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Generalized Langevin Equation and the Prabhakar Derivative

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  • Trifce Sandev

    (Radiation Safety Directorate, Partizanski Odredi 143, P.O. Box 22, 1020 Skopje, Macedonia
    Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University in Skopje, P.O. Box 162, 1001 Skopje, Macedonia
    Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia)

Abstract

We consider a generalized Langevin equation with regularized Prabhakar derivative operator. We analyze the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. We further introduce the so-called tempered regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term represented through the tempered derivative. Various diffusive behaviors are observed. We show the importance of the three parameter Mittag-Leffler function in the description of anomalous diffusion in complex media. We also give analytical results related to the generalized Langevin equation for a harmonic oscillator with generalized friction. The normalized displacement correlation function shows different behaviors, such as monotonic and non-monotonic decay without zero-crossings, oscillation-like behavior without zero-crossings, critical behavior, and oscillation-like behavior with zero-crossings. These various behaviors appear due to the friction of the complex environment represented by the Mittag-Leffler and tempered Mittag-Leffler memory kernels. Depending on the values of the friction parameters in the system, either diffusion or oscillations dominate.

Suggested Citation

  • Trifce Sandev, 2017. "Generalized Langevin Equation and the Prabhakar Derivative," Mathematics, MDPI, vol. 5(4), pages 1-11, November.
    • Handle: RePEc:gam:jmathe:v:5:y:2017:i:4:p:66-:d:119597
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    References listed on IDEAS

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    1. Sandev, Trifce & Tomovski, Živorad & Dubbeldam, Johan L.A., 2011. "Generalized Langevin equation with a three parameter Mittag-Leffler noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3627-3636.
    2. Liemert, André & Sandev, Trifce & Kantz, Holger, 2017. "Generalized Langevin equation with tempered memory kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 356-369.
    3. Sandev, Trifce & Sokolov, Igor M. & Metzler, Ralf & Chechkin, Aleksei, 2017. "Beyond monofractional kinetics," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 210-217.
    4. Pottier, Noëlle, 2003. "Aging properties of an anomalously diffusing particule," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(3), pages 371-382.
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    Cited by:

    1. Pece Trajanovski & Petar Jolakoski & Ljupco Kocarev & Trifce Sandev, 2023. "Ornstein–Uhlenbeck Process on Three-Dimensional Comb under Stochastic Resetting," Mathematics, MDPI, vol. 11(16), pages 1-28, August.

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