GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s48>GLYPH<s48> 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<s76>GLYPH<s61>GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s76>GLYPH<s61>GLYPH<s51> GLYPH<s76>GLYPH<s61>GLYPH<s49> GLYPH<s76>GLYPH<s61>GLYPH<s49>GLYPH<s47>GLYPH<s50> GLYPH<s32> GLYPH<s60> GLYPH<s124> GLYPH<s68> GLYPH<s67> GLYPH<s124> GLYPH<s62> GLYPH<s108>GLYPH<s97>GLYPH<s103>GLYPH<s32>GLYPH<s116>GLYPH<s105>GLYPH<s109> GLYPH<s101>GLYPH<s32> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s54> GLYPH<s48>GLYPH<s46>GLYPH<s55> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s57> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s45>GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s48>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s48>GLYPH<s48>GLYPH<s53> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s49>GLYPH<s47>GLYPH<s50> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s49> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s51> GLYPH<s32>GLYPH<s76>GLYPH<s61>GLYPH<s49>GLYPH<s48>GLYPH<s48> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s60> GLYPH<s68> GLYPH<s67> GLYPH<s62> GLYPH<s32> FIG. 10. (Color online) (a) Displacement correlation (DC) versus lag time Δ for L = 1 / 2, 1, 3, and 100 (from bottom to top). (b) Absolute value of the displacement correlation as a function of Δ for the given values of L . The two slopes correspond to the limiting behavior Δ 2 and Δ 2 − 2 $^{H}$. In (a) and (b), the ensemble averaged curves were obtained from 200 different time averaged displacement correlation curves. Same parameter values as in Fig. 8. In the case when the memory tensor is diagonalized, each coordinate motion is independent and FLE mo- tion exhibits qualitatively the same behavior as shown in the case of FBM with increasing dimensionality. From the mean squared displacement curves, the same scal- ing behavior is expected with more elevated amplitude for higher dimensionality. In fact, when each coordinate motion is decoupled, a d -dimensional motion effectively increases the number of single trajectories d times com- pared to the one-dimensional case. Therefore, the scatter in the mean square displacement curves decreases with increasing dimensionality and the ergodicity breaking pa- rameter E$_{B}$ is expected to follow the relation Eq. (4.7). Motivated by recent single particle tracking experi- ments in biological cells, in which confinement due to the rather small cell size becomes relevant, we studied In this appendix we derive analytical expressions for the displacement correlations, Eqs. (4.5) and (5.5). For a stochastic variable x , we define The displacement correlation is then given by We thank Stas Burov and Eli Barkai for helpful and enjoyable discussions. Financial support from the DFG is acknowledged. The displacement correlation function introduced here is a useful quantity that can be easily obtained from sin- gle particle trajectories. It can be used as a tool to dis- criminate one stochastic model from another. For subd- iffusive motion governed by FBM and FLE motion, the displacement correlation should be negative and saturate in the long measurement time limit due to the confine- ment. Notably, the negative decrease ( ∼ − Δ $^{α}$) with lag time Δ and anomalous diffusion exponent α is an intrin- sic property of FBM and FLE displacement correlations which is clearly distinguished from that of CTRW subdif- fusion. In the latter case, the subdiffusive motion occurs due to the long waiting time distribution between suc- cessive jumps and there is no spatial correlation between them, so that displacement correlation only fluctuates around zero with time. FLE motion can be distinguished from FBM motion since the displacement correlation has a positive value at short times due to the ballistic motion in the FLE model. FBM and FLE motions in confined space. In particular we analyzed the effects of confinement and dimensional- ity on the stochastic and ergodic properties of the two processes. Interestingly for both stochastic models, the confinement tends to decrease the value of the ergodicity breaking parameter E$_{B}$ compared to that in free space. The same trend is observed for increasing dimensionality. Correspondingly the scatter of time averaged quantities between individual trajectories is quite small, apart from regimes when the lag time Δ becomes close to the over- all measurement time T and the sampling statistics for the corresponding time average become poor. The relax- ation of the ergodicity breaking parameter as function of measurement time is quite similar to previous results in free space. We conclude that neither confinement nor di- mensionality effects lead to the appearance of significant ergodicity breaking or scatter between single trajectories. 10 ACKNOWLEDGMENTS D. Dimensionality VI. CONCLUSION Appendix A: Derivation of the displacement correlation function δx ( t, Δ) = x ( t + Δ) − x ( t ) . (A1) 〈 δx ( t, Δ) δx ( t − Δ , Δ) 〉 = 〈 x ( t + 2Δ) x ( t + Δ) 〉 − 〈 x ( t + 2Δ) x ( t ) 〉 −〈 x ( t + Δ) $^{2}$〉 + 〈 x ( t + Δ) x ( t ) 〉 . (A2)