3 Finally, the two point correlation behaves as Here 〈·〉 represents the ensemble average. It is con- venient to introduce fractional Gaussian noise (FGN), ξ $^{H}$( t ), from which the FBM is generated by Note that | x $^{H}$( t ) | cannot satisfy these properties, thus it is not an FBM. A few remarks on this multidimensional extension of FBM are in order. We note that, albeit intuitive due to the Gaussian nature of FBM, this multidimensional extension is not necessarily unique. In mathematical lit- erature higher dimensional FBM in the above sense was defined in Refs. [43, 44]. In physics literature an analo- gous extension to higher dimensions was used in Ref. [13] based on the Weierstrass-Mandelbrot method. To ver- ify that this d -dimensional extension is meaningful, we checked from our simulations of d -dimensional FBM that the fractal dimension of FBM, d$_{f}$ = 1 /H for H > 1 /d [45], is preserved in higher dimensions. Moreover we used an alternative method to create FBM in d -dimensions, namely, to use 1D FBM to choose the length of a radius and then choose the space angle randomly. The results were equivalent to the above definition to use indepen- dent FBMs for every Cartesian coordinate. We are there- fore confident that our definition of FBM in d dimensions is a proper extension of regular 1D FBM. FGN has the properties of zero mean and autocorrelation [39, 40] as can be seen by differentiation of Eq. (2.3) with re- spect to t$_{1}$ and t$_{2}$ . Here we see that K$_{H}$ plays the role of a noise strength. For subdiffusion (0 < H < 1 / 2) the autocorrelation is negative for t$_{1}$ = t$_{2}$ , i.e., the process is anti-correlated or antipersistent [41]. In contrast, for 1 / 2 < H < 1, the noise is positively correlated (persis- tent) and the motion becomes superdiffusive. For nor- mal diffusion ( H = 1 / 2), the noise is uncorrelated, i.e., 〈 ξ $^{H}$( t$_{1}$ ) ξ $^{H}$( t$_{2}$ ) 〉 = 2 K$_{H}$δ ( t$_{1}$ − t$_{2}$ ). For further details com- pare the discussions in Ref. [32, 42]. We define d -dimensional FBM as a superposition of independent FBMs for each Cartesian coordinate, such that where ˆ x$_{i}$ is the Cartesian coordinate of the i th component and ξ H i is FGN which satisfies and From this definition, d -dimensional FBM x $^{H}$( t ) has the properties of zero mean variance and autocorrelation Here, γ is a generalized friction coefficient. We also define the coupling constant imposed by the fluctuation dissipation theorem, and the Caputo fractional derivative [52] where m is the test particle mass, K is the memory kernel [49–51] which satisfies the fluctuation-dissipation theorem 〈 ξ ( t ) ξ ( t $^{′}$) 〉 = k$_{B}$T γ K ( t − t $^{′}$). When ξ is the FGN introduced above, K decays algebraically, and Eq. (2.14) becomes the fractional Langevin equation (FLE) where ξ ( t ) corresponds to white Gaussian noise. When the random noise ξ ( t ) is non-white, the resulting motion is described by the generalized Langevin equation (GLE) An alternative approach to Brownian motion is based on the Langevin equation [46–48] B. Fractional Langevin equation motion 〈 x $^{H}$( t$_{1}$ ) x $^{H}$( t$_{2}$ ) 〉 = K$_{H}$ ( t 2 H 1 + t 2 H 2 − | t$_{1}$ − t$_{2}$ | 2 $^{H}$) . (2.3) x $^{H}$( t ) = ∫ t 0 dt $^{′}$ξ $^{H}$( t $^{′}$) . (2.4) 〈 ξ $^{H}$( t ) 〉 = 0 (2.5) 〈 ξ $^{H}$( t$_{1}$ ) ξ $^{H}$( t$_{2}$ ) 〉 = 2 K$_{H}$H (2 H − 1) | t$_{1}$ − t$_{2}$ | 2 H − 2 + 4 K$_{H}$H | t$_{1}$ − t$_{2}$ | 2 H − $^{1}$δ ( t$_{1}$ − t$_{2}$ ) , (2.6) m d $^{2}$y ( t ) dt 2 = − γ dy ( t ) dt + ξ ( t ) , (2.13) m d $^{2}$y ( t ) dt 2 = − γ ∫ t 0 K ( t − t $^{′}$) dy dt $_{′}$dt ′ + ξ ( t ) , (2.14) x $^{H}$( t ) = d ∑ i =1 ∫ t 0 dt $^{′}$ξ H i ( t $^{′}$)ˆ x$_{i}$, (2.7) 〈 ξ H i ( t ) 〉 = 0 (2.8) 〈 ξ H i ( t$_{1}$ ) ξ H j ( t$_{2}$ ) 〉 = 2 K$_{H}$H (2 H − 1) | t$_{1}$ − t$_{2}$ | 2 H − $^{2}$δ$_{ij}$ + 4 K$_{H}$H | t$_{1}$ − t$_{2}$ | 2 H − $^{1}$δ ( t$_{1}$ − t$_{2}$ ) δ$_{ij}$. (2.9) m d $^{2}$y ( t ) dt 2 = − γ ∫ t 0 ( t − t $^{′}$) 2 H − 2 dy dt $_{′}$dt ′ + ηξ $^{H}$( t ) = − γ Γ(2 H − 1) d 2 − 2 H dt 2 − 2 $_{H}$y ( t ) + ηξ $^{H}$( t ) . (2.15) 〈 x $^{H}$( t ) 〉 = 0 , (2.10) 〈 x $^{H}$( t ) $^{2}$〉 = 2 dK$_{H}$t 2 $^{H}$, (2.11) 〈 x $^{H}$( t$_{1}$ ) · x $^{H}$( t$_{2}$ ) 〉 = dK$_{H}$ ( t 2 H 1 + t 2 H 2 − | t$_{1}$ − t$_{2}$ | 2 $^{H}$) . (2.12) η = √ k$_{B}$T γ 2 K$_{H}$H (2 H − 1) (2.16) d 2 − 2 H dt 2 − 2 $_{H}$y ( t ) = 1 Γ(2 H − 1) ∫ t 0 dt $^{′}$( t − t $^{′}$) 2 H − 2 dy dt $_{′}$. (2.17)