TY  - JOUR
A1  - Awad, Emad
A1  - Metzler, Ralf
T1  - Crossover dynamics from superdiffusion to subdiffusion
BT  - models and solutions
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - The Cattaneo or telegrapher's equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the dth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function.
KW  - Cattaneo equation
KW  - telegrapher's equation
KW  - crossover dynamics
KW  - fractional dynamic equations
KW  - anomalous diffusion
KW  - superdiffusion and
KW  - subdiffusion
KW  - Fox H-functions
Y1  - 2020
U6  - https://doi.org/10.1515/fca-2020-0003
SN  - 1311-0454
SN  - 1314-2224
VL  - 23
IS  - 1
SP  - 55
EP  - 102
PB  - De Gruyter
CY  - Berlin ; Boston
ER  - 
TY  - JOUR
A1  - Jeon, Jae-Hyung
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
JF  - Physical chemistry, chemical physics : PCCP
N2  - Anomalous diffusion is frequently described by scaled Brownian motion (SBM){,} a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is ?x2(t)? [similar{,} equals] 2K(t)t with K(t) [similar{,} equals] t[small alpha]-1 for 0 < [small alpha] < 2. SBM may provide a seemingly adequate description in the case of unbounded diffusion{,} for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely{,} we demonstrate that under confinement{,} the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments{,} in particular{,} under confinement inside cellular compartments or when optical tweezers tracking methods are used.
KW  - single-particle tracking
KW  - living cells
KW  - random-walks
KW  - subdiffusion
KW  - dynamics
KW  - nonergodicity
KW  - coefficients
KW  - transport
KW  - membrane
KW  - behavior
Y1  - 2014
U6  - https://doi.org/10.1039/C4CP02019G
VL  - 30
IS  - 16
SP  - 15811
EP  - 15817
PB  - The Royal Society of Chemistry
CY  - Cambridge
ER  -