IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v601y2022ics0378437122003995.html
   My bibliography  Save this article

On the correlation between Kappa and Lévy stable distributions

Author

Listed:
  • Tawfik, Ashraf M.
  • Elkamash, I.S.

Abstract

This article investigates the correlation between the Kappa and Lévy distributions via two approaches of the Klein–Kramers equation. The first approach illustrates the velocity distribution functions via the solution of the fractional Klein–Kramers equation obtained using the Riesz fractional derivative. In contrast, the second approach shows the velocity distribution functions according to steady-state Kappa distribution, which arises from the solution of the Klein–Kramers equation with variable coefficients dependent on velocity. We find a unique and straightforward formula representing the relation between the Kappa exponent and the fractality index (Lévy stable index). The results indicate a viable probability distribution obtained from the fractional equations as an alternative to the Kappa distribution. Hence, our results may shed light on the stationary power-law distribution in non extensive statistics and introduce a new correlation between the order of the fractional derivative (α) and the nonthermal index (κ) of the distribution function. Our results also show exact matching with the probability distributions illustrated in the literature.

Suggested Citation

  • Tawfik, Ashraf M. & Elkamash, I.S., 2022. "On the correlation between Kappa and Lévy stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    • Handle: RePEc:eee:phsmap:v:601:y:2022:i:c:s0378437122003995
    DOI: 10.1016/j.physa.2022.127576
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437122003995
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2022.127576?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Yanovsky, V.V. & Chechkin, A.V. & Schertzer, D. & Tur, A.V., 2000. "Lévy anomalous diffusion and fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 282(1), pages 13-34.
    2. Metzler, Ralf & Chechkin, Aleksei V. & Gonchar, Vsevolod Yu. & Klafter, Joseph, 2007. "Some fundamental aspects of Lévy flights," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 129-142.
    3. Bazzani, Armando & Bassi, Gabriele & Turchetti, Giorgio, 2003. "Diffusion and memory effects for stochastic processes and fractional Langevin equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(3), pages 530-550.
    4. Sandev, Trifce & Schulz, Alexander & Kantz, Holger & Iomin, Alexander, 2018. "Heterogeneous diffusion in comb and fractal grid structures," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 551-555.
    5. Gorenflo, Rudolf & Mainardi, Francesco & Vivoli, Alessandro, 2007. "Continuous-time random walk and parametric subordination in fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 87-103.
    6. Du, Jiulin, 2012. "Transition state theory: A generalization to nonequilibrium systems with power-law distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1718-1728.
    7. Guo, Ran & Du, Jiulin, 2014. "Are power-law distributions an equilibrium distribution or a stationary nonequilibrium distribution?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 406(C), pages 281-286.
    8. Langlands, T.A.M., 2006. "Solution of a modified fractional diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 136-144.
    9. Plastino, A.R. & Plastino, A., 1995. "Non-extensive statistical mechanics and generalized Fokker-Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 222(1), pages 347-354.
    Full references (including those not matched with items on IDEAS)

    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. Pereira, A.P.P. & Fernandes, J.P. & Atman, A.P.F. & Acebal, J.L., 2018. "Parameter calibration between models and simulations: Connecting linear and non-linear descriptions of anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 369-382.
    2. Hirsch, Christian & Jahnel, Benedikt & Cali, Elie, 2022. "Percolation and connection times in multi-scale dynamic networks," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 490-518.
    3. Golder, J. & Joelson, M. & Néel, M.C., 2011. "Mass transport with sorption in porous media," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(10), pages 2181-2189.
    4. Awad, Emad & Sandev, Trifce & Metzler, Ralf & Chechkin, Aleksei, 2021. "Closed-form multi-dimensional solutions and asymptotic behaviors for subdiffusive processes with crossovers: I. Retarding case," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    5. Ván, P, 2004. "One- and two-component fluids: restrictions from the Second Law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 418-426.
    6. Bassi, Gabriele & Bazzani, Armando & Mais, Helmut & Turchetti, Giorgio, 2005. "Stochastic continuity equation and related processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 347(C), pages 17-37.
    7. Ván, P., 2006. "Unique additive information measures—Boltzmann–Gibbs–Shannon, Fisher and beyond," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(1), pages 28-33.
    8. Lubashevsky, Ihor, 2013. "Equivalent continuous and discrete realizations of Lévy flights: A model of one-dimensional motion of an inertial particle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(10), pages 2323-2346.
    9. Awad, Emad, 2019. "On the time-fractional Cattaneo equation of distributed order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 518(C), pages 210-233.
    10. Drozdov, A.D., 2007. "Fractional oscillator driven by a Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 237-245.
    11. Guo, Wei & Liu, Ying-Zhou & Huang, Fei-Jie & Shi, Hong-Da & Du, Lu-Chun, 2023. "Brownian particles in a periodic potential corrugated by disorder: Anomalous diffusion and ergodicity breaking," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    12. Khalili Golmankhaneh, Alireza & Ontiveros, Lilián Aurora Ochoa, 2023. "Fractal calculus approach to diffusion on fractal combs," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    13. Qi, Haitao & Jiang, Xiaoyun, 2011. "Solutions of the space-time fractional Cattaneo diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(11), pages 1876-1883.
    14. Oraby, T. & Suazo, E. & Arrubla, H., 2023. "Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    15. Guo, Gang & Chen, Bin & Zhao, Xinjun & Zhao, Fang & Wang, Quanmin, 2015. "First passage time distribution of a modified fractional diffusion equation in the semi-infinite interval," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 433(C), pages 279-290.
    16. Lenzi, E.K. & Ribeiro, M.A. & Fuziki, M.E.K. & Lenzi, M.K. & Ribeiro, H.V., 2018. "Nonlinear diffusion equation with reaction terms: Analytical and numerical results," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 254-265.
    17. Marjorie Hahn & Kei Kobayashi & Sabir Umarov, 2012. "SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations," Journal of Theoretical Probability, Springer, vol. 25(1), pages 262-279, March.
    18. Liu, Lin & Chen, Siyu & Bao, Chunxu & Feng, Libo & Zheng, Liancun & Zhu, Jing & Zhang, Jiangshan, 2023. "Analysis of the absorbing boundary conditions for anomalous diffusion in comb model with Cattaneo model in an unbounded region," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    19. Moretto, Enrico & Pasquali, Sara & Trivellato, Barbara, 2016. "Option pricing under deformed Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 446(C), pages 246-263.
    20. Zeng, Lingzao & Li, Jianlong & Shi, Jiachun, 2012. "M-ary signal detection via a bistable system in the presence of Lévy noise," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 378-382.

    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:eee:phsmap:v:601:y:2022:i:c:s0378437122003995. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.