The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Testing the Minimum System Entropy and the Quantum of Entropy

Experimental and theoretical results about entropy limits for macroscopic and single-particle systems are reviewed. All experiments confirm the minimum system entropy S⩾kln2. We clarify in which cases it is possible to speak about a minimum system entropykln2 and in which cases about a quantum of entropy. Conceptual tensions with the third law of thermodynamics, with the additivity of entropy, with statistical calculations, and with entropy production are resolved. Black hole entropy is surveyed. Claims for smaller system entropy values are shown to contradict the requirement of observability, which, as possibly argued for the first time here, also implies the minimum system entropy kln2. The uncertainty relations involving the Boltzmann constant and the possibility of deriving thermodynamics from the existence of minimum system entropy enable one to speak about a general principle that is valid across nature.

Magic Numbers and Mixing Degree in Many-Fermion Systems

We consider an N fermion system at low temperature T in which we encounter special particle number values Nm exhibiting special traits. These values arise when focusing attention upon the degree of mixture (DM) of the pertinent quantum states. Given the coupling constant of the Hamiltonian, the DMs stay constant for all N-values but experience sudden jumps at the Nm. For a quantum state described by the matrix ρ, its purity is expressed by Trρ2 and then the degree of mixture is given by 1-Trρ2, a quantity that coincides with the entropy Sq for q=2. Thus, Tsallis entropy of index two faithfully represents the degree of mixing of a state, that is, it measures the extent to which the state departs from maximal purity. Macroscopic manifestations of the degree of mixing can be observed through various physical quantities. Our present study is closely related to properties of many-fermion systems that are usually manipulated at zero temperature. Here, we wish to study the subject at finite temperature. The Gibbs ensemble is appealed to. Some interesting insights are thereby gained.

Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy

Nonequilibrium systems in chemistry and physics are generally modeled with the Boltzmann, Fokker-Planck, and Master equations. There has been a considerable interest in the nonequilibrium distributions of electrons and ions in space physics in different environments as well as in other systems. An often-used empirical model to characterize these distributions, especially in space physics, is the Kappa distribution. There have been numerous efforts to provide a theoretical basis for the Kappa distribution that include the Fokker-Planck equation with specific drift and diffusion coefficients. Alternatively, the maximization of the Tsallis nonextensive entropy provides the desired Kappa distribution. This paper examines three families of Fokker-Planck equations that provide a steady-state Kappa distribution as well as a myriad of other nonequilibrium distributions. The relationship of these works with analogous studies of distributions with asymptotic high-energy tails is also considered. It is clear that the many different nonequilibrium distribution functions that can occur cannot all be rationalized with Gibbs-Boltzmann statistical mechanics, which uniquely gives equilibrium distributions, or with the Tsallis nonextensive entropy, which gives uniquely the Kappa distribution. The current research is directed towards an improved understanding of the origin of nonequilibrium distributions in several specific systems.

Tsallis q-Statistics in Seismology

Non-extensive statistical mechanics (or q-statistics) is based on the so-called non-additive Tsallis entropy. Since its introduction by Tsallis, in 1988, as a generalization of the Boltzmann-Gibbs equilibrium statistical mechanics, it has steadily gained ground as a suitable theory for the description of the statistical properties of non-equilibrium complex systems. Therefore, it has been applied to numerous phenomena, including real seismicity. In particular, Tsallis entropy is expected to provide a guiding principle to reveal novel aspects of complex dynamical systems with catastrophes, such as seismic events. The exploration of the existing connections between Tsallis formalism and real seismicity has been the focus of extensive research activity in the last two decades. In particular, Tsallis q-statistics has provided a unified framework for the description of the collective properties of earthquakes and faults. Despite this progress, our present knowledge of the physical processes leading to the initiation of a rupture, and its subsequent growth through a fault system, remains quite limited. The aim of this paper was to provide an overview of the non-extensive interpretation of seismicity, along with the contributions of the Tsallis formalism to the statistical description of seismic events.

Entropy Optimization, Generalized Logarithms, and Duality Relations

Several generalizations or extensions of the Boltzmann-Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies Sq≡k1-∑ipiqq-1(q∈R;S1=SBG≡-k∑ipilnpi) have harvested the largest number of successful applications. The specific structural features of the Sq thermostatistics, therefore, are worthy of close scrutiny. In the present work, we analyze one of these features, according to which the q-logarithm function lnqx≡x1-q-11-q(ln1x=lnx) associated with the Sq entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations.

Order Parameter and Entropy of Seismicity in Natural Time before Major Earthquakes: Recent Results

A lot of work in geosciences has been completed during the last decade on the analysis in the new concept of time, termed natural time, introduced in 2001. The main advances are presented, including, among others, the following: First, the direct experimental verification of the interconnection between a Seismic Electric Signals (SES) activity and seismicity, i.e., the order parameter fluctuations of seismicity exhibit a clearly detectable minimum when an SES activity starts. These two phenomena are also linked closely in space. Second, the identification of the epicentral area and the occurrence time of an impending major earthquake (EQ) by means of the order parameter of seismicity and the entropy change of seismicity under time reversal as well as the extrema of their fluctuations. An indicative example is the M9 Tohoku EQ in Japan on 11 March 2011. Third, to answer the crucial question—when a magnitude 7 class EQ occurs—whether it is a foreshock or a mainshock. This can be answered by means of the key quantities already mentioned, i.e., the order parameter of seismicity and the entropy change of seismicity under time reversal along with their fluctuations. The explanation of the experimental findings identified before major EQs is given in a unified way on the basis of a physical model already proposed in the 1980s.

The Principle of Maximum Entropy and the Distribution of Mass in Galaxies

We do not have a final answer to the question of why galaxies choose a particular internal mass distribution. Here we examine whether the distribution is set by thermodynamic equilibrium (TE). Traditionally, TE is discarded for a number of reasons including the inefficiency of two-body collisions to thermalize the mass distribution in a Hubble time, and the fact that the mass distribution maximizing the classical Boltzmann–Gibbs entropy is unphysical. These arguments are questionable. In particular, when the Tsallis entropy that describes self-gravitating systems is used to define TE, the mass distributions that result (i.e., the polytropes) are physically sensible. This work spells out this and other arguments for TE and presents the polytropes and their properties. It puts forward empirical evidence for the mass distribution observed in galaxies to be consistent with polytropes. It compares polytropes with Sérsic functions and it shows how the DM halos resulting from cosmological numerical simulations become polytropes when efficient collisions are allowed. It also discusses pathways to thermalization bypassing two-body collisions. It finally outlines future developments including deciphering whether or not DM particles collide efficiently.

The Sharma–Mittal Model’s Implications on FRW Universe in Chern–Simons Gravity

The Sharma–Mittal holographic dark energy model is investigated in this paper using the Chern–Simons modified gravity theory. We investigate several cosmic parameters, including the deceleration, equation of state, square of sound speed, and energy density. According to the deceleration parameter, the universe is in an decelerating and expanding phase known as de Sitter expansion. The Sharma–Mittal HDE model supports a deceleration to acceleration transition that is compatible with the observational data. The EoS depicts the universe’s dominance era through a number of components, such as ω=0, 13, 1, which indicate that the universe is influenced by dust, radiation, and stiff fluid, while −1<ω<13, ω=−1, and ω<−1 are conditions for quintessence DE, ΛCDM, and Phantom era dominance. Our findings indicate that the universe is in an accelerated expansion phase, and this is similar to the observational data.

On Conditional Tsallis Entropy

There is no generally accepted definition for conditional Tsallis entropy. The standard definition of (unconditional) Tsallis entropy depends on a parameter α that converges to the Shannon entropy as α approaches 1. In this paper, we describe three proposed definitions of conditional Tsallis entropy suggested in the literature—their properties are studied and their values, as a function of α, are compared. We also consider another natural proposal for conditional Tsallis entropy and compare it with the existing ones. Lastly, we present an online tool to compute the four conditional Tsallis entropies, given the probability distributions and the value of the parameter α.

Fokker-Planck equation for Coulomb relaxation and wave-particle diffusion: Spectral solution and the stability of the Kappa distribution to Coulomb collisions

The present paper considers the time evolution of a charged test particle of mass m in a constant temperature heat bath of a second charged particle of mass M. The time dependence of the distribution function of the test particles is given by a Fokker-Planck equation with a diffusion coefficient for Coulomb collisions as well as a diffusion coefficient for wave-particle interactions. For the mass ratio m/M→0, the steady distribution is a Kappa distribution which has been employed in space physics to fit observed particle energy spectra. The time dependence of the distribution functions with some initial value is expressed in terms of the eigenvalues and eigenfunctions of the linear Fokker-Planck operator and also interpreted with the transformation to a Schrödinger equation. We also consider the explicit time dependence of the distribution function with a discretization of the Fokker-Planck equation. We study the stability of the Kappa distribution to Coulomb collisions.

Tsallis Entropy for Assessing Spatial Uncertainty Associated with Mean Annual Runoff of Quaternary Catchments of the Middle Vaal Basin in South Africa

This study assesses mainly the uncertainty of the mean annual runoff (MAR) for quaternary catchments (QCs) considered as metastable nonextensive systems (from Tsalllis entropy) in the Middle Vaal catchment. The study is applied to the surface water resources (WR) of the South Africa 1990 (WR90), 2005 (WR2005) and 2012 (WR2012) data sets. The q-information index (from the Tsalllis entropy) is used here as a deviation indicator for the spatial evolution of uncertainty for the different QCs, using the Shannon entropy as a baseline. It enables the determination of a (virtual) convergence point, zone of positive and negative uncertainty deviation, zone of null deviation and chaotic zone for each data set. Such a determination is not possible on the basis of the Shannon entropy alone as a measure for the MAR uncertainty of QCs, i.e., when they are viewed as extensive systems. Finally, the spatial distributions for the zones of the q-uncertainty deviation (gain or loss in information) of the MAR are derived and lead to iso q-uncertainty deviation maps.

Entropy-Based Strategies for Rapid Pre-Processing and Classification of Time Series Data from Single-Molecule Force Experiments

Recent advances in single-molecule science have revealed an astonishing number of details on the microscopic states of molecules, which in turn defined the need for simple, automated processing of numerous time-series data. In particular, large datasets of time series of single protein molecules have been obtained using laser optical tweezers. In this system, each molecular state has a separate time series with a relatively uneven composition from the point of view-point of local descriptive statistics. In the past, uncertain data quality and heterogeneity of molecular states were biased to the human experience. Because the data processing information is not directly transferable to the black-box-framework for an efficient classification, a rapid evaluation of a large number of time series samples simultaneously measured may constitute a serious obstacle. To solve this particular problem, we have implemented a supervised learning method that combines local entropic models with the global Lehmer average. We find that the methodological combination is suitable to perform a fast and simple categorization, which enables rapid pre-processing of the data with minimal optimization and user interventions.

Thermodynamics from first principles: Correlations and nonextensivity

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 Rapid Communication, we use the fundamental principles of ergodicity (via Liouville's 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 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.

The Design of Global Correlation Quantifiers and Continuous Notions of Statistical Sufficiency

Using first principles from inference, we design a set of functionals for the purposes of ranking joint probability distributions with respect to their correlations. Starting with a general functional, we impose its desired behavior through the Principle of Constant Correlations (PCC), which constrains the correlation functional to behave in a consistent way under statistically independent inferential transformations. The PCC guides us in choosing the appropriate design criteria for constructing the desired functionals. Since the derivations depend on a choice of partitioning the variable space into n disjoint subspaces, the general functional we design is the n-partite information (NPI), of which the total correlation and mutual information are special cases. Thus, these functionals are found to be uniquely capable of determining whether a certain class of inferential transformations, ρ → ∗ ρ ' , preserve, destroy or create correlations. This provides conceptual clarity by ruling out other possible global correlation quantifiers. Finally, the derivation and results allow us to quantify non-binary notions of statistical sufficiency. Our results express what percentage of the correlations are preserved under a given inferential transformation or variable mapping.

Nonlinear Fokker-Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions

Nonlinear Fokker–Planck equations (NLFPEs) constitute useful effective descriptions of some interacting many-body systems. Important instances of these nonlinear evolution equations are closely related to the thermostatistics based on the Sq power-law entropic functionals. Most applications of the connection between the NLFPE and the Sq entropies have focused on systems interacting through short-range forces. In the present contribution we re-visit the NLFPE approach to interacting systems in order to clarify the role played by the range of the interactions, and to explore the possibility of developing similar treatments for systems with long-range interactions, such as those corresponding to Newtonian gravitation. In particular, we consider a system of particles interacting via forces following the inverse square law and performing overdamped motion, that is described by a density obeying an integro-differential evolution equation that admits exact time-dependent solutions of the q-Gaussian form. These q-Gaussian solutions, which constitute a signature of Sq-thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker–Planck equation.

On the Statistical Significance of the Variability Minima of the Order Parameter of Seismicity by Means of Event Coincidence Analysis

Natural time analysis has led to the introduction of an order parameter for seismicity when considering earthquakes as critical phenomena. The study of the fluctuations of this order parameter has shown that its variability exhibits minima before strong earthquakes. In this paper, we evaluate the statistical significance of such minima by using the recent method of event coincidence analysis. Our study includes the variability minima identified before major earthquakes in Japan and Eastern Mediterranean as well as in global seismicity.

Möbius Transforms, Cycles and q-triplets in Statistical Mechanics

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving unity (q=1 corresponds to the BG theory). Such transforms have the form q↦(aq+1−a)/[(1+a)q−a], where a is a real number; the particular cases a=−1 and a=0 yield, respectively, q↦(2−q) and q↦1/q, currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

One-parameter functionals of the Rényi R ρ , γ ( α ) and Tsallis T ρ , γ ( α ) types are calculated both in the position (subscript ρ ) and momentum ( γ ) spaces for the azimuthally symmetric 2D nanoring that is placed into the combination of the transverse uniform magnetic field B and the Aharonov–Bohm (AB) flux ϕ A B and whose potential profile is modeled by the superposition of the quadratic and inverse quadratic dependencies on the radius r. Position (momentum) Rényi entropy depends on the field B as a negative (positive) logarithm of ω e f f ≡ ω 0 2 + ω c 2 / 4 1 / 2 , where ω 0 determines the quadratic steepness of the confining potential and ω c is a cyclotron frequency. This makes the sum R ρ n m ( α ) + R γ n m ( α 2 α − 1 ) a field-independent quantity that increases with the principal n and azimuthal m quantum numbers and satisfies the corresponding uncertainty relation. In the limit α → 1 , both entropies in either space tend to their Shannon counterparts along, however, different paths. Analytic expression for the lower boundary of the semi-infinite range of the dimensionless coefficient α where the momentum entropies exist reveals that it depends on the ring geometry, AB intensity, and quantum number m. It is proved that there is the only orbital for which both Rényi and Tsallis uncertainty relations turn into the identity at α = 1 / 2 , which is not necessarily the lowest-energy level. At any coefficient α , the dependence of the position of the Rényi entropy on the AB flux mimics the energy variation with ϕ A B , which, under appropriate scaling, can be used for the unique determination of the associated persistent current. Similarities and differences between the two entropies and their uncertainty relations are discussed as well.

Dynamical Properties of Dark Energy Models in Fractal Universe

In this paper, we consider the flat FRW spacetime filled with interacting dark energy and dark matter in fractal universe. We work with the three models of dark energy named as Tsallis, Renyi and Sharma–Mittal. We investigate different cosmological implications such as equation of state parameter, squared speed of sound, deceleration parameter, statefinder parameters, ω e f f - ω e f f ´ (where prime indicates the derivative with respect to ln a , and a is cosmic scale factor) plane and Om diagnostic. We explore these parameters graphically to study the evolving universe. We compare the consistency of dark energy models with the accelerating universe observational data. All three models are stable in fractal universe and support accelerated expansion of the universe.

Entropic Characterization of Quantum States with Maximal Evolution under Given Energy Constraints

A measure D [ t 1 , t 2 ] for the amount of dynamical evolution exhibited by a quantum system during a time interval [ t 1 , t 2 ] is defined in terms of how distinguishable from each other are, on average, the states of the system at different times. We investigate some properties of the measure D showing that, for increasing values of the interval's duration, the measure quickly reaches an asymptotic value given by the linear entropy of the energy distribution associated with the system's (pure) quantum state. This leads to the formulation of an entropic variational problem characterizing the quantum states that exhibit the largest amount of dynamical evolution under energy constraints given by the expectation value of the energy.

Shannon, Rényi, Tsallis Entropies and Onicescu Information Energy for Low-Lying Singly Excited States of Helium

Knowledge of the electronic structures of atomic and molecular systems deepens our understanding of the desired system. In particular, several information-theoretic quantities, such as Shannon entropy, have been applied to quantify the extent of electron delocalization for the ground state of various systems. To explore excited states, we calculated Shannon entropy and two of its one-parameter generalizations, Rényi entropy of order α and Tsallis entropy of order α , and Onicescu Information Energy of order α for four low-lying singly excited states (1s2s 1 S e , 1s2s 3 S e , 1s3s 1 S e , and 1s3s 3 S e states) of helium. This paper compares the behavior of these three quantities of order 0.5 to 9 for the ground and four excited states. We found that, generally, a higher excited state had a larger Rényi entropy, larger Tsallis entropy, and smaller Onicescu information energy. However, this trend was not definite and the singlet–triplet reversal occurred for Rényi entropy, Tsallis entropy and Onicescu information energy at a certain range of order α .

Classification of MRI Brain Images Using DNA Genetic Algorithms Optimized Tsallis Entropy and Support Vector Machine

As a non-invasive diagnostic tool, Magnetic Resonance Imaging (MRI) has been widely used in the field of brain imaging. The classification of MRI brain image conditions poses challenges both technically and clinically, as MRI is primarily used for soft tissue anatomy and can generate large amounts of detailed information about the brain conditions of a subject. To classify benign and malignant MRI brain images, we propose a new method. Discrete wavelet transform (DWT) is used to extract wavelet coefficients from MRI images. Then, Tsallis entropy with DNA genetic algorithm (DNA-GA) optimization parameters (called DNAGA-TE) was used to obtain entropy characteristics from DWT coefficients. At last, DNA-GA optimized support vector machine (called DNAGA-KSVM) with radial basis function (RBF) kernel, is applied as a classifier. In our experimental procedure, we use two kinds of images to validate the availability and effectiveness of the algorithm. One kind of data is the Simulated Brain Database and another kind of image is real MRI images which downloaded from Harvard Medical School website. Experimental results demonstrate that our method (DNAGA-TE+KSVM) obtained better classification accuracy.

Analytic Study of Complex Fractional Tsallis' Entropy with Applications in CNNs

In this paper, we study Tsallis' fractional entropy (TFE) in a complex domain by applying the definition of the complex probability functions. We study the upper and lower bounds of TFE based on some special functions. Moreover, applications in complex neural networks (CNNs) are illustrated to recognize the accuracy of CNNs.

The Complexity Measures Associated with the Fluctuations of the Entropy in Natural Time before the Deadly México M8.2 Earthquake on 7 September 2017

We analyse seismicity during the 6-year period 2012-2017 in the new time domain termed natural time in the Chiapas region where the M8.2 earthquake occurred, Mexico's largest earthquake in more than a century, in order to study the complexity measures associated with fluctuations of entropy as well as with entropy change under time reversal. We find that almost three months before the M8.2 earthquake, i.e., on 14 June 2017, the complexity measure associated with the fluctuations of entropy change under time reversal shows an abrupt increase, which, however, does not hold for the complexity measure associated with the fluctuations of entropy in forward time. On the same date, the entropy change under time reversal has been previously found to exhibit a minimum [Physica A 506, 625-634 (2018)]; we thus find here that this minimum is also accompanied by increased fluctuations of the entropy change under time reversal. In addition, we find a simultaneous increase of the Tsallis entropic index q.

Generalized Information Theory Meets Human Cognition: Introducing a Unified Framework to Model Uncertainty and Information Search

Searching for information is critical in many situations. In medicine, for instance, careful choice of a diagnostic test can help narrow down the range of plausible diseases that the patient might have. In a probabilistic framework, test selection is often modeled by assuming that people's goal is to reduce uncertainty about possible states of the world. In cognitive science, psychology, and medical decision making, Shannon entropy is the most prominent and most widely used model to formalize probabilistic uncertainty and the reduction thereof. However, a variety of alternative entropy metrics (Hartley, Quadratic, Tsallis, Rényi, and more) are popular in the social and the natural sciences, computer science, and philosophy of science. Particular entropy measures have been predominant in particular research areas, and it is often an open issue whether these divergences emerge from different theoretical and practical goals or are merely due to historical accident. Cutting across disciplinary boundaries, we show that several entropy and entropy reduction measures arise as special cases in a unified formalism, the Sharma-Mittal framework. Using mathematical results, computer simulations, and analyses of published behavioral data, we discuss four key questions: How do various entropy models relate to each other? What insights can be obtained by considering diverse entropy models within a unified framework? What is the psychological plausibility of different entropy models? What new questions and insights for research on human information acquisition follow? Our work provides several new pathways for theoretical and empirical research, reconciling apparently conflicting approaches and empirical findings within a comprehensive and unified information-theoretic formalism.

Kappa and other nonequilibrium distributions from the Fokker-Planck equation and the relationship to Tsallis entropy

This paper considers two nonequilibrium model systems described by linear Fokker-Planck equations for the time-dependent velocity distribution functions that yield steady state Kappa distributions for specific system parameters. The first system describes the time evolution of a charged test particle in a constant temperature heat bath of a second charged particle. The time dependence of the distribution function of the test particle is given by a Fokker-Planck equation with drift and diffusion coefficients for Coulomb collisions as well as a diffusion coefficient for wave-particle interactions. A second system involves the Fokker-Planck equation for electrons dilutely dispersed in a constant temperature heat bath of atoms or ions and subject to an external time-independent uniform electric field. The momentum transfer cross section for collisions between the two components is assumed to be a power law in reduced speed. The time-dependent Fokker-Planck equations for both model systems are solved with a numerical finite difference method and the approach to equilibrium is rationalized with the Kullback-Leibler relative entropy. For particular choices of the system parameters for both models, the steady distribution is found to be a Kappa distribution. Kappa distributions were introduced as an empirical fitting function that well describe the nonequilibrium features of the distribution functions of electrons and ions in space science as measured by satellite instruments. The calculation of the Kappa distribution from the Fokker-Planck equations provides a direct physically based dynamical approach in contrast to the nonextensive entropy formalism by Tsallis [J. Stat. Phys. 53, 479 (1988)JSTPBS0022-471510.1007/BF01016429].

Entropic nonadditivity, H theorem, and nonlinear Klein-Kramers equations

We use the H theorem to establish the entropy and the entropic additivity law for a system composed of subsystems, with the dynamics governed by the Klein-Kramers equations, by considering relations among the dynamics of these subsystems and their entropies. We start considering the subsystems governed by linear Klein-Kramers equations and verify that the Boltzmann-Gibbs entropy is appropriated to this dynamics, leading us to the standard entropic additivity, S_{BG}^{(1∪2)}=S_{BG}^{1}+S_{BG}^{2}, consistent with the fact that the distributions of the subsystem are independent. We then extend the dynamics of these subsystems to independent nonlinear Klein-Kramers equations. For this case, the results show that the H theorem is verified for a generalized entropy, which does not preserve the standard entropic additivity for independent distributions. In this scenario, consistent results are obtained when a suitable coupling among the nonlinear Klein-Kramers equations is considered, in which each subsystem modifies the other until an equilibrium state is reached. This dynamics, for the subsystems, results in the Tsallis entropy for the system and, consequently, verifies the relation S_{q}^{(1∪2)}=S_{q}^{1}+S_{q}^{2}+(1-q)S_{q}^{1}S_{q}^{2}/k, which is a nonadditive entropic relation.

Mechanisms of self-organized criticality in social processes of knowledge creation

In online social dynamics, a robust scale invariance appears as a key feature of collaborative efforts that lead to new social value. The underlying empirical data thus offers a unique opportunity to study the origin of self-organized criticality (SOC) in social systems. In contrast to physical systems in the laboratory, various human attributes of the actors play an essential role in the process along with the contents (cognitive, emotional) of the communicated artifacts. As a prototypical example, we consider the social endeavor of knowledge creation via Questions and Answers (Q&A). Using a large empirical data set from one of such Q&A sites and theoretical modeling, we reveal fundamental characteristics of SOC by investigating the temporal correlations at all scales and the role of cognitive contents to the avalanches of the knowledge-creation process. Our analysis shows that the universal social dynamics with power-law inhomogeneities of the actions and delay times provides the primary mechanism for self-tuning towards the critical state; it leads to the long-range correlations and the event clustering in response to the external driving by the arrival of new users. In addition, the involved cognitive contents (systematically annotated in the data and observed in the model) exert important constraints that identify unique classes of the knowledge-creation avalanches. Specifically, besides determining a fine structure of the developing knowledge networks, they affect the values of scaling exponents and the geometry of large avalanches and shape the multifractal spectrum. Furthermore, we find that the level of the activity of the communities that share the knowledge correlates with the fluctuations of the innovation rate, implying that the increase of innovation may serve as the active principle of self-organization. To identify relevant parameters and unravel the role of the network evolution underlying the process in the social system under consideration, we compare the social avalanches to the avalanche sequences occurring in the field-driven physical model of disordered solids, where the factors contributing to the collective dynamics are better understood.

Repulsive particles under a general external potential: Thermodynamics by neglecting thermal noise

A recent proposal of an effective temperature θ, conjugated to a generalized entropy s_{q}, typical of nonextensive statistical mechanics, has led to a consistent thermodynamic framework in the case q=2. The proposal was explored for repulsively interacting vortices, currently used for modeling type-II superconductors. In these systems, the variable θ presents values much higher than those of typical room temperatures T, so that the thermal noise can be neglected (T/θ≃0). The whole procedure was developed for an equilibrium state obtained after a sufficiently long-time evolution, associated with a nonlinear Fokker-Planck equation and approached due to a confining external harmonic potential, ϕ(x)=αx^{2}/2 (α>0). Herein, the thermodynamic framework is extended to a quite general confining potential, namely ϕ(x)=α|x|^{z}/z (z>1). It is shown that the main results of the previous analyses hold for any z>1: (i) The definition of the effective temperature θ conjugated to the entropy s_{2}. (ii) The construction of a Carnot cycle, whose efficiency is shown to be η=1-(θ_{2}/θ_{1}), where θ_{1} and θ_{2} are the effective temperatures associated with two isothermal transformations, with θ_{1}>θ_{2}. The special character of the Carnot cycle is indicated by analyzing another cycle that presents an efficiency depending on z. (iii) Applying Legendre transformations for a distinct pair of variables, different thermodynamic potentials are obtained, and furthermore, Maxwell relations and response functions are derived. The present approach shows a consistent thermodynamic framework, suggesting that these results should hold for a general confining potential ϕ(x), increasing the possibility of experimental verifications.

Pathological brain detection in MRI scanning by wavelet packet Tsallis entropy and fuzzy support vector machine

An computer-aided diagnosis system of pathological brain detection (PBD) is important for help physicians interpret and analyze medical images. We proposed a novel automatic PBD to distinguish pathological brains from healthy brains in magnetic resonance imaging scanning in this paper. The proposed method simplified the PBD problem to a binary classification task. We extracted the wavelet packet Tsallis entropy (WPTE) from each brain image. The WPTE is the Tsallis entropy of the coefficients of the discrete wavelet packet transform. The, the features were submitted to the fuzzy support vector machine (FSVM). We tested the proposed diagnosis method on 3 benchmark datasets with different sizes. A ten runs of K-fold stratified cross validation was carried out. The results demonstrated that the proposed WPTE + FSVM method excelled 17 state-of-the-art methods w.r.t. classification accuracy. The WPTE is superior to discrete wavelet transform. The Tsallis entropy performs better than Shannon entropy. The FSVM excels standard SVM. In closing, the proposed method “WPTE + FSVM” is effective in PBD.

The dynamics of meaningful social interactions and the emergence of collective knowledge

Collective knowledge as a social value may arise in cooperation among actors whose individual expertise is limited. The process of knowledge creation requires meaningful, logically coordinated interactions, which represents a challenging problem to physics and social dynamics modeling. By combining two-scale dynamics model with empirical data analysis from a well-known Questions &Answers system Mathematics, we show that this process occurs as a collective phenomenon in an enlarged network (of actors and their artifacts) where the cognitive recognition interactions are properly encoded. The emergent behavior is quantified by the information divergence and innovation advancing of knowledge over time and the signatures of self-organization and knowledge sharing communities. These measures elucidate the impact of each cognitive element and the individual actor's expertise in the collective dynamics. The results are relevant to stochastic processes involving smart components and to collaborative social endeavors, for instance, crowdsourcing scientific knowledge production with online games.

Statistical scaling, Shannon entropy, and Generalized space-time q-entropy of rainfall fields in tropical South America

We study diverse scaling and information theory characteristics of Mesoscale Convective Systems (MCSs) as seen by the Tropical Rainfall Measuring Mission (TRMM) over continental and oceanic regions of tropical South America, and 2-D radar rainfall fields from Amazonia. The bi-dimensional Fourier spectra of MCSs exhibit inverse power laws with respect to the spatial scale, whose scaling exponents, β, capture the type of spatial correlation of rainfall among the study regions, including those over the Andes of Colombia as well as over oceanic and Amazonian regions. The moment-scaling analysis evidences that the structure function deviates from simple scaling at order q > 1.0, thus signaling the multi-scaling nature of rainfall fields within MCSs in tropical South America, with departures from simple scaling associated with the physical characteristics of MCSs over the different study regions. Entropy is estimated for a large set of radar rainfall fields during the distinctive atmospheric regimes (Easterly and Westerly events) in this part of Amazonia. Results evidence that there are significant differences in the dynamics of rainfall among regimes. No clear-cut relationship is found between entropy and the first two statistical moments, but power fits in space and time, S(γ) ∼ γ(-η) for skewness and, S(κ) ∼ κ(-ϵ) for kurtosis. The exponents η and ϵ are statistically different between Easterly and Westerly events, although the significance of fits is less when L-moments are used to estimate skewness and kurtosis. Interesting differences are identified between the time and space generalized q-entropy functions of Amazonian rainfall fields. In both cases, the functions are a continuous set of power laws (analogous to the structure function in turbulence), S(T, q) ∼ T(β), and, S(λ, q) ∼ λ(β), covering a broad range of temporal and spatial scales. Both time and space generalized q-entropy functions exhibit linear growth in the range -1.0 < q < -0.5, and saturation of the exponent β for q ≥ 1.0. In the case of the spatial analysis, the exponent saturates at 〈β〉∼1.0, whereas at 〈β〉=0.5 for the temporal case. Results are similar for time series extracted from the S-POL radar and time series of rainfall in tropical Andes. Additionally, differences in values of 〈β〉 for q ≥ 1.0 between Easterly and Westerly events are not statistically significant.

Dynamic properties of small-scale solar wind plasma fluctuations

The paper presents the latest results of the studies of small-scale fluctuations in a turbulent flow of solar wind (SW) using measurements with extremely high temporal resolution (up to 0.03 s) of the bright monitor of SW (BMSW) plasma spectrometer operating on astrophysical SPECTR-R spacecraft at distances up to 350,000 km from the Earth. The spectra of SW ion flux fluctuations in the range of scales between 0.03 and 100 s are systematically analysed. The difference of slopes in low- and high-frequency parts of spectra and the frequency of the break point between these two characteristic slopes was analysed for different conditions in the SW. The statistical properties of the SW ion flux fluctuations were thoroughly analysed on scales less than 10 s. A high level of intermittency is demonstrated. The extended self-similarity of SW ion flux turbulent flow is constantly observed. The approximation of non-Gaussian probability distribution function of ion flux fluctuations by the Tsallis statistics shows the non-extensive character of SW fluctuations. Statistical characteristics of ion flux fluctuations are compared with the predictions of a log-Poisson model. The log-Poisson parametrization of the structure function scaling has shown that well-defined filament-like plasma structures are, as a rule, observed in the turbulent SW flows.

Nonlinear Kramers equation associated with nonextensive statistical mechanics

Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results.

Consistent thermodynamic framework for interacting particles by neglecting thermal noise

An effective temperature θ, conjugated to a generalized entropy s(q), was introduced recently for a system of interacting particles. Since θ presents values much higher than those of typical room temperatures T≪θ, the thermal noise can be neglected (T/θ≃0) in these systems. Moreover, the consistency of this definition, as well as of a form analogous to the first law of thermodynamics, du=θds(q)+δW, were verified lately by means of a Carnot cycle, whose efficiency was shown to present the usual form, η=1-(θ(2)/θ(1)). Herein we explore further the heat contribution δQ=θds(q) by proposing a way for a heat exchange between two such systems, as well as its associated thermal equilibrium. As a consequence, the zeroth principle is also established. Moreover, we consolidate the first-law proposal by following the usual procedure for obtaining different potentials, i.e., applying Legendre transformations for distinct pairs of independent variables. From these potentials we derive the equation of state, Maxwell relations, and define response functions. All results presented are shown to be consistent with those of standard thermodynamics for T>0.

Second law and entropy production in a nonextensive system

A model of superconducting vortices under overdamped motion is currently used for describing type-II superconductors. Recently, this model has been identified to a nonlinear Fokker-Planck equation and associated to an entropic form characteristic of nonextensive statistical mechanics, S(2)(t)≡S((q)=2)(t). In the present work, we consider a system of superconducting vortices under overdamped motion, following an irreversible process, so that by using the corresponding nonlinear Fokker-Planck equation, the entropy time rate [dS(2)(t)/dt] is investigated. Both entropy production and entropy flux from the system to its surroundings are analyzed. Molecular dynamics simulations are carried for this process, showing a good agreement between the numerical and analytical results. It is shown that the second law holds within the present framework, and we exhibit the increase of S(2)(t) with time, up to its stationary-state value.

Comment on "Vortex distribution in a confining potential"

A system of interacting vortices is considered as an appropriate model for describing properties of type-II superconductors, and it has been shown lately to be deeply associated with nonextensive statistical mechanics. Herein we comment on a recent investigation of this model [M. Girotto, A. P. dos Santos, and Y. Levin, Phys. Rev. E 88, 032118 (2013)], which tried to contradict this assertion, based on a mean-field type of solution, compared with numerical-simulation data that correspond typically to a regime characterized by low concentrations of particles, as well as very high temperatures. It is shown that the physical situations analyzed differ significantly from those of a real superconducting phase. The analytical solution obtained from such a mean-field approximation shows a discrepancy with respect to the results of molecular-dynamics numerical simulations, which increases as the temperature is lowered towards the superconducting phase, as expected. We demonstrate that these results, when interpreted properly by means of an analytical solution within the framework of nonextensive statistical mechanics, present a remarkable agreement between molecular-dynamics simulations and theoretical results, for all temperatures, specially for those temperatures associated with the existence of type-II superconductivity.

Probability distributions and associated nonlinear Fokker-Planck equation for the two-index entropic form S(q,δ)

The probability distributions and associated Fokker-Planck equation of the recently postulated entropic form, S(q,δ), are investigated. This entropy was proposed as an unification of the well-known S(q) of nonextensive-statistical mechanics and S(δ), which appeared lately as a possibly appropriate candidate for the black-hole entropy. The connection between S(q,δ) and a nonlinear Fokker-Planck equation, such as to satisfy an H-theorem, is explored. The stationary-state probability distribution follows a transcendental equation, which is solved numerically for typical values of q and δ. The same transcendental equation is obtained through the maximum-entropy principle, showing that the two procedures are equivalent.

Carnot cycle for interacting particles in the absence of thermal noise

A thermodynamic formalism is developed for a system of interacting particles under overdamped motion, which has been recently analyzed within the framework of nonextensive statistical mechanics. It amounts to expressing the interaction energy of the system in terms of a temperature θ, conjugated to a generalized entropy s(q), with q = 2. Since θ assumes much higher values than those of typical room temperatures T ≪ θ, the thermal noise can be neglected for this system (T/θ ≃ 0). This framework is now extended by the introduction of a work term δW which, together with the formerly defined heat contribution (δ Q = θ ds(q)), allows for the statement of a proper energy conservation law that is analogous to the first law of thermodynamics. These definitions lead to the derivation of an equation of state and to the characterization of s(q) adiabatic and θ isothermic transformations. On this basis, a Carnot cycle is constructed, whose efficiency is shown to be η = 1-(θ(2)/θ(1)), where θ(1) and θ(2) are the effective temperatures of the two isothermic transformations, with θ(1)>θ(2). The results for a generalized thermodynamic description of this system open the possibility for further physical consequences, like the realization of a thermal engine based on energy exchanges gauged by the temperature θ.

Probability distributions extremizing the nonadditive entropy S(δ) and stationary states of the corresponding nonlinear Fokker-Planck equation

Under the assumption that the physically appropriate entropy of generic complex systems satisfies thermodynamic extensivity, we investigate the recently introduced entropy S(δ) (which recovers the usual Boltzmann-Gibbs form for δ=1) and establish the microcanonical and canonical extremizing distributions. Using a generalized version of the H theorem, we find the nonlinear Fokker-Planck equation associated with that entropic functional and calculate the stationary-state probability distributions. We demonstrate that both approaches yield one and the same equation, which in turn uniquely determines the probability distribution. We show that the equilibrium distributions asymptotically behave like stretched exponentials, and that, in appropriate probability-energy variables, an interesting return occurs at δ=4/3. As a mathematically simple illustration, we consider the one-dimensional harmonic oscillator and calculate the generalized chemical potential for different values of δ.

First-passage time for subdiffusion: the nonadditive entropy approach versus the fractional model

We study the similarities and differences between different models concerning subdiffusion. More particularly, we calculate first passage time (FPT) distributions for subdiffusion, derived from Greens' functions of nonlinear equations obtained from Sharma-Mittal's, Tsallis's, and Gauss's nonadditive entropies. Then we compare these with FPT distributions calculated from a fractional model using a subdiffusion equation with a fractional time derivative. All of Greens' functions give us exactly the same standard relation ((Δx)(2) )=2D(α)t(α) which characterizes subdiffusion (0 < α <1), but generally FPT distributions are not equivalent to one another. We will show here that the FPT distribution for the fractional model is asymptotically equal to the Sharma-Mittal model over the long time limit only if in the latter case one of the three parameters describing Sharma-Mittal entropy r depends on α, and satisfies the specific equation derived in this paper, whereas the other two models mentioned above give different FPT distributions with the fractional model. Greens' functions obtained from the Sharma-Mittal and fractional models, for r obtained from this particular equation, are very similar to each other. We will also discuss the interpretation of subdiffusion models based on nonadditive entropies and the possibilities of the experimental measurement of subdiffusion models' parameters.