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On the correlation between Kappa and Lévy stable distributions

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    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)

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