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Mathematics 2017, 5(4), 66; doi:10.3390/math5040066

Generalized Langevin Equation and the Prabhakar Derivative

1
Radiation Safety Directorate, Partizanski Odredi 143, P.O. Box 22, 1020 Skopje, Macedonia
2
Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University in Skopje, P.O. Box 162, 1001 Skopje, Macedonia
3
Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
Received: 15 October 2017 / Revised: 11 November 2017 / Accepted: 15 November 2017 / Published: 20 November 2017
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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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. View Full-Text
Keywords: generalized Langevin equation; regularized Prabhakar derivative; tempered memory kernel; Mittag-Leffler functions generalized Langevin equation; regularized Prabhakar derivative; tempered memory kernel; Mittag-Leffler functions
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Sandev, T. Generalized Langevin Equation and the Prabhakar Derivative. Mathematics 2017, 5, 66.

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