IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v5y2017i4p66-d119597.html
   My bibliography  Save this article

Generalized Langevin Equation and the Prabhakar Derivative

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/5/4/66/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/5/4/66/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    2. Lini Qiu & Guitian He & Yun Peng & Huijun Lv & Yujie Tang, 2023. "Average amplitudes analysis for a phenomenological model under hydrodynamic interactions with periodic perturbation and multiplicative trichotomous noise," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(4), pages 1-20, April.
    3. 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.
    4. dos Santos, Maike A.F., 2019. "Analytic approaches of the anomalous diffusion: A review," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 86-96.
    5. Paekivi, Sander & Mankin, Romi, 2019. "Bimodality of the interspike interval distributions for subordinated diffusion models of integrate-and-fire neurons," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).
    6. Wang, Zhaoyang & Lin, Ping & Wang, Erhui, 2021. "Modeling multiple anomalous diffusion behaviors on comb-like structures," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    7. Ram K. Saxena & Zivorad Tomovski & Trifce Sandev, 2015. "Analytical Solution of Generalized Space-Time Fractional Cable Equation," Mathematics, MDPI, vol. 3(2), pages 1-18, April.
    8. Alexander Apelblat, 2020. "Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach," Mathematics, MDPI, vol. 8(5), pages 1-22, April.
    9. Trifce Sandev & Irina Petreska & Ervin K. Lenzi, 2016. "Effective Potential from the Generalized Time-Dependent Schrödinger Equation," Mathematics, MDPI, vol. 4(4), pages 1-9, September.
    10. Baleanu, D. & Shiri, B., 2018. "Collocation methods for fractional differential equations involving non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 136-145.
    11. Sandev, Trifce & Sokolov, Igor M. & Metzler, Ralf & Chechkin, Aleksei, 2017. "Beyond monofractional kinetics," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 210-217.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:5:y:2017:i:4:p:66-:d:119597. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.