Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1−q−11−q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as ⟨x⟩q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers ⟨n⟩q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡⟨ζ(s)⟩q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζqΣ(s)≡∑n=1∞1⟨n⟩qs=1+1⟨2⟩qs+⋯ and ζqΠ(s)≡∏pprime11−⟨p⟩q−s=11−⟨2⟩q−s11−⟨3⟩q−s11−⟨5⟩q−s⋯, which, for q<1, generically satisfy ζqΣ(s)<ζqΠ(s), in variance with the q=1 case, where of course ζ1Σ(s)=ζ1Π(s).

Thermodynamic Definitions of Temperature and Kappa and Introduction of the Entropy Defect

This paper develops explicit and consistent definitions of the independent thermodynamic properties of temperature and the kappa index within the framework of nonextensive statistical mechanics and shows their connection with the formalism of kappa distributions. By defining the “entropy defect” in the composition of a system, we show how the nonextensive entropy of systems with correlations differs from the sum of the entropies of their constituents of these systems. A system is composed extensively when its elementary subsystems are independent, interacting with no correlations; this leads to an extensive system entropy, which is simply the sum of the subsystem entropies. In contrast, a system is composed nonextensively when its elementary subsystems are connected through long-range interactions that produce correlations. This leads to an entropy defect that quantifies the missing entropy, analogous to the mass defect that quantifies the mass (energy) associated with assembling subatomic particles. We develop thermodynamic definitions of kappa and temperature that connect with the corresponding kinetic definitions originated from kappa distributions. Finally, we show that the entropy of a system, composed by a number of subsystems with correlations, is determined using both discrete and continuous descriptions, and find: (i) the resulted entropic form expressed in terms of thermodynamic parameters; (ii) an optimal relationship between kappa and temperature; and (iii) the correlation coefficient to be inversely proportional to the temperature logarithm.

Scaling properties of the Mw7.0 Samos (Greece), 2020 aftershock sequence.

&lt;p&gt;On October 30, 2020 a strong shallow earthquake of magnitude Mw=7.0 occurred on the Eastern edge of Aegean Sea. The epicenter was located on the North offshore of the Greek island of Samos. The aim of our work is to present a first analysis of the scaling properties observed in the aftershock sequence as reported until December 31, 2020, as numerous seismic clusters activated. Our analysis is focused on the main of the clusters observed in the East area of the activated fault zone and strongly related with the main shock&amp;#8217;s fault. The aftershock sequence follows the Omori law with a value of p&amp;#8776;1.01 for the main cluster which is remarkably close to a logarithmic evolution. The analysis of interevent times distribution, based on non-extensive statistical physics indicates a system in an anomalous equilibrium with a cross over from anomalous (q&gt;1) to normal (q=1) statistical mechanics, for great interevent times. A discussion of the cross over observed, in terms of superstatistics is given. In addition the obtained value q&amp;#8776;1.67 suggests a system with one degree of freedom. Furthermore, an scaling of the migration of aftershock zone as a function of the logarithm of time is discussed in terms of rate strengthening rheology that govern the evolution of afterslip process.&lt;/p&gt;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;Tsallis, C. Introduction to Nonextensive Statistical Mechanics-Approaching a Complex World; Springer: New York, USA, 2009; pp. 1&amp;#8211;382.&lt;/p&gt;&lt;p&gt;Perfettini, H.,Frank, W. B., Marsan, D., and Bouchon, M. (2018). A model of aftershock migration driven by afterslip. Geophys. Res. Let., 45, 2283&amp;#8211;2293.&lt;/p&gt;&lt;p&gt;Vallianatos, F.; Papadakis, G.; Michas, G. (2016). Generalized statistical mechanics approaches to earthquakes and tectonics. Proc. R. Soc. A Math. Phys. Eng. Sci. &lt;strong&gt;2016&lt;/strong&gt;,472, 20160497&lt;/p&gt;&lt;p&gt;&lt;strong&gt;Acknowledgements&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;We acknowledge support of this work by the project &amp;#8220;HELPOS &amp;#8211; Hellenic System for Lithosphere Monitoring&amp;#8221; (MIS 5002697) which is implemented under the Action &amp;#8220;Reinforcement of the Research and Innovation Infrastructure&amp;#8221;, funded by the Operational Programme &quot;Competitiveness, Entrepreneurship and Innovation&quot; (NSRF 2014-2020) and co-financed by Greece and the European Union (European Regional Development Fund).&lt;/p&gt;

A new structural entropy measurement of networks based on the nonextensive statistical mechanics and hub repulsion

<abstract><p>The structure properties of complex networks are an open issue. As the most important parameter to describe the structural properties of the complex network, the structure entropy has attracted much attention. Recently, the researchers note that hub repulsion plays an role in structural entropy. In this paper, the repulsion between nodes in complex networks is simulated when calculating the structure entropy of the complex network. Coulomb's law is used to quantitatively express the repulsive force between two nodes of the complex network, and a new structural entropy based on the Tsallis nonextensive statistical mechanics is proposed. The new structure entropy synthesizes the influence of repulsive force and betweenness. We study several construction networks and some real complex networks, the results show that the proposed structure entropy can describe the structural properties of complex networks more reasonably. In particular, the new structural entropy has better discrimination in describing the complexity of the irregular network. Because in the irregular network, the difference of the new structure entropy is larger than that of degree structure entropy, betweenness structure entropy and Zhang's structure entropy. It shows that the new method has better discrimination for irregular networks, and experiments on Graph, Centrality literature, US Aire lines and Yeast networks confirm this conclusion.</p></abstract>

Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities

The concept of duality of probability distributions constitutes a fundamental “brick” in the solid framework of nonextensive statistical mechanics—the generalization of Boltzmann–Gibbs statistical mechanics under the consideration of the q-entropy. The probability duality is solving old-standing issues of the theory, e.g., it ascertains the additivity for the internal energy given the additivity in the energy of microstates. However, it is a rather complex part of the theory, and certainly, it cannot be trivially explained along the Gibb’s path of entropy maximization. Recently, it was shown that an alternative picture exists, considering a dual entropy, instead of a dual probability. In particular, the framework of nonextensive statistical mechanics can be equivalently developed using q- and 1/q- entropies. The canonical probability distribution coincides again with the known q-exponential distribution, but without the necessity of the duality of ordinary-escort probabilities. Furthermore, it is shown that the dual entropies, q-entropy and 1/q-entropy, as well as, the 1-entropy, are involved in an identity, useful in theoretical development and applications.

Tsallis Entropy, Likelihood, and the Robust Seismic Inversion

The nonextensive statistical mechanics proposed by Tsallis have been successfully used to model and analyze many complex phenomena. Here, we study the role of the generalized Tsallis statistics on the inverse problem theory. Most inverse problems are formulated as an optimisation problem that aims to estimate the physical parameters of a system from indirect and partial observations. In the conventional approach, the misfit function that is to be minimized is based on the least-squares distance between the observed data and the modelled data (residuals or errors), in which the residuals are assumed to follow a Gaussian distribution. However, in many real situations, the error is typically non-Gaussian, and therefore this technique tends to fail. This problem has motivated us to study misfit functions based on non-Gaussian statistics. In this work, we derive a misfit function based on the q-Gaussian distribution associated with the maximum entropy principle in the Tsallis formalism. We tested our method in a typical geophysical data inverse problem, called post-stack inversion (PSI), in which the physical parameters to be estimated are the Earth’s reflectivity. Our results show that the PSI based on Tsallis statistics outperforms the conventional PSI, especially in the non-Gaussian noisy-data case.