Variations in the temporal evolution of seismicity pointed out by non-extensive statistical physics approach
Authors: Rotondi Renata , Varini Elisa 
Published: March 2020

&lt;p&gt;&lt;span&gt;Physics-based models focus on the generation process of individual earthquakes but the &lt;/span&gt;&lt;span&gt;strong&lt;/span&gt;&lt;span&gt; space-time interaction existing among the events of seismic sequences requires that the seismic&lt;/span&gt;&lt;span&gt;&lt;span&gt; activity be studied as a whole &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;through statistics-based models&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;span&gt;in order&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; to forecast &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;its&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; future trend. Properties such as long-range interactions, power law distributions, fractal geometries are common to all complex systems and are also shared by earthquakes and fault systems. Over recent years, on the one hand many studies have shown the inability of classical statistical mechanics to treat complex systems in exhaustive manner, and, on the other hand, the application of the Tsallis entropy &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;em&gt;S&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;sub&gt;&lt;span&gt;&lt;span&gt;&lt;em&gt;q &lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;span&gt;&lt;span&gt;- a generalization of the classical Boltzmann-Gibbs entropy&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;span&gt;in&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; non-extensive sense &amp;#8211; has led to long-tailed power-law distributions typical of complex systems (Vallianatos&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;em&gt; et al&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;. 201&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;8&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;); it seems hence that the non-extensive statistical physics can offer an appropriate framework of investigation for complex phenomena. In this work we follow this approach giving a detailed statistical treatment of its application to &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;Italian &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;earthquake sequences&lt;/span&gt;&lt;/span&gt; &lt;span&gt;&lt;span&gt;covering&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; a period of some years; &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;each data set &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;was&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; partitioned in moving &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;time windows.&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;Given a continuous variable X with probability distribution &lt;/span&gt;&lt;span&gt;&lt;em&gt;f(X)&lt;/em&gt;&lt;/span&gt;&lt;span&gt;, by maximizing the Tsallis entropy under appropriate constraints, such as the generalized expectation value and the normalization constant, it turns out that &lt;/span&gt;&lt;span&gt;&lt;em&gt;f(X)&lt;/em&gt;&lt;/span&gt;&lt;span&gt; is a &lt;/span&gt;&lt;span&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;span&gt;-exponential distribution. The &lt;/span&gt;&lt;span&gt;&lt;em&gt;q &lt;/em&gt;&lt;/span&gt;&lt;span&gt;entropic index can assume &lt;/span&gt;&lt;span&gt;positive &lt;/span&gt;&lt;span&gt;values less or larger than 1: in the former case&lt;/span&gt; &lt;span&gt;the system is in a super-additive state and &lt;/span&gt;&lt;span&gt;&lt;em&gt;f(X)&lt;/em&gt;&lt;/span&gt;&lt;span&gt; is defined on a finite domain depending on model parameters, in the latter case the system is in a sub-additive state and&lt;/span&gt;&lt;span&gt;&lt;em&gt; f(X)&lt;/em&gt;&lt;/span&gt;&lt;span&gt; is defined on R&lt;/span&gt;&lt;sup&gt;&lt;span&gt;+&lt;/span&gt;&lt;/sup&gt;&lt;span&gt;. &lt;/span&gt;&lt;span&gt;Through a variable transformation &lt;/span&gt;&lt;span&gt;required by&lt;/span&gt;&lt;span&gt; the fragment-asperity model for earthquake generation, &lt;/span&gt;&lt;span&gt;one&lt;/span&gt; &lt;span&gt;derives&lt;/span&gt; &lt;span&gt;the probability distribution of the magnitude&lt;/span&gt; &lt;span&gt;from the two versions of &lt;/span&gt;&lt;span&gt;&lt;em&gt;f(X)&lt;/em&gt;&lt;/span&gt;&lt;span&gt;. &lt;/span&gt;&lt;span&gt;Following the Bayesian approach we have &lt;/span&gt;&lt;span&gt;estima&lt;/span&gt;&lt;span&gt;ted&lt;/span&gt; &lt;span&gt;the&lt;/span&gt;&lt;span&gt; parameters &lt;/span&gt;&lt;span&gt;by generating rando&lt;/span&gt;&lt;span&gt;m&lt;/span&gt;&lt;span&gt; samples from the posterior distributions &lt;/span&gt;&lt;span&gt;through &lt;/span&gt;&lt;span&gt;the Metropolis-Hasting&lt;/span&gt; &lt;span&gt;algorithm; &lt;/span&gt;&lt;span&gt;moreover, in each time window, we have evaluated the Tsallis entropy and compared the performance of the &lt;/span&gt;&lt;span&gt;two&lt;/span&gt;&lt;span&gt; versions of the magnitude distribution in terms of marginal posterior likelihood.&lt;/span&gt; &lt;span&gt;The t&lt;/span&gt;&lt;span&gt;emporal&lt;/span&gt;&lt;span&gt; v&lt;/span&gt;&lt;span&gt;ariations of the &lt;/span&gt;&lt;span&gt;&lt;em&gt;q-&lt;/em&gt;&lt;/span&gt;&lt;span&gt;index and of the entropy &lt;/span&gt;&lt;span&gt;&lt;em&gt;S&lt;/em&gt;&lt;/span&gt;&lt;sub&gt;&lt;span&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;/sub&gt;&lt;span&gt;can be helpful in identifying in which dynamics regime the system &lt;/span&gt;&lt;span&gt;is&lt;/span&gt;&lt;span&gt;, and &lt;/span&gt;&lt;span&gt;therefore &lt;/span&gt;&lt;span&gt;in improving our ability to forecast seismicity evo&lt;/span&gt;&lt;span&gt;lution.&lt;/span&gt; &lt;span&gt;S&lt;/span&gt;&lt;span&gt;&lt;span&gt;ome of the results achieved partially disagree with th&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;ose&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; present&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; in the literature &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;(Vallianatos &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;em&gt;et al. &lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;201&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;8&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;)&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;; &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;what seems reasonable is to consider &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;the &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;change &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;of one of these&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt; variable&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;s, &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;rather than &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;a &lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;specific trend, as index of a phase change of the physical system.&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;References &lt;/strong&gt;&lt;/p&gt;&lt;p&gt;Vallianatos F., Michas G. and G. Papadakis (2018) &lt;em&gt;Nonextensive statistical seismology: An overview&lt;/em&gt;, from: &lt;strong&gt;Complexity and Seismic Time Series. Measurement and Application&lt;/strong&gt;, eds. Chelidze T., Vallianatos F., Telesca L., Elsevier, 25-59&lt;/p&gt;