Senses along Which the Entropy <i>S<sub>q</sub></i> Is Unique
The Boltzmann–Gibbs–von Neumann–Shannon <i>additive</i> entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mrow><mi>B</mi><mi>G&l...
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MDPI AG
2023-05-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/5/743 |
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| author | Constantino Tsallis |
| author_facet | Constantino Tsallis |
| author_sort | Constantino Tsallis |
| collection | DOAJ |
| description | The Boltzmann–Gibbs–von Neumann–Shannon <i>additive</i> entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mrow><mi>B</mi><mi>G</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>k</mi><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub><mo form="prefix">ln</mo><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the <i>nonextensive statistical mechanics</i>—as currently referred to—grounded on the <i>nonadditive</i> entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>q</mi></msub><mo>=</mo><mi>k</mi><mfrac><mrow><mn>1</mn><mo>−</mo><msub><mo>∑</mo><mi>i</mi></msub><msubsup><mi>p</mi><mi>i</mi><mi>q</mi></msubsup></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></semantics></math></inline-formula> as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula> plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity—<i>plectics</i>, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely <i>In what senses is entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula> unique?</i> The present effort is dedicated to a—surely non exhaustive—mathematical answer to this basic question. |
| first_indexed | 2024-03-11T03:46:08Z |
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| language | English |
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| spelling | doaj.art-de3a68dae6e04848b21e7b79272cf0b12023-11-18T01:15:45ZengMDPI AGEntropy1099-43002023-05-0125574310.3390/e25050743Senses along Which the Entropy <i>S<sub>q</sub></i> Is UniqueConstantino Tsallis0Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, BrazilThe Boltzmann–Gibbs–von Neumann–Shannon <i>additive</i> entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mrow><mi>B</mi><mi>G</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>k</mi><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub><mo form="prefix">ln</mo><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the <i>nonextensive statistical mechanics</i>—as currently referred to—grounded on the <i>nonadditive</i> entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>q</mi></msub><mo>=</mo><mi>k</mi><mfrac><mrow><mn>1</mn><mo>−</mo><msub><mo>∑</mo><mi>i</mi></msub><msubsup><mi>p</mi><mi>i</mi><mi>q</mi></msubsup></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></semantics></math></inline-formula> as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula> plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity—<i>plectics</i>, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely <i>In what senses is entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>q</mi></msub></semantics></math></inline-formula> unique?</i> The present effort is dedicated to a—surely non exhaustive—mathematical answer to this basic question.https://www.mdpi.com/1099-4300/25/5/743Boltzmann–Gibbs statistical mechanicsnonadditive entropiesnonextensive statistical mechanicsentropic uniqueness theorems |
| spellingShingle | Constantino Tsallis Senses along Which the Entropy <i>S<sub>q</sub></i> Is Unique Entropy Boltzmann–Gibbs statistical mechanics nonadditive entropies nonextensive statistical mechanics entropic uniqueness theorems |
| title | Senses along Which the Entropy <i>S<sub>q</sub></i> Is Unique |
| title_full | Senses along Which the Entropy <i>S<sub>q</sub></i> Is Unique |
| title_fullStr | Senses along Which the Entropy <i>S<sub>q</sub></i> Is Unique |
| title_full_unstemmed | Senses along Which the Entropy <i>S<sub>q</sub></i> Is Unique |
| title_short | Senses along Which the Entropy <i>S<sub>q</sub></i> Is Unique |
| title_sort | senses along which the entropy i s sub q sub i is unique |
| topic | Boltzmann–Gibbs statistical mechanics nonadditive entropies nonextensive statistical mechanics entropic uniqueness theorems |
| url | https://www.mdpi.com/1099-4300/25/5/743 |
| work_keys_str_mv | AT constantinotsallis sensesalongwhichtheentropyissubqsubiisunique |