Scaled Brownian motion as a mean-field model for continuous-time random walks

Abstract

We consider scaled Brownian motion (sBm), a random process described by a diffusion equation with explicitly time-dependent diffusion coefficient D(t) = D0t α−1 (Batchelor's equation) which, for α < 1, is often used for fitting experimental data for subdiffusion of unclear genesis. We show that this process is a close relative of subdiffusive continuous-time random walks and describes the motion of the center of mass of a cloud of independent walkers. It shares with subdiffusive CTRW its non-stationary and non-ergodic properties. The non-ergodicity of sBm does not however go hand in hand with strong difference between its different realizations: its heterogeneity ("ergodicity breaking") parameter tends to zero for long trajectories.PACS numbers: 05.40. Fb,05.10.Gg Anomalous diffusion is a generic name for a class of transport processes which are close to diffusion in their origin (i.e. can be represented via generalized random walk schemes or Langevin equations) but do not lead to the mean squared displacement growing as the first power of time(with D being the diffusion coefficient and d the dimension of space), as predicted by the Fick's laws. Within the random walk schemes such deviations from the normal diffusion picture can arise either due to broad distributions of the waiting times between the steps (continuous time random walk models, CTRW), or due to slow decay of correlations between steps, or both, see [1] for a review, leading to the change of the power law in the time dependence of the mean squared displacement,The processes with α < 1 are called subdiffusion, the ones with α > 1 are termed superdiffusion. In the first case the formal diffusion coefficient D in Eq.(1) vanishes in the long time limit; in the second case it diverges. The single trajectory dynamics in normal diffusion is described by the Langevin equatioṅwith white, delta-correlated Gaussian noise ξ(t), ξ(t) = 0, ξ(t)ξ(t ) = δ(t−t ); the time-dependence of the probability density function (PDF) of the process or the one of its transition probabilities is given by the Fick's second law (diffusion equation)(both equations given here in one dimension). The description of anomalous diffusion of different origins often follows by modification of one of the equations above.In experiments, many processes of anomalous diffusion of unknown origin, i.e. when the observable of interest which cannot be fitted to the solutions of Eq. (1), are fitted to the results obtained for the so-called scaled Brownian motion (sBm) [2], a diffusion process with explicitly time-dependent diffusion coefficient D(t) = D 0 t α−1 . Numerical simulations of Ref. [3] show that, at least for the case of FRAP (fluorescence recovery after photobleaching), the fits may be astonishingly good, independent on the true nature of the simulated process (percolation, CTRW, etc.). This means that the form of FRAP recovery curves, if they hint onto anomalous diffusion, hardly depends on the origin of the corresponding anomaly. Using sBm model for calculating other properties ma...

Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

“…In addition, the nonergodic properties of SBM systems have been recently studied in detail in Refs. [55,56] and could be used here.…”

Ultraslow scaled Brownian motion

“…A diffusion equation with a time dependent diffusivity proportional to t 2 was originally introduced by Batchelor [57] to describe the anomalous Richardson relative diffusion [58] in turbulent atmospheric systems. SBM with diffusivity D t t ( ) 1 ≃ α− was studied extensively during the last few years [59][60][61][62][63]. In particular, the weakly non-ergodic disparity between ensemble and time averages in SBM as well as its ageing behaviour were analysed [60][61][62][63], see also below.…”

Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel

“…A diffusion process on a matrix with a fractal dimension such as a Sierpinski gasket or a percolation cluster near criticality turns anomalous due to the abundance of bottlenecks and dead ends on all scales [48,55,56,40,70,96,97,109]. Finally, stochastic processes with space or time dependent diffusivities give rise to anomalous diffusion [16,17,18,19,25,26,49,60,69,117]. A contemporary summary of a rich variety of anomalous diffusion processes is provided in Ref.…”