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-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>&#949;</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>&#8733;</mo> <msub> <mi>&#949;</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.

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