Tsallis statistics and magnetospheric self-organization

In this paper we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality of the stable Lévy distribution.The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior.

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“…Tsallis statistics is now widely applied e.g. to solar and space plasmas such as the heliosphere magnetic field and the solar wind [28][29][30] .…”

Section: Introductionmentioning

confidence: 99%

A fractional Fokker-Planck model for anomalous diffusion

2014

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“…Studies concerning the nonlinear analysis of various time series derived from Earth's magnetosphere, solar wind, solar flares and sunspots others revealed the complex behavior of those systems, which include among others chaotic, SOC and intermittent turbulent dynamics, non-equilibrium phase transitions, as well as Tsallis non-Gaussian statistics [53,[64][65][66][67][68][69][70][71].…”

Section: Space Plasmasmentioning

confidence: 99%

Complex Systems: Phenomenology, Modeling, Analysis

Iliopoulos¹

2016

Int J Appl Exp Math

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In this paper a short overview on complex systems and their basic features, as well as the models and mathematical tools developed for their analysis, is given. This review is formed according to the related experience, activity and scientific interests of the author, namely focused on mainly in nonlinear time series analysis and statistics and their applications on different physical systems. In particular, rough outlines are given concerning the phenomenology of complex systems, e.g. far from equilibrium thermodynamics and Tsallis statistics, power law scaling, multi-fractality, low dimensional chaos, SOC, strange kinetics and anomalous diffusion and turbulent intermittency. In addition, a non-complete list of models, based on equations or agent based, is briefly described such as Kuramoto -Sivashinsky equation, cubic complex Ginzburg-Landau Equation, reaction-diffusion Equation, fractional equations, cellular automata, complex networks and artificial neural networks. A more extended review is provided concerning the nonlinear time series analysis complex systems, describing tools like mutual information, flatness coefficient, structure functions, Tsallis q-triplet, correlation dimension and Lyapunov exponents in the reconstructed phase space, which can provide valuable information for the complex system's dynamics. Finally, applications of nonlinear time series analysis on various physical systems, such as earthquakes, Earth's magnetosphere, solar plasma and solar wind, plastic deformation of materials, epilepsy, economical indices and DNA structure, are provided. Mini ReviewOpen Access Complex SystemsIn general, even though there is no precise definition of complex systems, a complex system can be thought of as a collection of nonlinearly interacting elements summing up as a whole which is characterized by novel large scale effects. These effects arise as emergent properties related with nonlinear-complex behavior, which is the most distinguishing feature of complex systems [1]. A nonexhaustive list of complex systems contains among others (in fact most systems can be considered as complex systems): geophysical processes (earthquakes), biophysics (brain dynamics, DNA), space plasmas (solar wind, solar flares, Earth's magnetosphere), plastic deformation of materials, economy (stock indices), socio-technical systems and many others. The aforementioned complex systems, even though they are completely different in many aspects (e.g. different elements), share common characteristics such as:1. They consist of a large number of interacting elements. This fact allows the description of these systems in two different hierarchical levels, namely microscopic and macroscopic. 2. Their subsystems and their interactions are nonlinear. 3. They exhibit emergent behavior, namely a self-organizing collective behavior, which is not predictable from the knowledge of the element's behavior. 4. These systems are open, namely they interact with their environment. This allows these systems to be driven to metastable state...

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“…To estimate q rel , we fit the regressions of ln q C(τ ) on τ , for each of the values of q in the interval [1, 1.5] with δ q = 0.01, and choose the q-value for which the corresponding coefficient of determination is maximized (see Ferri et al [8]; Pavlos et al [7]). A high value for q rel is a symptom of long-range memory.…”

Section: Q-relmentioning

confidence: 99%

“…The q-statcoefficient reflects the sensibility of the volatility to the occurrence of higher variations, such as crashes, resulting from self-organization in the market (see Pavlos et al [7] and Ferri et al [8]). …”

Section: Q-statmentioning

confidence: 99%

“…It consists of a threefold determination of an entropic coefficient, q, in the context of: (1) the sensitivity to initial conditions in a dynamical system, which may be seen as reflecting unpredictability; (2) relaxation of macroscopic variable towards a stationary state, which in a time series, may be taken to detect the existence of long memory; (3) the stationary distribution obtained after the constrained optimization of an entropy function, where the departure of this distribution from the Gaussian distribution is the result of self-organization in the market leading to a rougher path. For several empirical approaches, see Pavlos et al [7], Ferri et al [8], de Freitas et al [9], De Sousa and Rostirolla [10] and, in the finance area, Queirós et al [11].…”

Section: Introductionmentioning

confidence: 99%

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Market Efficiency, Roughness and Long Memory in PSI20 Index Returns: Wavelet and Entropy Analysis

Pascoal

Monteiro

2014

Entropy

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In this study, features of the financial returns of the PSI20index, related to market efficiency, are captured using wavelet-and entropy-based techniques. This characterization includes the following points. First, the detection of long memory, associated with low frequencies, and a global measure of the time series: the Hurst exponent estimated by several methods, including wavelets. Second, the degree of roughness, or regularity variation, associated with the Hölder exponent, fractal dimension and estimation based on the multifractal spectrum. Finally, the degree of the unpredictability of the series, estimated by approximate entropy. These aspects may also be studied through the concepts of non-extensive entropy and distribution using, for instance, the Tsallis q-triplet. They allow one to study the existence of efficiency in the financial market. On the other hand, the study of local roughness is performed by considering wavelet leader-based entropy. In fact, the wavelet coefficients are computed from a multiresolution analysis, and the wavelet leaders are defined by the local suprema of these coefficients, near the point that we are considering. The resulting entropy is more accurate in that detection than the Hölder exponent. These procedures enhance the capacity to identify the occurrence of financial crashes.

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Tsallis statistics and magnetospheric self-organization

Cited by 35 publications

References 43 publications

A fractional Fokker-Planck model for anomalous diffusion

A fractional Fokker-Planck model for anomalous diffusion

Complex Systems: Phenomenology, Modeling, Analysis

Market Efficiency, Roughness and Long Memory in PSI20 Index Returns: Wavelet and Entropy Analysis

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