Multi-Additivity in Kaniadakis Entropy
It is known that Kaniadakis entropy, a generalization of the Shannon–Boltzmann–Gibbs entropic form, is always super-additive for any bipartite statistically independent distributions. In this paper, we show that when imposing a suitable constraint, there exist classes of maximal entropy distribution...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/26/1/77 |
| _version_ | 1827372322597634048 |
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| author | Antonio M. Scarfone Tatsuaki Wada |
| author_facet | Antonio M. Scarfone Tatsuaki Wada |
| author_sort | Antonio M. Scarfone |
| collection | DOAJ |
| description | It is known that Kaniadakis entropy, a generalization of the Shannon–Boltzmann–Gibbs entropic form, is always super-additive for any bipartite statistically independent distributions. In this paper, we show that when imposing a suitable constraint, there exist classes of maximal entropy distributions labeled by a positive real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℵ</mo><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> that makes Kaniadakis entropy multi-additive, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mrow><mi mathvariant="normal">A</mi><mo>∪</mo><mi mathvariant="normal">B</mi></mrow></msup><mo>]</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>ℵ</mo><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mfenced separators="" open="(" close=")"><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mo>]</mo></mrow><mo>+</mo><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mo>]</mo></mrow></mfenced></mrow></semantics></math></inline-formula>, under the composition of two statistically independent and identically distributed distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mrow><mi mathvariant="normal">A</mi><mo>∪</mo><mi mathvariant="normal">B</mi></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, with reduced distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> belonging to the same class. |
| first_indexed | 2024-03-08T10:56:56Z |
| format | Article |
| id | doaj.art-14664ae4e0804defbd46b7d1bab3e8f9 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-08T10:56:56Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-14664ae4e0804defbd46b7d1bab3e8f92024-01-26T16:23:15ZengMDPI AGEntropy1099-43002024-01-012617710.3390/e26010077Multi-Additivity in Kaniadakis EntropyAntonio M. Scarfone0Tatsuaki Wada1Istituto dei Sistemi Complessi—Consiglio Nazionale delle Ricerche (ISC-CNR), c/o Dipartimento di Scienza Applicata e Tecnologia del Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, ItalyRegion of Electrical and Electronic Systems Engineering, Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi 316-8511, Ibaraki, JapanIt is known that Kaniadakis entropy, a generalization of the Shannon–Boltzmann–Gibbs entropic form, is always super-additive for any bipartite statistically independent distributions. In this paper, we show that when imposing a suitable constraint, there exist classes of maximal entropy distributions labeled by a positive real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℵ</mo><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> that makes Kaniadakis entropy multi-additive, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mrow><mi mathvariant="normal">A</mi><mo>∪</mo><mi mathvariant="normal">B</mi></mrow></msup><mo>]</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>ℵ</mo><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mfenced separators="" open="(" close=")"><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mo>]</mo></mrow><mo>+</mo><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mo>]</mo></mrow></mfenced></mrow></semantics></math></inline-formula>, under the composition of two statistically independent and identically distributed distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mrow><mi mathvariant="normal">A</mi><mo>∪</mo><mi mathvariant="normal">B</mi></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, with reduced distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> belonging to the same class.https://www.mdpi.com/1099-4300/26/1/77<i>κ</i>-entropypseudo-additivitypower-law distributions |
| spellingShingle | Antonio M. Scarfone Tatsuaki Wada Multi-Additivity in Kaniadakis Entropy Entropy <i>κ</i>-entropy pseudo-additivity power-law distributions |
| title | Multi-Additivity in Kaniadakis Entropy |
| title_full | Multi-Additivity in Kaniadakis Entropy |
| title_fullStr | Multi-Additivity in Kaniadakis Entropy |
| title_full_unstemmed | Multi-Additivity in Kaniadakis Entropy |
| title_short | Multi-Additivity in Kaniadakis Entropy |
| title_sort | multi additivity in kaniadakis entropy |
| topic | <i>κ</i>-entropy pseudo-additivity power-law distributions |
| url | https://www.mdpi.com/1099-4300/26/1/77 |
| work_keys_str_mv | AT antoniomscarfone multiadditivityinkaniadakisentropy AT tatsuakiwada multiadditivityinkaniadakisentropy |