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Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel

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Abstract

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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Authors and Affiliations

  1. Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187, Dresden, Germany

    Trifce Sandev, Aleksei Chechkin & Holger Kantz

  2. Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020, Skopje, Macedonia

    Trifce Sandev

  3. Akhiezer Institute for Theoretical Physics, Akademicheskaya St. 1, Kharkov, 61108, Ukraine

    Aleksei Chechkin

  4. Institute for Physics and Astronomy, Karl-Liebknecht-St. 24/25, D-14776, Potsdam-Golm, Germany

    Aleksei Chechkin & Ralf Metzler

  5. Department of Physics, Tampere University of Technology, Korkeakoulunkatu 3, FI-33101, Tampere, Finland

    Ralf Metzler

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  1. Trifce Sandev
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  2. Aleksei Chechkin
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  3. Holger Kantz
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  4. Ralf Metzler
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Sandev, T., Chechkin, A., Kantz, H. et al. Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel. FCAA 18, 1006–1038 (2015). https://doi.org/10.1515/fca-2015-0059

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