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

Katarzyna Górska and Andrzej Horzela
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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

Mathematics, 2021, vol. 9, issue 5, 1-13

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.

Keywords: non-Debye relaxations; positive definite functions; Sonine equation; Laplace (Lévy) exponent (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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