlighting the aging effect, in excellent agreement with theory which predicts $A\left(t_{\mathrm{m}}\right)~t_{\mathrm{m}}^{-\left(1-\alpha \right)}$.