Fractional Brownian motion and generalized Langevin equation motion in confined geometries Jae-Hyung Jeon 1, ∗ and Ralf Metzler 1, † $^{1}$Department of Physics, Technical University of Munich, James-Franck Straße, 85747 Garching, Germany Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time averaged mean squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume, and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single particle trajectory data. Anomalous diffusion denominates deviations from the regular linear growth of the mean squared displacement 〈 r $^{2}$( t ) 〉 ≃ K$_{1}$t as a function of time t , where the propor- tionality factor K$_{1}$ is the diffusion constant of dimension cm $^{2}$/ sec. Often these deviations are of power-law form, and in this case the mean squared displacement in d di- mensions describes subdiffusion when the anomalous diffusion ex- ponent is in the range 0 < α < 1, and superdiffusion for α > 1 [1]. The generalized diffusion constant K$_{α}$ has di- mension cm $^{2}$/ sec $^{α}$. We here concentrate on subdiffusion phenomena. Power-law mean squared displacements of the form (1.1) with 0 < α < 1 have been observed in a multitude of systems such as amorphous semiconductors [2], subsurface tracer dispersion [3], or in financial mar- ket dynamics [4]. Subdiffusion is quite abundant in small systems. These include the motion of small probe beads in actin networks [5], local dynamics in polymer melts [6], or the motion of particles in colloidal glasses [7]. In vivo , crowding-induced subdiffusion has been reported for RNA motion in E.coli cells [8], the diffusion of lipid granules embedded in the cytoplasm [9, 10], the propa- gation of virus shells in cells [11], the motion of telomeres in mammalian cells [12], as well as the diffusion of mem- brane proteins and of dextrane probes in HeLa cells [13]. While the mean squared displacement Eq. (1.1) is com- monly used to classify a process as subdiffusive, it does not provide any information on the physical mechanism (iii) Fractional Brownian motion (FBM) and the frac- tional Langevin equation (FLE) that will be in the fo- cus of this work and will be defined in Sec. II. The un- derstanding of these types of stochastic motions is up to date somewhat fragmentary. Thus the first passage behavior of FBM is known analytically only in one di- mension on a semi-infinite domain [17]; the escape from potential wells in the framework of FLE has been stud- ied analytically [18, 19] and numerically [20, 21]; and a priori unexpected critical exponents have been identified for the FLE [22]. Here we address a fundamental ques- tion related to FBM and FLE. Namely, what is their behavior under confinement? Two main aspects of this question will be addressed. One is the study of the re- laxation towards stationarity in a finite box by means of (ii) A random walk on a fractal support meets bot- tlenecks and dead ends on all scales and is subdiffusive. The resulting subdiffusion is also of the form (1.1), and the anomalous diffusion exponent is related to the fractal and spectral dimensions, d$_{f}$ and d$_{s}$ , characteristic of the fractal, through α = d$_{s}$/d$_{f}$ [15]. A typical example is the subdiffusion on a percolation cluster near criticality that was actually verified experimentally [16]. (i) The continuous time random walk (CTRW) [2] in which the step length has a finite variance 〈 δ r $^{2}$〉 of jump lengths, but the waiting time τ elapsing between succes- sive jumps is distributed as a power law ψ ( τ ) ≃ τ α 0 /τ 1+ $^{α}$, with 0 < α < 1. The diverging characteristic waiting time gives rise to subdiffusion of the form (1.1) [2]. The subdiffusive CTRW in the diffusion limit is equivalent to the fractional Fokker-Planck equation, that directly shows the long-ranged memory intrinsic to the process [1]. Waiting times of the form ψ ( τ ) were, for instance, observed in the motion of tracer beads in an actin net- work [5]. We note that recently subdiffusion was also demonstrated in a coupled CTRW model [14]. underlying this subdiffusion. In fact there are several pathways along which this subdiffusion may emerge. The most common are: 〈 r $^{2}$( t ) 〉 = 2 dK$_{α}$ Γ(1 + α ) t α (1.1) I. INTRODUCTION arXiv:1001.0681v2 [cond-mat.stat-mech] 6 Jan 2010 ∗ jae-hyung.jeon@ph.tum.de † metz@ph.tum.de