Fig. 3. Power spectral density of numerical simulations of fBM with heavy-tailed immobilization times.

a Simulations for five different measurement times with α = 0.4 and H = 0.3. The number of realizations is N = 10,000. Given that the fBM is subdiffusive (H < 1/2), the PSD is predicted to scale as $⟨S\left(\omega ,t_{\mathrm{m}}\right)⟩~t_{\mathrm{m}}^{-\left(1-\alpha \right)}{\omega }^{-2+\alpha -2\alpha H}$ as in Eq. (13). The dashed line shows the scaling ω^{−2+α−2αH} and the arrow indicates the decay in the PSD as the measurement time t_m increases. b Simulations for five different measurement times with α = 0.4 and H = 0.7, N = 10,000 realizations. The fBM is superdiffusive (H > 1/2) and the PSD is, thus, predicted to scale as $⟨S\left(\omega ,t_{\mathrm{m}}\right)⟩~t_{\mathrm{m}}^{-\left(1-2\alpha H\right)}{\omega }^{-2}$ (Eq. (14)). The dashed line shows the scaling ω⁻² and the arrow shows the decay in the PSD with measurement time t_m. c Simulations for five different measurement times with α = 0.8 and H = 0.75, N = 5,000 realizations. Given that 2αH > 1, the power spectrum increases with measurement time as indicated by the arrow. The dashed black line indicates ω⁻². d The shaded region (regime III) indicates the set of values for α and H that yields a PSD 〈S(ω, t_m)〉 that increases with measurement time. In the rest of the plane, the power spectrum decays with t_m. Within this part of the plane, regime I is characterized by $⟨S\left(\omega ,t_{\mathrm{m}}\right)⟩~t_{\mathrm{m}}^{-\left(1-\alpha \right)}{\omega }^{-2+\alpha -2\alpha H}$ and regime II by $⟨S\left(\omega ,t_{\mathrm{m}}\right)⟩~t_{\mathrm{m}}^{-\left(1-2\alpha H\right)}{\omega }^{-2}$. The red crosses indicate the pairs (H, α) used in the examples in a–c, and the CTRW in Fig. 2.