Tsallis Non-Extensive Statistics

In this study the general theoretical framework of Complexity theory is presented, viewed through non-extensive statistical theory introduced by Constantino Tsallis in 1988. Nonlinear dynamics, fractional dynamics, thermodynamics and statistical physics constitute the basic elements of Complexity theory which can be applied to processes near, as well as, far from equilibrium. Moreover, the nonextensive statistics can be the fundamental linchpin of Complexity theory from microscopical to macroscopical level. Finally, a brief review of results concerning nonlinear analysis of time series corresponding to different complex systems is given.

During the last two decades our scientic group has developed new nonlinear methods of analysis applied to various physical systems. In this review study we present our scientic contribution to nonlinear science, including also some novel concepts as for the constructive role of complexity in modern physical theory. The experimental verification of chaos existence in physical systems remains one of the
most significant problems of non linear science and complexity. The extended chaotic algorithm presented in the following as well as the results concerning its application at different experimental time series reveal the universal character of the complexity theory for the far from equilibrium dynamics of spatially extended physical systems. The
developed methodology that was used compromises different types of computational tools as well as theoretical concepts for the physical interpretation of the experimental information. As we present here the strong dispute and criticism of chaos hypothesis in physical systems during the last two decades was fruitful and challenged us to develop a novel composition of experimental and theoretical knowledge of universal character for the far from equilibrium dynamics. The solar and magnetospheric dynamics included in space plasma processes, the environmental and seismic dynamics, the human brain or the on{chip workload are distinct systems which were studied by our group revealing common chaotic characteristics and chaotic phase transition
processes. Moreover, the intellectual struggle for the comprehension of the theoretical presuppositions of the experimentally observed universal chaotic character of spatially distributed systems lead us to the fundamentals of complexity theory as manifested at the macroscopic and microscopic level of physical reality. From this point of view, some common characteristics of macroscopic and microscopic complexity included in the scientific knowledge of the recent two or three decades can be used as a road for the physical theory unification. That is complexity, scaling, chaos, quanticity and
fractality could be supported as different manifestations of a unified physical law from the microscopic to the macroscopic and cosmological level. As we can argue, determinism and probabilism can also be unied through chaoticity. Moreover, the rising of new physical knowledge reveals that under the macroscopic or the microscopic
physical phenomena there exist a fundamental and multilevel acting unit physical process that produces physical reality rather than a fundamental essence or simple substance from which cosmos can be build.

We investigate Tsallis qq-statistics for various complex physical systems. In particular, the Tsallis qq-triplet is estimated for space plasmas, atmospheric dynamics, seismicity, and brain and cardiac activity. Remarkable agreement between theoretical Tsallis predictions and experimental estimations was observed in all cases. We also present theoretical extensions of Tsallis qq-statistics for intermittent turbulence theory, non-equilibrium statistics, and fractal extension of dynamics.

Tsallis q-extension of statistics and fractal generalization of dynamics are two faces of the same physical reality, as well as the Kernel modern complexity theory. The fractal generalization dynamics is based at the multiscale – multifractal characters of complex dynamics in the physical space-time and the complex system’s dynamical phase space. Tsallis q-triplet of non-extensive statistics can be used for the experiment test of q-statistic as well as of the fractal dynamics. In this study we present indicative experimental verifications of Tsallis theory in various complex systems such as solar plasmas, (planetic magnetospheres, cosmic stars and cosmic rays), atmospheric dynamics, seismogenesis and brain dynamics.

In this study, the non-linear analysis of the sunspot index is embedded in the non-extensive statistical theory of Tsallis (1988, 2004, 2009) [7,9,10]. The q-triplet of Tsallis, as well as the correlation dimension and the Lyapunov exponent spectrum were estimated for the
SVD components of the sunspot index timeseries. Also the multifractal scaling exponent spectrum f (a), the generalized Renyi dimension spectrum D(q) and the spectrum J(p) of the structure function exponents were estimated experimentally and theoretically by
using the q-entropy principle included in Tsallis non-extensive statistical theory, following Arimitsu and Arimitsu (2001, 2000) [76,77]. Our analysis showed clearly the following: (a) a phase transition process in the solar dynamics from high dimensional non-Gaussian SOC state to a low dimensional non-Gaussian chaotic state, (b) strong intermittent solar turbulence and anomalous (multifractal) diffusion solar process, which is strengthened as the solar dynamics makes a phase transition to low dimensional chaos in accordance
to Ruzmaikin, Zeleny and Milovanov’s studies (Zelenyi and Milovanov (1991) [21]); Milovanov and Zelenyi (1993) [22]; Ruzmakin et al. (1996) [26]) (c) faithful agreement of Tsallis non-equilibrium statistical theory with the experimental estimations of (i) non-Gaussian probability distribution function P(x), (ii) multifractal scaling exponent spectrum f (a) and generalized Renyi dimension spectrum Dq, (iii) exponent spectrum J(p) of the structure functions estimated for the sunspot index and its underlying non equilibrium solar dynamics.

In two previous studies [Karakatsanis & Pavlos , 2008], [Karakatsanis et al. , 2010] we showed the coexistence of SOC and low dimensional chaos at solar activity concerning the sunspot index and the solar flares time series. This work complements the previous studies with new results obtained by using the intermittent turbulence theory, the non-extensive q-statistics of Tsallis as well as the singular value decomposition analysis. Particularly, we show the independent
dynamics of sunspot system related to the convection zone of sun and the solar flare system related to the lower solar atmosphere. However, both systems reveal non-equilibrium phase transition process from a high-dimensional intermittent turbulence state with SOC profile to
a low-dimensional and chaotic intermittent turbulence state. The high-dimensional SOC state in both dynamical systems underlying the sunspot and solar flare signal is related with low q values and low Flatness values (F) while the low-dimensional chaotic state is related with higher q-values and Flatness F-values. The higher q- and F-values reveal strong character of long-range correlations corresponding to system wide global process while the lower q- and F- values reveal
scale invariant local avalanche process. Also, the high-dimensional SOC state corresponds to second order non-equilibrium critical phase transition process while the low-dimensional chaotic state corresponds to first order non-equilibrium phase transition process. Finally, both dynamics underlying sunspot index and solar flare and in their both states of phase transition process the multi-scale and multi-fractal character was found to exist but with different profile or strength.

The concepts of non-extensive statistical physics, which have recently been applied in the study of complex systems, have been used here to analyse stress-induced electric current data in triaxially deformed Carrara marble samples. The fluctuations of electric current appear to follow a q-Gaussian distribution, with the PDF exhibiting ‘fat tails’. The application of super-statistical techniques to these electric current fluctuations shows to good approximation that they can be described by local Gaussian processes with fluctuating variance.

In this study we present results of the evaluation of q − triplet of Tsallis non-extensive statistics concerning two time stock market time series, Standart & Poor's 500 (S & P) 500 and TVIX. The analysis of the results, support the hypothesis in which economic dynamics from physical point of view correspond to far from equilibrium spatial distributed non-linear dynamics. In particular, the analysis of the stock market time series revealed underlying complex dynamics, indicating clearly that the statistics of the dynamics in the multifractal phase space can be described by Tsallis distribution functions of power law and heavy tails forms. The non-extensive character of the underlying dynamics is related to the existence of long range interactions in space and time, as well as the interaction in many scales. In addition, the detailed analysis of S & P index unraveled the existence of non-equilibrium phase transitions depicted clearly in the variations of Tsallis q-triplet values. These phase transitions are connected with non-equilibrium stationary states of economical dynamics derived from processes of strong self organization which correspond to local maxima of Tsallis entropy, while the changes in the control parameters can induce new phase transitions and shifts to new metaequilibrium steady states of maximizing Tsallis entropy. These phase transitions lead to changes in Tsallis qtriplet values which correspond to multi-fractal changes in the formation of the phase state and an alteration in the phenomenology of the economical system. Finally, these characteristics also indicate the existence of fractional dynamics in the phase space which can be described through fractional Fokker-Planck equations and anomalous diffusion equations. The solutions of these equations are fractional spatiotemporal functions and non-Gaussian distributions functions which fall into the category of Levy distributions and Tsallis distributions.

We study the behavior of the numbers in 412 light curves of cataclysmic variables, x-ray
binary systems, galaxies, pulsars, supernovae remnants and other x-ray sources present
in the public data collected by the instrument All Sky Monitor on board of the satellite
Rossi x-ray timing explorer. The temporal light curves were analyzed applying Newcomb-
Benford Law. The first digit of the x-ray light curves coming from astrophysical systems
obeys the Newcomb-Benford Law as an intrinsic behavior. The nonextensive statistical
mechanics behavior of astrophysical sources seem to be the cause for these sources to
obey the Newcomb-Benford law. Some x-ray binary systems, however, do not follow this
behavior. These systems obey either a gaussian or a bimodal distribution.

In this work, we present the coexistence of self-organized criticality (SOC) and low-dimensional chaos at solar activity with results obtained by using the intermittent turbulence theory, the nonextensive q-statistics of Tsallis as well as the singular value decomposition analysis. Particularly, we show the independent dynamics of sunspot system related to the convection zone of sun and the solar flare system related to the lower solar atmosphere. However, both systems reveal nonequilibrium phase transition process from a high-dimensional intermittent turbulence state with SOC profile to a low-dimensional and chaotic intermittent turbulence state. The high-dimensional SOC state in both dynamical systems underlying the sunspot and solar flare signal is related with low q-values and low Flatness values (F ) while the low-dimensional chaotic state is related with higher q-values and Flatness F -values. The higher qand F -values reveal strong character of long-range correlations corresponding to system-wide global process while the lower qand F -values reveal scale invariant local avalanche process. Also, the high-dimensional SOC state corresponds to second order nonequilibrium critical phase transition process while the low-dimensional chaotic state corresponds to first order nonequilibrium phase transition process. Finally, for both dynamics underlying sunspot index and solar flare, at both states of phase transition process, the multiscale and multifractal character was found to exist but with different profile or strength.

Este artículo propone una relación biunívoca entre el parámetro de no extensividad q y la función de densidad de probabilidad estacionaria f correspondientes a un observable u para algunos sistemas super–aditivos en los que la Entropía de Tsallis sea aplicable. Dicha relación se da en términos de comparación entre funciones de enlace que caracterizan la falta de memoria de ciertas variables aleatorias asociadas al parámetro q y a la densidad estacionaria f. Finalmente, a partir de los resultados anteriores, se propone un método que permite aproximar el parámetro q, mediante una estimación de f, cuando la energía efectiva asociada a u sea la energía cinética efectiva.

During the last two decades, low dimensional chaotic or selforganized criticality (SOC) processes have been observed by our group in many different physical systems such as space plasmas, the solar or the magnetospheric dynamics, the atmosphere, earthquakes, the brain activity as well as in informational systems. All these systems are complex systems living far from equilibrium with strong self-organization and phase transition character. The theoretical interpretation of these natural phenomena needs a deeper insight into the fundamentals of complexity theory. In this study, we try to give a synoptic description of complexity theory both at the microscopic and at the macroscopic level of the physical reality. Also, we propose that the self-organization observed macroscopically is a phenomenon that reveals the strong unifying character of the complex dynamics which includes thermodynamical and dynamical characteristics in all levels of the physical reality. From this point of view, macroscopical deterministic and stochastic processes are closely related to the microscopical chaos and self-organization. In this study the scientific work of scientists such as Wilson, Nicolis, Prigogine, Hooft, Nottale, El Naschie, Castro, Tsallis, Chang and others is used for the development of a unified physical comprehension of complex dynamics from the microscopic to the macroscopic level.

Strong evidence is provided for significant far from equilibrium complex processes in the seismogenic layer of the North Aegean region (Greece), after applying modern nonlinear methods
to various seismicity time series. The data used are subsets of the regional catalogue compiled in the central Seismological Station of Geophysics Department, Aristotle University of Thessaloniki
and concern 4367 earthquakes of magnitude greater than 3.8, which took place during the period of 1968–2008. We present results, derived from the application of nonlinear algorithms, concerning the estimation of correlation dimension, mutual information, largest Lyapunov exponent, flatness coefficient and q-value which correspond to Tsallis nonextensive statistics. These quantities are estimated for two seismic time series corresponding to the basic focal parameters of earthquakes, namely origin time and magnitude. The obtained results can be associated with novel far from equilibrium complex dynamics such as low dimensional chaos, Self Organized
Criticality (SOC) and intermittent turbulence. Furthermore, in this study, new information is provided about the nonlinear turbulent character of the Hellenic lithospheric dynamics related to the Tsallis nonextensive statistical theory. Our analysis indicates the coexistence of two different lithospheric processes, one low dimensional (chaotic) and the other high dimensional (SOC), revealing the strongly turbulent character of the Greek lithospheric system. In particular, the low
dimensional chaotic process corresponds to the temporal manifestation of earthquakes, whereas the high dimensional nonlinear (SOC) process corresponds to the burst energy releases, a result
that has significant implications concerning the ability of earthquake prediction.

In this study which is the continuation of the first part (Pavlos et al. 2012) [1], the nonlinear analysis of the solar flares index is embedded in the non-extensive statistical theory of Tsallis (1988) [3]. The q-triplet of Tsallis, as well as the correlation dimension and the
Lyapunov exponent spectrum were estimated for the singular value decomposition (SVD) components of the solar flares timeseries. Also the multifractal scaling exponent spectrum f (a), the generalized Renyi dimension spectrum D(q) and the spectrum J(p) of the structure
function exponents were estimated experimentally and theoretically by using the q-entropy principle included in Tsallis non-extensive statistical theory, following Arimitsu and Arimitsu (2000) [25]. Our analysis showed clearly the following: (a) a phase transition
process in the solar flare dynamics from a high dimensional non-Gaussian self-organized critical (SOC) state to a low dimensional also non-Gaussian chaotic state, (b) strong intermittent solar corona turbulence and an anomalous (multifractal) diffusion solar corona
process, which is strengthened as the solar corona dynamics makes a phase transition to low dimensional chaos, (c) faithful agreement of Tsallis non-equilibrium statistical theory with the experimental estimations of the functions: (i) non-Gaussian probability distribution
function P(x), (ii) f (a) and D(q), and (iii) J(p) for the solar flares timeseries and its underlying non-equilibrium solar dynamics, and (d) the solar flare dynamical profile is revealed similar to the dynamical profile of the solar corona zone as far as the phase transition
process from self-organized criticality (SOC) to chaos state. However the solar low corona (solar flare) dynamical characteristics can be clearly discriminated from the dynamical characteristics of the solar convection zone.

We propose a two-parametric non-distributive algebraic structure that follows from $(q,q')$-logarithm and $(q,q')$-exponential functions. Properties of generalized $(q,q')$-operators are analyzed. We also generalize the proposal into a multi-parametric structure (generalization of logarithm and exponential functions and the corresponding algebraic operators). All $n$-parameter expressions recover $(n-1)$-generalization when the corresponding $q_n\to1$. Nonextensive statistical mechanics has been the source of successive generalizations of entropic forms and mathematical structures, in which this work is a consequence.

We study the behavior of the numbers in 412 light curves of cataclysmic variables, x-ray binary systems, galaxies, pulsars, supernovae remnants and other x-ray sources present in the public data collected by the instrument All Sky Monitor on board of the satellite Rossi x-ray timing explorer. The temporal light curves were analyzed applying Newcomb-Benford Law. The first digit of the x-ray light curves coming from astrophysical systems obeys the Newcomb-Benford Law as an intrinsic behavior. The nonextensive statistical mechanics behavior of astrophysical sources seem to be the cause for these sources to obey the Newcomb-Benford law. Some x-ray binary systems, however, do not follow this behavior. These systems obey either a gaussian or a bimodal distribution.

We have studied the role of long-range interactions on the thermodynamics of magnetic systems. We have simulated, through the Monte Carlo method, magnetization curves of a two-dimensional classical Ising model including a long-range dipole-dipole-like interaction, where the range of interaction is tuned by a parameter. Based on the conjectures of Tsallis statistics, we show that, for

This paper proposes non-extensive cross entropy econometrics (NCEE) methodology for econometric modelling, particularly in the case of nonlinear, non-ergodic inverse problem models. The methodology is a coincident junction of recent non-additive Tsallis statistics, Kullback-Leibler statistical information theory, and the traditional Cowles Commission related-econometrics. The added value of NCEE emerges from the capacity of Tsallis non-additive statistics to describe with higher accuracy long-memory dynamics of complex systems of the real world. These do not diverge from complex socioeconomic manmade structures once they are observed at the same time-space frequency. To conclude, we present outcomes of this technique through parameter estimation using time series and space series constant elasticity of technical substitution (CETS) models of national production. A comparison is carried out with outputs from traditional econometric techniques (GMM, NLS, ML).

In this study we use Tsallis non-extensive statistics for a new understanding the magnetospheric dynamics and the magnetospheric self-organization during quiet and intensive superstorm periods. The qsensqsens, qstatqstat, and qrelqrel indices set known as the Tsallis qq-triplet was estimated during both quiet and strongly active periods, as well as the correlation dimensions and Lyapunov exponents spectrum for magnetospheric bulk plasma flows data. The results obtained by our analysis clearly indicate the magnetospheric phase transition process from a high-dimensional quiet SOC state to a low-dimensional global chaotic state when superstorm events are developed. During such a phase transition process the non-extensive statistical character of the magnetospheric plasma is strengthened as the values of the qq-triplet indices changes obtaining higher values than their values during the quiet periods.