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Velocity and displacement correlation functions for fractional generalized Langevin equations

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Abstract

We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.

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Authors and Affiliations

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

    Trifce Sandev

  2. Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476, Potsdam - Golm, Germany

    Ralf Metzler

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

    Ralf Metzler

  4. Faculty of Natural Sciences and Mathematics, Institute of Mathematics Saints Cyril and Methodius University, 1000, Skopje, Macedonia

    Živorad Tomovski

Authors
  1. Trifce Sandev
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  2. Ralf Metzler
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  3. Živorad Tomovski
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Correspondence to Trifce Sandev.

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Sandev, T., Metzler, R. & Tomovski, Ž. Velocity and displacement correlation functions for fractional generalized Langevin equations. fcaa 15, 426–450 (2012). https://doi.org/10.2478/s13540-012-0031-2

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