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.
| Author details: | Trifce SandevORCiD, Aleksei V. ChechkinORCiDGND, Holger KantzORCiD, Ralf MetzlerORCiDGND |
|---|---|
| DOI: | https://doi.org/10.1515/fca-2015-0059 |
| ISSN: | 1311-0454 |
| ISSN: | 1314-2224 |
| Title of parent work (English): | Fractional calculus and applied analysis : an international journal for theory and applications |
| Publisher: | De Gruyter |
| Place of publishing: | Berlin |
| Publication type: | Article |
| Language: | English |
| Year of first publication: | 2015 |
| Publication year: | 2015 |
| Release date: | 2017/03/27 |
| Tag: | Fokker-Planck-Smoluchowski equation; Mittag-Leffler functions; anomalous diffusion; continuous time random walk (CTRW); multi-scaling |
| Volume: | 18 |
| Issue: | 4 |
| Number of pages: | 33 |
| First page: | 1006 |
| Last Page: | 1038 |
| Funding institution: | IMU Berlin Einstein Foundation; Academy of Finland within the FiDiPro programme |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
| Peer review: | Referiert |
