Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character
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 con...
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/5/477 |
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| author | Katarzyna Górska Andrzej Horzela |
| author_facet | Katarzyna Górska Andrzej Horzela |
| author_sort | Katarzyna Górska |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>. 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. |
| first_indexed | 2024-03-09T00:30:12Z |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T00:30:12Z |
| publishDate | 2021-02-01 |
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| series | Mathematics |
| spelling | doaj.art-215cd1185a6a404db30489b87ca6429c2023-12-11T18:32:06ZengMDPI AGMathematics2227-73902021-02-019547710.3390/math9050477Non-Debye Relaxations: Two Types of Memories and Their Stieltjes CharacterKatarzyna Górska0Andrzej Horzela1Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31342 Kraków, PolandInstitute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31342 Kraków, PolandIn 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>. 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.https://www.mdpi.com/2227-7390/9/5/477non-Debye relaxationspositive definite functionsSonine equationLaplace (Lévy) exponent |
| spellingShingle | Katarzyna Górska Andrzej Horzela Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character Mathematics non-Debye relaxations positive definite functions Sonine equation Laplace (Lévy) exponent |
| title | Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character |
| title_full | Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character |
| title_fullStr | Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character |
| title_full_unstemmed | Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character |
| title_short | Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character |
| title_sort | non debye relaxations two types of memories and their stieltjes character |
| topic | non-Debye relaxations positive definite functions Sonine equation Laplace (Lévy) exponent |
| url | https://www.mdpi.com/2227-7390/9/5/477 |
| work_keys_str_mv | AT katarzynagorska nondebyerelaxationstwotypesofmemoriesandtheirstieltjescharacter AT andrzejhorzela nondebyerelaxationstwotypesofmemoriesandtheirstieltjescharacter |