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  - https://doi.org/10.1515/fca-2015-0059
SN  - 1311-0454
SN  - 1314-2224
VL  - 18
IS  - 4
SP  - 1006
EP  - 1038
PB  - De Gruyter
CY  - Berlin
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  - https://doi.org/10.1103/PhysRevE.92.042117
SN  - 1539-3755
SN  - 1550-2376
VL  - 92
IS  - 4
PB  - American Physical Society
CY  - College Park
ER  -