TY  - JOUR
A1  - Schulz, Johannes H. P.
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Correlated continuous time random walks - combining scale-invariance with long-range memory for spatial and temporal dynamics
JF  - Journal of physics : A, Mathematical and theoretical
N2  - Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through scale-free forms of the jump length and/or waiting time distributions by virtue of the generalized central limit theorem. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful for describing recent experimental single particle tracking data that feature a combination of CTRW and FBM properties.
Y1  - 2013
U6  - https://doi.org/10.1088/1751-8113/46/47/475001
SN  - 1751-8113
SN  - 1751-8121
VL  - 46
IS  - 47
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Cherstvy, Andrey G.
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
JF  - New journal of physics : the open-access journal for physics
N2  - We demonstrate the non-ergodicity of a simple Markovian stochastic process with space-dependent diffusion coefficient D(x). For power-law forms D(x) similar or equal to vertical bar x vertical bar(alpha), this process yields anomalous diffusion of the form < x(2)(t)> similar or equal to t(2/(2-alpha)). Interestingly, in both the sub- and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time-averaged mean-squared displacement <(delta(2)(Delta))over bar> remains linear in the lag time Delta and thus differs from the corresponding ensemble average < x(2)(t)>. We analyse the non-ergodic behaviour of this process in terms of the time-averaged mean- squared displacement (delta(2)) over bar and its random features, i.e. the statistical distribution of (delta(2)) over bar and the ergodicity breaking parameters. The heterogeneous diffusion model represents an alternative approach to non- ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.
Y1  - 2013
U6  - https://doi.org/10.1088/1367-2630/15/8/083039
SN  - 1367-2630
VL  - 15
IS  - 15
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  -