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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.

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Author:Trifce Sandev, Aleksei V. ChechkinORCiDGND, Holger Kantz, Ralf MetzlerORCiDGND
ISSN:1311-0454 (print)
ISSN:1314-2224 (online)
Parent Title (English):Fractional calculus and applied analysis : an international journal for theory and applications
Publisher:De Gruyter
Place of publication:Berlin
Document Type:Article
Year of first Publication:2015
Year of Completion:2015
Release Date:2017/03/27
Tag:Fokker-Planck-Smoluchowski equation; Mittag-Leffler functions; anomalous diffusion; continuous time random walk (CTRW); multi-scaling
First Page:1006
Last Page:1038
Funder:IMU Berlin Einstein Foundation; Academy of Finland within the FiDiPro programme
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Peer Review:Referiert