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Register a new userNonextensivity in Geological Faults?
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G. S. Franc{c}a
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Geological fault systems, as the San Andreas fault (SAF) in USA, constitute typical examples of self-organizing systems in nature. In this paper, we have considered some geophysical properties of the SAF system to test the viability of the nonextensive models for earthquakes developed in [Phys. Rev. E {bf 73}, 026102, 2006]. To this end, we have used 6188 earthquakes events ranging in the magnitude interval $2 < m < 8$ that were taken from the Network Earthquake International Center catalogs (NEIC, 2004-2006) and the Bulletin of the International Seismological Centre (ISC, 1964-2003). For values of the Tsallis nonextensive parameter $q simeq 1.68$, it is shown that the energy distribution function deduced in above reference provides an excellent fit to the NEIC and ISC SAF data.
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The San Andreas fault (SAF) in the USA is one of the most investigated self-organizing systems in nature. In this paper, we studied some geophysical properties of the SAF system in order to analyze the behavior of earthquakes in the context of Tsalliss $q$--Triplet. To that end, we considered 134,573 earthquake events in magnitude interval $2leq m<8$, taken from the Southern Earthquake Data Center (SCEDC, 1932 - 2012). The values obtained ($q$--Triplet$equiv$${$$q$$_{stat}$,$q$$_{sen}$,$q$$_{rel}$$}$) reveal that the $q_{stat}$--Gaussian behavior of the aforementioned data exhibit long-range temporal correlations. Moreover, $q_{sen}$ exhibits quasi-monofractal behavior with a Hurst exponent of 0.87.
We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds that of the tuned onset of chaos in say logistic-like onedimensional maps. We also calculate the dynamical exponent z.
The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this letter, we use the fundamental principles of ergodicity (via Liouvilles theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Boltzmann-Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics.
An updated review [1] of nonextensive statistical mechanics and thermodynamics is colloquially presented. Quite naturally the possibility emerges for using the value of q-1 (entropic nonextensivity) as a simple and efficient manner to provide, at least for some classes of systems, some characterization of the degree of what is currently referred to as complexity [2]. A few historical digressions are included as well.
A proof of the relativistic $H$-theorem by including nonextensive effects is given. As it happens in the nonrelativistic limit, the molecular chaos hypothesis advanced by Boltzmann does not remain valid, and the second law of thermodynamics combined with a duality transformation implies that the q-parameter lies on the interval [0,2]. It is also proved that the collisional equilibrium states (null entropy source term) are described by the relativistic $q$-power law extension of the exponential Juttner distribution which reduces, in the nonrelativistic domain, to the Tsallis power law function. As a simple illustration of the basic approach, we derive the relativistic nonextensive equilibrium distribution for a dilute charged gas under the action of an electromagnetic field $F^{{mu u}}$. Such results reduce to the standard ones in the extensive limit, thereby showing that the nonextensive entropic framework can be harmonized with the space-time ideas contained in the special relativity theory.
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