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Fractional Klein-Kramers Equations: Subdiffusive and Superdiffusive Cases

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Abstract

Brownian diffusion processes in phase space are described by the Klein-Kramers equation governing the time evolution of the probability density W(x, v, t) to find the test particle with velocity v at position x at time t. We here summarise generalisations of this equation to anomalous diffusion processes. These fractional Klein-Kramers equations describe either subdiffusive or superdiffusive processes.

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Notes

  1. 1.

    Here and in the following we denote the Laplace transform of a function through the explicit dependence on the Laplace variable, i.e.,

    $$f(u) = \mathcal{L}\{f(t)\} ={ \int \nolimits \nolimits }_{0}^{\infty }f(t)\exp (-ut)dt.$$

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Acknowledgements

Funding from the Academy of Finland within the FiDiPro scheme is gratefully acknowledged.

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Authors and Affiliations

  1. Institute for Physics and Astronomy, University of Potsdam, 14476, Potsdam-Golm, Germany

    Ralf Metzler

  2. Physics Department, Tampere University of Technology, FI-33101, Tampere, Finland

    Ralf Metzler

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  1. Ralf Metzler
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Correspondence to Ralf Metzler .

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Editors and Affiliations

  1. , Lab. de Mathématiques et Physique, Université de Perpignan, Avenue Paul Alduy 52, Perpignan, 66860, France

    Yuri P. Kalmykov

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Metzler, R. (2013). Fractional Klein-Kramers Equations: Subdiffusive and Superdiffusive Cases. In: Kalmykov, Y. (eds) Recent Advances in Broadband Dielectric Spectroscopy. NATO Science for Peace and Security Series B: Physics and Biophysics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5012-8_13

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