TY - JOUR
A1 - Godec, Aljaz
A1 - Chechkin, Aleksei V.
A1 - Barkai, Eli
A1 - Kantz, Holger
A1 - Metzler, Ralf
T1 - Localisation and universal fluctuations in ultraslow diffusion processes
JF - Journal of physics : A, Mathematical and theoretical
N2 - We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) < x(2)(t)> similar or equal to log(gamma)t. Comparison of annealed (renewal) continuous time random walks (CTRWs) with logarithmic waiting time distribution psi(tau) similar or equal to 1/(tau log(1+gamma)tau) and Sinai diffusion in quenched random landscapes reveals striking similarities, despite the great differences in their physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time-averaged and ensemble-averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal, with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble-averaged MSD and time-averaged MSD.
KW - Sinai diffusion
KW - anomalous diffusion
KW - quenched energy landscape
Y1 - 2014
U6 - http://dx.doi.org/10.1088/1751-8113/47/49/492002
SN - 1751-8113 (print)
SN - 1751-8121 (online)
VL - 47
IS - 49
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Sandev, Trifce
A1 - Chechkin, Aleksei V.
A1 - Korabel, Nickolay
A1 - Kantz, Holger
A1 - Sokolov, Igor M.
A1 - Metzler, Ralf
T1 - Distributed-order diffusion equations and multifractality: Models and solutions
JF - Physical review : E, Statistical, nonlinear and soft matter physics
N2 - We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.
Y1 - 2015
U6 - http://dx.doi.org/10.1103/PhysRevE.92.042117
SN - 1539-3755 (print)
SN - 1550-2376 (online)
VL - 92
IS - 4
PB - American Physical Society
CY - College Park
ER -
TY - JOUR
A1 - Sandev, Trifce
A1 - Chechkin, Aleksei V.
A1 - Kantz, Holger
A1 - Metzler, Ralf
T1 - Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel
JF - Fractional calculus and applied analysis : an international journal for theory and applications
N2 - We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.
KW - continuous time random walk (CTRW)
KW - Fokker-Planck-Smoluchowski equation
KW - Mittag-Leffler functions
KW - anomalous diffusion
KW - multi-scaling
Y1 - 2015
U6 - http://dx.doi.org/10.1515/fca-2015-0059
SN - 1311-0454 (print)
SN - 1314-2224 (online)
VL - 18
IS - 4
SP - 1006
EP - 1038
PB - De Gruyter
CY - Berlin
ER -