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

  • Z. Z. AlisultanovEmail author
  • A. M. Agalarov
  • A. A. Potapov
  • G. B. Ragimkhanov


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.


Fractional liouville derivative Fractional riesz derivative Green’s functions Maxwell equations Many-particle systems Instability in the gas discharge 



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).


  1. 1.
    I.M. Lifshitz, M.I. Kaganov, Sov. Phys. Usp. 2, 831–855 (1960)ADSCrossRefGoogle Scholar
  2. 2.
    F.G. Bass, E.A. Rubinstein, Sov. Phys.: Solid State 19, 800 (1977)Google Scholar
  3. 3.
    F.G. Bass, V.V. Konotop, A.P. Panchekha, JETP 99(6), 1055–1060 (1989)Google Scholar
  4. 4.
    F.G. Bass, A.D. Panchekha, JETP 99(6), 1711–1717 (1991)Google Scholar
  5. 5.
    R. Hilfer (ed.), Applications of Fractional Calculus in Physics (World Scientific, 2000), p. 463Google Scholar
  6. 6.
    V.V. Uchaikin, The method of fractional derivatives (Publishing house “Artichoke”, Ulyanovsk, 2008), p. 512Google Scholar
  7. 7.
    V.V. Uchaikin, Anomalous diffusion and fractional stable distributions. JETP 97(4), 810–825 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    A.A. Stanislavsky, Probability interpretation of the integral of fractional order. Theor. Math. Phys. 138(3), 418–431 (2004)CrossRefGoogle Scholar
  9. 9.
    R.P. Meilanov, M.R. Shabanova, Peculiarities of solutions to the heat conduction equation in fractional derivatives. Tech. Phys. 56(7), 903–908 (2011)CrossRefGoogle Scholar
  10. 10.
    R.T. Sibatov, V.V. Uchaikin, Fractional differential approach to dispersive transport in semiconductors. Phys. Usp. 52(10), 1019–1043 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    SSh Rekhviashvili, Application of fractional integro-differentiation to the calculation of the thermodynamic properties of surfaces. Phys. Solid State 49(4), 796–799 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    X.-B. Wang, J.-X. Li, Q. Jiang, Z.-H. Zhang, Phys. Rev. 49(14), 9778–9781 (1994)CrossRefGoogle Scholar
  13. 13.
    A.V. Milovanov, J.J. Rasmussen, Phys. Rev. B 66, 134505 (2002)ADSCrossRefGoogle Scholar
  14. 14.
    F.J. Dyson, Commun. Math. Phys. 12(2), 91–107 (1969)Google Scholar
  15. 15.
    F.J. Dyson, Commun. Math. Phys. 12(3), 212–215 (1969)Google Scholar
  16. 16.
    F.J. Dyson, Commun. Math. Phys. 21(4), 269–283 (1971)Google Scholar
  17. 17.
    N. Laskin, G. Zaslavsky, Phys. A 368, 38–54 (2006)CrossRefGoogle Scholar
  18. 18.
    V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-adic Analysis and Mathematical Physics (M.: Nauka, 1994), p. 352 sGoogle Scholar
  19. 19.
    N. Laskin, Phys. Lett. A 268, 298–305 (2000)Google Scholar
  20. 20.
    N. Laskin, Phys. Rev. E 62(3), 3135–3145 (2000)Google Scholar
  21. 21.
    V.M. Eleonsky, V.G. Korolev, N.E. Kulagin, Lett. JETP 76(12), 859–862 (2002)Google Scholar
  22. 22.
    V.E. Tarasov, Phys. Rev. E 71(1), 011102 (2005)Google Scholar
  23. 23.
    V.E. Tarasov, Chaos 16(3), 033108 (2006)Google Scholar
  24. 24.
    V.E. Tarasov, Mod. Phys. Lett. B 21(5), 237–248 (2007)Google Scholar
  25. 25.
    A.A. Potapov, Fractals and Scaling in the Radar: A Look from 2015, Book of Abstracts 8nd International Conference (CHAOS’ 2015) on Chaotic Modeling, Simulation and Applications, Henri Poincaré Institute, Paris, 26–29 May 2015, pp. 101, 102Google Scholar
  26. 26.
    A.A. Potapov, V.A. German, Detection of artificial objects with fractal signatures. Pattern Recogn. Image Anal. 8(2), 226–229 (1998)Google Scholar
  27. 27.
    A.A. Potapov, Y.V. Gulyaev, S.A. Nikitov, A.A. Pakhomov, V.A. German, Newest Images Processing Methods, ed. by A.A. Potapov (FIZMATLIT, Moscow, 2008), p. 496Google Scholar
  28. 28.
    Z.Z. Alisultanov, R.P. Meilanov, Theor. Math. Phys. 171, 404 (2012)CrossRefGoogle Scholar
  29. 29.
    Z.Z. Alisultanov, R.P. Meilanov, Theor. Math. Phys. 173(1), 1445–1456 (2012)CrossRefGoogle Scholar
  30. 30.
    Z.Z. Alisultanov, G.B. Ragimkhanov, Fractional-differential approach to the study of instability in a gas discharge. Chaos, Solitons Fractals 107, 39–42 (2018)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    A.N. Bogolyubov, A.A. Potapov, S.Sh. Rekhviashvili, Introduction of fractional integro-differentiation in classical electrodynamics. WMU. Phys. Astron. Series 3, 64(4), 9–12 (2009)Google Scholar
  32. 32.
    A.M Nakhushev, Fractional Calculus and Its Application (Fizmatlit, Moscow, 2003), p. 272 secGoogle Scholar
  33. 33.
    N. Van Hieu, Fundamentals of the Method of Second Quantization (Energoatomizdat, Moscow, 1984Google Scholar
  34. 34.
    R. Mailanov, M.Y. Yanpolov, Lett. ZhTF 28(1), 67–73 (2001)Google Scholar
  35. 35.
    R.P. Meilanov, Z.Z. Alisultanov, International Russian-Bulgarian Symposium “Equations of the Mixed Type” (Nalchik, 2010), p. 161Google Scholar
  36. 36.
    A. Potapov, Fractals in Radiophysics and Radar: Sampling Topology (The University Book, Moscow, 2005), p. 848Google Scholar
  37. 37.
    M. Naber, Time fractional Schrodinger equation Department of Mathematics, Monroe, Michigan, 48161-9746 (2004)Google Scholar
  38. 38.
    L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 2 (Nauka, Moscow, 1978)Google Scholar
  39. 39.
    G.B. Kadanov, Quantum Statistical Mechanics (Mir, Moscow, 1964)Google Scholar
  40. 40.
    C.G. Samko, F.F. Kilbas, O.I. Marichev, Integrals and Derivatives of Fractional Order and Some Applications (Science and Technology, Minsk, 1987), p. 688Google Scholar
  41. 41.
    A.B. Alkhasov, R.P. Meilanov, M.R. Shabanova, Fiz 84(2), 309–317 (2011)Google Scholar
  42. 42.
    I.G. Kaplan, Introduction to the Theory of Intermolecular Interactions (Science, Moscow, 1982), p. 312Google Scholar
  43. 43.
    R. Balescu, Statistical Mechanics of Charged Particles (Mir, Moscow, 1967), p. 516Google Scholar
  44. 44.
    W. Weibull, J. Appl. Mech.-Trans. ASME. 18(3), 293–297 (1951)Google Scholar
  45. 45.
    L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 1 (Nauka, Moscow, 1978)Google Scholar
  46. 46.
    I.A. Kvasnikov, Thermodynamics and Statistical Physics, vol. 4. Quantum statistics. - M.: Editorial URSS (2002)Google Scholar
  47. 47.
    A.N. Kozlov, V.N. Flerov, Lett. JETP. 25(1), 54–58 (1977)Google Scholar
  48. 48.
    V.F. Grin, D.S. Lepsveridze, EA Sal’kov et al., Pis’ma Zh. 87(8), 415–418 (1975)Google Scholar
  49. 49.
    E.G. Maksimov, Sh Van, M.V. Magnitskaya, S.V. Ebert, Pis’ma Zh. T. 21(7), 507–510 (2008)Google Scholar
  50. 50.
    Y.E. Lozovik, S.P. Merkulova, A.A. Sokolik, UFN. 178, 757–776 (2008)Google Scholar
  51. 51.
    G. Savini, A.C. Ferrari, F. Giustino, Phys. Rev. Lett. 105(037002), 1–4 (2010)Google Scholar
  52. 52.
    A.V. Milovanov, J.J. Rasmussen, arXiv:cond-mat/0201504v3 [cond-mat.supr-con] 8 Oct 2002
  53. 53.
    S.I. Yakovlenko, Tech. Phys. Lett. 2005. T.31. AT 4, 76–82Google Scholar
  54. 54.
    V.I. Karelin, A.A. Trenkin, Technical physics. Russ. J. Appl. Phys. 78(3), 29–35 (2008)Google Scholar
  55. 55.
    A.H. Nielsen, H.L. Pécseli, J. Juul, Phys. Plasmas 3, 1530 (1996)ADSCrossRefGoogle Scholar
  56. 56.
    B.A. Carreras, B. van Milligen, M.A. Pedrosa, R. Balbín, C. Hidalgo, D.E. Newman, E. Sánchez, M. Frances, I. García-Cortés, J. Bleuel, M. Endler, S. Davies, G.F. Matthews, Phys. Rev. Lett. 80, 4438 (1998)ADSCrossRefGoogle Scholar
  57. 57.
    B.A. Carreras, B. van Milligen, C. Hidalgo, R. Balbin, E. Sanchez, I. Garcia-Cortes, M.A. Pedrosa, J. Bleuel, M. Endler, Phys. Rev. Lett. 83, 3653 (1999)ADSCrossRefGoogle Scholar
  58. 58.
    V. Naulin, A. H. Nielsen, J. Juul Rasmussen, Phys. Plasmas 6, 4575 (1999)Google Scholar
  59. 59.
    G.M. Zaslavsky, M. Edelman, H. Weitzner, B. Carreras, G. McKee, R. Bravenec, R. Fonck, Phys. Plasmas 7, 3691 (2000)ADSCrossRefGoogle Scholar
  60. 60.
    J. Kigami, Analysis on Fractals (Cambridge University Press, 2001)Google Scholar
  61. 61.
    R.S. Strichartz, Differential Equations on Fractals (Princeton University Press, Princeton and Oxford, 2006)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Z. Z. Alisultanov
    • 1
    Email author
  • A. M. Agalarov
    • 1
  • A. A. Potapov
    • 2
    • 3
  • G. B. Ragimkhanov
    • 4
  1. 1.Department of Theoretical PhysicsAmirkhanov Institute of Physics of Dagestan Scientific Center of Russian Academy of SciencesMakhachkalaRussia
  2. 2.Kotel’nikov Institute of Radio Engineering and Electronics of Russian Academy of SciencesMoscowRussia
  3. 3.JNU-IRAE RAS Joint Laboratory of Information Technology and Fractal Processing of SignalsJiNan UniversityGuangzhouChina
  4. 4.Faculty of PhysicsDagestan State UniversityMakhachkalaRussia

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