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Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

Author

Listed:
  • Katarzyna Górska

    (Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31342 Kraków, Poland)

  • Andrzej Horzela

    (Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31342 Kraków, Poland)

Abstract

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M ( t ) , which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M ( t ) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k ( t ) . Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

Suggested Citation

  • Katarzyna Górska & Andrzej Horzela, 2021. "Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character," Mathematics, MDPI, vol. 9(5), pages 1-13, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:477-:d:506244
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    References listed on IDEAS

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    1. Rudolf Hilfer, 2017. "Composite continuous time random walks," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(12), pages 1-4, December.
    2. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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