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Some Applications of Fractional Derivatives in Many-Particle Disordered Large Systems

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Fractional Dynamics, Anomalous Transport and Plasma Science

Abstract

In this chapter we consider several possibilities for introducing fractional derivatives with respect to time and spatial coordinate into the equations of a many-particle system. We considered the possibility of introducing a fractional time derivative in the quantum-mechanical Heisenberg equation, as well as elucidating the physical conditions for the appearance of fractional derivatives with respect to the spatial coordinate in the equation for the quantum Green’s functions. For the latter, the Hartree-Fock approximation is used to calculate the interparticle interaction potential. Finally, we applied the fractional derivative approach for specific tasks in plasma science. Namely, using an approach based on a fractional-order kinetic equation on the time variable, we investigated two types of instability in a gas discharge: the instability of the electron avalanche and the sticking instability in a non-self-sustaining discharge.

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Notes

  1. 1.

    We used the property of the fractional Riemann-Liouville derivative: \(\partial_{ - \infty x}^{\beta } {\text{e}}^{ax} = a^{\beta } {\text{e}}^{ax}\).

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Acknowledgements

AZZ, AAM and RGB dedicate this work to the memory of a friend and teacher Prof. Ruslan Meilanov.

AZZ thanks: President grant MK-2130.2017.2, RFBR (# 18-02-01022a).

AAP is grateful to the China grant “Leading Talent Program in Guangdong Province” (No. 00201502, 2016-2020) JiNan University (China, Guangzhou).

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Authors and Affiliations

  1. Department of Theoretical Physics, Amirkhanov Institute of Physics of Dagestan Scientific Center of Russian Academy of Sciences, Makhachkala, 367015, Russia

    Z. Z. Alisultanov & A. M. Agalarov

  2. Kotel’nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Moscow, 125009, Russia

    A. A. Potapov

  3. JNU-IRAE RAS Joint Laboratory of Information Technology and Fractal Processing of Signals, JiNan University, Guangzhou, 510632, China

    A. A. Potapov

  4. Faculty of Physics, Dagestan State University, 367000, Makhachkala, Dagestan Republic, Russia

    G. B. Ragimkhanov

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  1. Z. Z. Alisultanov
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  2. A. M. Agalarov
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  3. A. A. Potapov
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  4. G. B. Ragimkhanov
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Correspondence to Z. Z. Alisultanov .

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  1. ManLab, Technical University of Crete, Chania, Crete, Greece

    Christos H. Skiadas

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Alisultanov, Z.Z., Agalarov, A.M., Potapov, A.A., Ragimkhanov, G.B. (2018). Some Applications of Fractional Derivatives in Many-Particle Disordered Large Systems. In: Skiadas, C. (eds) Fractional Dynamics, Anomalous Transport and Plasma Science. Springer, Cham. https://doi.org/10.1007/978-3-030-04483-1_7

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