Associating an Entropy with Power-Law Frequency of Events
Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy <inline-formu...
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doaj-e5bd2898301743578f14096c975f99522020-11-24T21:23:01ZengMDPI AGEntropy1099-43002018-12-01201294010.3390/e20120940e20120940Associating an Entropy with Power-Law Frequency of EventsEvaldo M. F. Curado0Fernando D. Nobre1Angel Plastino2Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, BrazilCentro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, BrazilLa Plata National University and Argentina’s National Research Council (IFLP-CCT-CONICET)-C. C. 727, La Plata 1900, ArgentinaEvents occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula>. The relevant parameters become: (i) The entropic index <i>q</i>, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy <inline-formula> <math display="inline"> <semantics> <msub> <mi>ε</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <msub> <mi>T</mi> <mi>q</mi> </msub> <mo>∝</mo> <msub> <mi>ε</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> (i.e., isothermal processes, for a given <i>q</i>), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index <i>q</i> and value of <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula> are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures.https://www.mdpi.com/1099-4300/20/12/940self-organized criticalitygeneralized entropiesnonextensive thermostatisticsinformation theory |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Evaldo M. F. Curado Fernando D. Nobre Angel Plastino |
spellingShingle |
Evaldo M. F. Curado Fernando D. Nobre Angel Plastino Associating an Entropy with Power-Law Frequency of Events Entropy self-organized criticality generalized entropies nonextensive thermostatistics information theory |
author_facet |
Evaldo M. F. Curado Fernando D. Nobre Angel Plastino |
author_sort |
Evaldo M. F. Curado |
title |
Associating an Entropy with Power-Law Frequency of Events |
title_short |
Associating an Entropy with Power-Law Frequency of Events |
title_full |
Associating an Entropy with Power-Law Frequency of Events |
title_fullStr |
Associating an Entropy with Power-Law Frequency of Events |
title_full_unstemmed |
Associating an Entropy with Power-Law Frequency of Events |
title_sort |
associating an entropy with power-law frequency of events |
publisher |
MDPI AG |
series |
Entropy |
issn |
1099-4300 |
publishDate |
2018-12-01 |
description |
Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula>. The relevant parameters become: (i) The entropic index <i>q</i>, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy <inline-formula> <math display="inline"> <semantics> <msub> <mi>ε</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <msub> <mi>T</mi> <mi>q</mi> </msub> <mo>∝</mo> <msub> <mi>ε</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> (i.e., isothermal processes, for a given <i>q</i>), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index <i>q</i> and value of <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>q</mi> </msub> </semantics> </math> </inline-formula> are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures. |
topic |
self-organized criticality generalized entropies nonextensive thermostatistics information theory |
url |
https://www.mdpi.com/1099-4300/20/12/940 |
work_keys_str_mv |
AT evaldomfcurado associatinganentropywithpowerlawfrequencyofevents AT fernandodnobre associatinganentropywithpowerlawfrequencyofevents AT angelplastino associatinganentropywithpowerlawfrequencyofevents |
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