38715
2015
2015
eng
1006
1038
33
4
18
article
De Gruyter
Berlin
1
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--
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Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel
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.
Fractional calculus and applied analysis : an international journal for theory and applications
10.1515/fca-2015-0059
1311-0454 (print)
1314-2224 (online)
wos:2015
WOS:000359161800010
Sandev, T (reprint author), Max Planck Inst Phys Komplexer Syst, Nothnitzer Str 38, D-01187 Dresden, Germany., sandev@pks.mpg.de; chechkin@pks.mpg.de; kantz@pks.mpg.de; rmetzler@uni-potsdam.de
IMU Berlin Einstein Foundation; Academy of Finland within the FiDiPro
programme
Trifce Sandev
Aleksei V. Chechkin
Holger Kantz
Ralf Metzler
eng
uncontrolled
continuous time random walk (CTRW)
eng
uncontrolled
Fokker-Planck-Smoluchowski equation
eng
uncontrolled
Mittag-Leffler functions
eng
uncontrolled
anomalous diffusion
eng
uncontrolled
multi-scaling
Institut für Physik und Astronomie
Referiert
38517
2015
2015
eng
19
4
92
article
American Physical Society
College Park
1
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Distributed-order diffusion equations and multifractality: Models and solutions
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.
Physical review : E, Statistical, nonlinear and soft matter physics
10.1103/PhysRevE.92.042117
26565178
1539-3755 (print)
1550-2376 (online)
wos:2015
042117
WOS:000362446200004
Sandev, T (reprint author), Max Planck Inst Phys Komplexer Syst, Nothnitzer Str 38, D-01187 Dresden, Germany.
IMU Berlin Einstein Foundation; Academy of Finland (Suomen Akatemia)
through the FiDiPro scheme
Trifce Sandev
Aleksei V. Chechkin
Nickolay Korabel
Holger Kantz
Igor M. Sokolov
Ralf Metzler
Institut für Physik und Astronomie
Referiert