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On the Cole-Cole relaxation function and related Mittag-Leffler distribution

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  • Weron, Karina
  • Kotulski, Marcin

Abstract

In the framework of the one-dimensional fractal time random walk (FTRW) relaxation model, we rigorously show that the frequency domain response takes, in both nonbiased and biased walks, the only possible Cole-Cole form. The underlying reason for this is the specific form of the relaxation function (the survival probability of a relaxing system) determined in this model by the Mittag-Leffler distribution. We provide also analytical formulas for the propagators of the nonbiased and biased FTRWs.

Suggested Citation

  • Weron, Karina & Kotulski, Marcin, 1996. "On the Cole-Cole relaxation function and related Mittag-Leffler distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 232(1), pages 180-188.
    • Handle: RePEc:eee:phsmap:v:232:y:1996:i:1:p:180-188
    DOI: 10.1016/0378-4371(96)00209-9
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    References listed on IDEAS

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    1. Hu, Y. M. & Woyczynski, W. A., 1995. "Limit Behavior of Quadratic Forms of Moving Averages and Statistical Solutions of the Burgers' Equation," Journal of Multivariate Analysis, Elsevier, vol. 52(1), pages 15-44, January.
    2. Aleksander Janicki & Aleksander Weron, 1994. "Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook9401.
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    Cited by:

    1. Kozubowski, Tomasz J. & Meerschaert, Mark M., 2009. "A bivariate infinitely divisible distribution with exponential and Mittag-Leffler marginals," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1596-1601, July.
    2. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    3. Aydiner, Ekrem, 2021. "Memory effects and KWW relaxation of the interacting magnetic nano-particles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 572(C).

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