FIG. 4. (Color online) Ergodicity breaking parameter E$_{B}$ versus overall measurement time T at given lag time Δ = 1, 100, and 1000. The dotted line depicts a power law with slope -1, representing the analytic behavior E$_{B}$ ≃ T − 1 in free space. For each Δ, the E$_{B}$ curves are drawn for different interval lengths L = 1, 3, 5, and 10. Each curve was obtained from 200 single trajectories, and the Hurst exponent was H = 1 / 4. FIG. 6. (Color online) Time-averaged mean squared displace- ment (MSD) curves versus lag time Δ for L = 1 in 1D, 2D, and 3D space (from bottom to top), with a slope Δ 2 H (dot- ted line). For each dimension, 5 trajectories were drawn with total measurement time T = 2 17 and H = 1 / 4. 7 from Eq. (4.5). However, at long lag times, we inter- estingly observe fluctuations of the correlations around a constant value, reflecting the confinement of the motion. To mimic the anomalous diffusion of particles in- side biological cells, we also simulate two- and three- dimensional FBM based on Eq. (2.7) in the presence of reflecting walls. In free space, the ensemble average of the time averaged mean squared displacement is simply given by i.e., it is additive as for the ensemble average. This be- havior is indeed observed in Fig. 6 where five different mean squared displacement curves are drawn for L = 1 in 1D, 2D, and 3D, respectively. Only the height of the curves are affected by the dimensionality. There is no noticeable difference in the scatter of the curves. We further investigate the effects of dimensionality on the scatter of the mean squared displacement curves, as possibly the strong scatter observed in experiments [8– 10] may also occur for FBM in higher dimensions. To see the effect of dimensionality on the ergodicity behavior we measure E$_{B}$ versus lag time for one-, two-, and three- dimensional embedding dimension for the same values of L and H . Interestingly, the result shows that E$_{B}$ tends to decrease with increasing dimensionality d , meaning that We present the time averaged displacement correlation for confined subdiffusive motion ( H = 1 / 4) in Fig. 5. Be- cause of the negativity of expression (4.5) the absolute value of the displacement correlation is drawn in the log- log representation. At short lag times the slope of the correlation functions is proportional to 2 H as expected FIG. 5. (Color online) Absolute value of the displacement correlation (DC) versus lag time Δ for L = 1, 2, and 3 (from bottom to top) with a slope proportional to Δ 2 H (dotted line). Total measurement time is T = 2 17 ≈ 1 . 3 · 10 $^{5}$, and the Hurst exponent is H = 1 / 4. Each curve was obtained via time averaging from a single particle trajectory. 〈 δ $^{2}$(Δ , T ) 〉 = 1 T − Δ ∫ T − Δ 0 〈 [ x $^{H}$( t + Δ) − x $^{H}$( t )] $^{2}$〉 dt, = 2 dK$_{H}$ Δ 2 $^{H}$, (4.6) D. Dimensionality GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s54> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s52> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s51> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s50> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s49> GLYPH<s126>GLYPH<s84> GLYPH<s45>GLYPH<s49> GLYPH<s61>GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s61>GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s49> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s51> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s53> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s49>GLYPH<s48> GLYPH<s32> GLYPH<s69> GLYPH<s66> GLYPH<s84> GLYPH<s61>GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s84> GLYPH<s105> GLYPH<s109> GLYPH<s101> GLYPH<s45> GLYPH<s97> GLYPH<s118> GLYPH<s101> GLYPH<s114> GLYPH<s97> GLYPH<s103> GLYPH<s101> GLYPH<s100> GLYPH<s32> GLYPH<s77> GLYPH<s83> GLYPH<s68> GLYPH<s108> GLYPH<s97>GLYPH<s103> GLYPH<s32>GLYPH<s116> GLYPH<s105> GLYPH<s109> GLYPH<s101>GLYPH<s32> GLYPH<s32> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s51> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s50> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s49> GLYPH<s124> GLYPH<s68> GLYPH<s67> GLYPH<s124> GLYPH<s108> GLYPH<s97>GLYPH<s103> GLYPH<s32>GLYPH<s116> GLYPH<s105> GLYPH<s109> GLYPH<s101>GLYPH<s32>