Nonextensivity in Geological Faults?

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.Comment: 9 pages, 1 figure, standard LaTeX fil

Nonextensive aspects of self-organized scale-free gas-like networks

We explore the possibility to interpret as a 'gas' the dynamical self-organized scale-free network recently introduced by Kim et al (2005). The role of 'momentum' of individual nodes is played by the degree of the node, the 'configuration space' (metric defining distance between nodes) being determined by the dynamically evolving adjacency matrix. In a constant-size network process, 'inelastic' interactions occur between pairs of nodes, which are realized by the merger of a pair of two nodes into one. The resulting node possesses the union of all links of the previously separate nodes. We consider chemostat conditions, i.e., for each merger there will be a newly created node which is then linked to the existing network randomly. We also introduce an interaction 'potential' (node-merging probability) which decays with distance d_ij as 1/d_ij^alpha; alpha >= 0). We numerically exhibit that this system exhibits nonextensive statistics in the degree distribution, and calculate how the entropic index q depends on alpha. The particular cases alpha=0 and alpha to infinity recover the two models introduced by Kim et al.Comment: 7 pages, 5 figure

Nonextensive statistical mechanics and complex scale-free networks

One explanation for the impressive recent boom in network theory might be that it provides a promising tool for an understanding of complex systems. Network theory is mainly focusing on discrete large-scale topological structures rather than on microscopic details of interactions of its elements. This viewpoint allows to naturally treat collective phenomena which are often an integral part of complex systems, such as biological or socio-economical phenomena. Much of the attraction of network theory arises from the discovery that many networks, natural or man-made, seem to exhibit some sort of universality, meaning that most of them belong to one of three classes: {\it random}, {\it scale-free} and {\it small-world} networks. Maybe most important however for the physics community is, that due to its conceptually intuitive nature, network theory seems to be within reach of a full and coherent understanding from first principles ..

Unified model for network dynamics exhibiting nonextensive statistics

We introduce a dynamical network model which unifies a number of network families which are individually known to exhibit $q$-exponential degree distributions. The present model dynamics incorporates static (non-growing) self-organizing networks, preferentially growing networks, and (preferentially) rewiring networks. Further, it exhibits a natural random graph limit. The proposed model generalizes network dynamics to rewiring and growth modes which depend on internal topology as well as on a metric imposed by the space they are embedded in. In all of the networks emerging from the presented model we find q-exponential degree distributions over a large parameter space. We comment on the parameter dependence of the corresponding entropic index q for the degree distributions, and on the behavior of the clustering coefficients and neighboring connectivity distributions.Comment: 11 pages 8 fig

Zubarev nonequilibrium statistical operator method in Renyi statistics. Reaction-diffusion processes

The Zubarev nonequilibrium statistical operator (NSO) method in Renyi statistics is discussed. The solution of $q$-parametrized Liouville equation within the NSO method is obtained. A statistical approach for a consistent description of reaction-diffusion processes in "gas-adsorbate-metal" system is proposed using the NSO method in Renyi statistics.Comment: 9 pages, no figure

Nonextensive entropy approach to space plasma fluctuations and turbulence

Spatial intermittency in fully developed turbulence is an established feature of astrophysical plasma fluctuations and in particular apparent in the interplanetary medium by in situ observations. In this situation the classical Boltzmann-Gibbs extensive thermo-statistics, applicable when microscopic interactions and memory are short ranged, fails. Upon generalization of the entropy function to nonextensivity, accounting for long-range interactions and thus for correlations in the system, it is demonstrated that the corresponding probability distributions (PDFs) are members of a family of specific power-law distributions. In particular, the resulting theoretical bi-kappa functional reproduces accurately the observed global leptokurtic, non-Gaussian shape of the increment PDFs of characteristic solar wind variables on all scales. Gradual decoupling is obtained by enhancing the spatial separation scale corresponding to increasing kappa-values in case of slow solar wind conditions where a Gaussian is approached in the limit of large scales. Contrary, the scaling properties in the high speed solar wind are predominantly governed by the mean energy or variance of the distribution. The PDFs of solar wind scalar field differences are computed from WIND and ACE data for different time-lags and bulk speeds and analyzed within the nonextensive theory. Consequently, nonlocality in fluctuations, related to both, turbulence and its large scale driving, should be related to long-range interactions in the context of nonextensive entropy generalization, providing fundamentally the physical background of the observed scale dependence of fluctuations in intermittent space plasmas.Comment: 21 pages, 8 figures, accepted for publication, to appear in Advances in Geosciences 2, chapter 04, 2006 (with minor corrections

A nonextensive entropy approach to solar wind intermittency

The probability distributions (PDFs) of the differences of any physical variable in the intermittent, turbulent interplanetary medium are scale dependent. Strong non-Gaussianity of solar wind fluctuations applies for short time-lag spacecraft observations, corresponding to small-scale spatial separations, whereas for large scales the differences turn into a Gaussian normal distribution. These characteristics were hitherto described in the context of the log-normal, the Castaing distribution or the shell model. On the other hand, a possible explanation for nonlocality in turbulence is offered within the context of nonextensive entropy generalization by a recently introduced bi-kappa distribution, generating through a convolution of a negative-kappa core and positive-kappa halo pronounced non-Gaussian structures. The PDFs of solar wind scalar field differences are computed from WIND and ACE data for different time lags and compared with the characteristics of the theoretical bi-kappa functional, well representing the overall scale dependence of the spatial solar wind intermittency. The observed PDF characteristics for increased spatial scales are manifest in the theoretical distribution functional by enhancing the only tuning parameter $\kappa$, measuring the degree of nonextensivity where the large-scale Gaussian is approached for $\kappa \to \infty$. The nonextensive approach assures for experimental studies of solar wind intermittency independence from influence of a priori model assumptions. It is argued that the intermittency of the turbulent fluctuations should be related physically to the nonextensive character of the interplanetary medium counting for nonlocal interactions via the entropy generalization.Comment: 17 pages, 7 figures, accepted for publication in Astrophys.