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<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<s32> GLYPH<s50> GLYPH<s40> GLYPH<s44>GLYPH<s84>GLYPH<s41> GLYPH<s32>GLYPH<s60>GLYPH<s121> GLYPH<s50> GLYPH<s40>GLYPH<s116>GLYPH<s41>GLYPH<s62> GLYPH<s32>GLYPH<s101>GLYPH<s120>GLYPH<s97>GLYPH<s99>GLYPH<s116>GLYPH<s32>GLYPH<s115>GLYPH<s111>GLYPH<s108>GLYPH<s117>GLYPH<s116>GLYPH<s105>GLYPH<s111>GLYPH<s110>GLYPH<s32>GLYPH<s69>GLYPH<s113>GLYPH<s46>GLYPH<s32>GLYPH<s40>GLYPH<s50>GLYPH<s46>GLYPH<s49>GLYPH<s57>GLYPH<s41> GLYPH<s32> GLYPH<s50> GLYPH<s40> GLYPH<s44>GLYPH<s84>GLYPH<s41>GLYPH<s32>GLYPH<s119> GLYPH<s105>GLYPH<s116>GLYPH<s104>GLYPH<s32>GLYPH<s100>GLYPH<s104>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s32>GLYPH<s60>GLYPH<s121> GLYPH<s50> GLYPH<s40>GLYPH<s116>GLYPH<s41>GLYPH<s62>GLYPH<s32>GLYPH<s119> GLYPH<s105>GLYPH<s116>GLYPH<s104>GLYPH<s32>GLYPH<s100>GLYPH<s104>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s101>GLYPH<s110>GLYPH<s115>GLYPH<s101>GLYPH<s109>GLYPH<s98>GLYPH<s108>GLYPH<s101>GLYPH<s32>GLYPH<s97>GLYPH<s110>GLYPH<s100>GLYPH<s32>GLYPH<s116>GLYPH<s105>GLYPH<s109>GLYPH<s101>GLYPH<s32>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<s115> GLYPH<s116>GLYPH<s32>GLYPH<s111>GLYPH<s114> GLYPH<s32> 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<s48>GLYPH<s46> GLYPH<s49> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s51> GLYPH<s52> GLYPH<s47>GLYPH<s51> GLYPH<s49> GLYPH<s47>GLYPH<s51> GLYPH<s76>GLYPH<s61>GLYPH<s51> GLYPH<s76>GLYPH<s61>GLYPH<s105>GLYPH<s110> GLYPH<s102>GLYPH<s105>GLYPH<s110> GLYPH<s105>GLYPH<s116>GLYPH<s121> GLYPH<s76>GLYPH<s61>GLYPH<s50> GLYPH<s76>GLYPH<s61>GLYPH<s49> GLYPH<s32> 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> 5 FIG. 1. (Color online) The mean squared displacements (MSD) for FLE motion in free space. The ensemble av- eraged and time averaged MSDs, 〈 y $^{2}$( t ) 〉 and δ 2 (Δ , T ), ob- tained from simulation are compared with the exact solution 2 t $^{2}$E 5 / 4 , 3 [ − 10Γ(5 / 4 − 1) t 5 / $^{4}$] given by Eq. (2.19). The ensem- ble averaged value was obtained from 200 simulated trajecto- ries. In the simulation, we chose the Hurst exponent H = 5 / 8, time increment dh = 0 . 01, particle mass m = 1, initial veloc- ity v 0 = 1, initial position y 0 = 0, friction coefficient γ = 10, and k$_{B}$T = 1. FIG. 2. (Color online) Time-averaged mean squared displace- ment (MSD) versus the lag time Δ for given values of L . The drawn line has slope 1 / 2, corresponding to the expected short lag time behavior for the used value H = 1 / 4 of the Hurst exponent. For L = 1, 2, and 3, five different trajectories each are drawn to be able to see whether the trajectories scatter. The simulation time is T = 2 17 ≈ 1 . 3 · 10 $^{5}$. We now turn to the investigation of the behavior of FBM under confinement, analyzing the mean squared displacement and potential ergodicity breaking. We then define the displacement correlation function, and finally study the influence of dimensionality. For FBM in free space 〈 x $^{H}$( t ) $^{2}$〉 can be estimated by the time averaged mean squared displacement Eq. (1.2) via the exact relation [25] 〈 δ $^{2}$(Δ , T ) 〉 = 2 K$_{H}$ Δ 2 $^{H}$. (4.1) Here 〈·〉 denotes the ensemble average. In contrast to CTRW subdiffusion, in FBM this quantity is ergodic. However, as mentioned above, the approach to ergod- icity is algebraically slow, and we want to explore here whether boundary conditions have an impact on the er- godic behavior. Let us now analyze the behavior in a box of size 2 L . In Fig. 2 we show typical curves for the time averaged mean squared displacement for Hurst exponent H = 1 / 4 In contrast to CTRW subdiffusion, FBM in free space is known to be ergodic [25]. The time averaged mean squared displacement traces displayed in Fig. 2 exhibit One can estimate the saturated value as a function of L . For long Δ and measurement time T , the probability p ( x ) to find the particle located at x is independent of x due to the equilibration between the reflecting walls, and thus ∫ L − $_{L}$x $^{2}$p ( x ) dx = L $^{2}$/ 3. The dotted lines in Fig. 2 represent these values. We observe that the scatter between different single trajectories becomes more pronounced when the interval length is increased. In fact the scatter is negligible for L = 1 while it is quite appreciable for L = 3, even though the slope of all curves at finite L converges to a horizon- tal slope, with an amplitude close to the predicted value L $^{2}$/ 3. We also note that the scatter depends on the total measurement time T . For given L it tends to be reduced as we increase T . This effect will be discussed quantita- tively in detail using the ergodicity breaking parameter. and three different interval lengths L . Regardless of the size of L , the confined environment does not affect the power law with exponent 2 H for short lag times. More- over at long lag times we observe saturation of the curves to a value that depends on L . This behavior is distinct from that of the CTRW case where 〈 δ $^{2}$(Δ , T ) 〉 shows a power law with slope 1 − α [27, 28]. B. Ergodicity breaking parameter A. Mean squared displacement IV. FRACTIONAL BROWNIAN MOTION IN CONFINED SPACE