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<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<s51>GLYPH<s68> GLYPH<s32>GLYPH<s50>GLYPH<s68> GLYPH<s32>GLYPH<s49>GLYPH<s68> GLYPH<s108>GLYPH<s97>GLYPH<s103>GLYPH<s32>GLYPH<s116>GLYPH<s105>GLYPH<s109> GLYPH<s101>GLYPH<s32> GLYPH<s100> GLYPH<s32> GLYPH<s69> GLYPH<s66> GLYPH<s40> GLYPH<s100> GLYPH<s41> GLYPH<s32> 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<s49>GLYPH<s48> GLYPH<s49> GLYPH<s76>GLYPH<s61>GLYPH<s49> GLYPH<s49> GLYPH<s47> GLYPH<s51> GLYPH<s76>GLYPH<s61>GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s76>GLYPH<s61>GLYPH<s51> GLYPH<s51> GLYPH<s49> GLYPH<s47> GLYPH<s49> GLYPH<s50> GLYPH<s76>GLYPH<s61>GLYPH<s49> GLYPH<s47> GLYPH<s50> 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> 8 FIG. 7. (Color online) Rescaled ergodicity breaking param- eter dE$_{B}$ versus lag time Δ for interval size L = 5 for 1D, 2D, and 3D space (from top to bottom), with Hurst exponent H = 1 / 4 and measurement time T = 2 $^{14}$. E$_{B}$ was evaluated from 200 trajectories with different initial positions. FIG. 8. (Color online) Time-averaged mean squared displace- ment (MSD) versus lag time Δ. The two dashed lines repre- sent the two asymptotic scaling behaviors 〈 δ 2 (Δ , T ) 〉 ≃ Δ 2 and 〈 δ 2 (Δ , T ) 〉 ≃ Δ 2 − 2 $^{H}$. For each given L = 1 / 2, 1, and 3, five different trajectories are drawn to visualize the scatter. As a reference curve for motion in free space, the curve for L = 100 (line) is drawn. In the simulation, we chose the Hurst exponent H = 5 / 8 [NB: for the FLE this means subdiffusion], 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. for FBM big scatter is not caused by higher dimensions in presence of reflecting walls. In fact, from Eqs. (4.2) and (4.6), we can analytically derive the relation which still holds in the case of confined motion (see ap- pendix B for the derivation). This relation is numerically confirmed in Fig. 7 where three E$_{B}$ curves collapse upon rescaling by dE$_{B}$ ( d ). According to this relation, we ex- pect that ergodic behavior obtained in one-dimensional confined motion (Figs. 3 and 4) will also be present in multiple dimensions with a factor of 1 /d . In this section we analyze FLE motion under confine- ment. Due to the different physical basis compared to FBM, in particular, the occurrence of inertia, we observe interesting variations on the properties studied in the pre- vious section. Using the correlation function [55] at long lag times, in free space. We numerically study how this scaling behavior is af- fected by the confinement. Figure 8 shows typical curves for the time averaged mean squared displacement, for in- terval sizes L = 1 / 2, 1, 3, and 100 (regarded as free space motion) with identical initial conditions and Hurst expo- nent H = 5 / 8. The results are summarized as follows: (1) We observe both scaling behaviors, 〈 δ $^{2}$(Δ , T ) 〉 ≃ Δ 2 at short lag time to the subdiffusive behavior Thus, the time averaged mean squared displacement turns over from a ballistic motion one can show analytically that, similarly to the FBM, the ensemble averaged second moment 〈 y $^{2}$( t ) 〉 is identical to its time averaged analog 〈 δ $^{2}$(Δ) 〉 , for all Δ in free space, namely 〈 δ $^{2}$(Δ , T ) 〉 = 2 k$_{B}$T m Δ $^{2}$E$_{2}$$_{H,}$$_{3}$ ( − γ Δ 2 $^{H}$) . (5.2) 〈 δ $^{2}$(Δ , T ) 〉 ≃ Δ 2 (5.3) 〈 δ $^{2}$(Δ , T ) 〉 ≃ Δ 2 − 2 H (5.4) 〈 y ( t$_{1}$ ) y ( t$_{2}$ ) 〉 = k$_{B}$T m [ t 2 $_{1}$E$_{2}$$_{H,}$$_{3}$ ( − γt 2 H 1 ) + t 2 $_{2}$E$_{2}$$_{H,}$$_{3}$ ( − γt 2 H 2 ) − ( t$_{2}$ − t$_{1}$ ) $^{2}$E$_{2}$$_{H,}$$_{3}$ ( − γ | t$_{2}$ − t$_{1}$ | 2 $^{H}$)] , (5.1) E$_{B}$ ( d ) = E$_{B}$ ( d = 1) d , (4.7) V. FRACTIONAL LANGEVIN EQUATION MOTION IN CONFINED SPACE A. Mean squared displacement