Nonextensive Statistical Mechanics Research Articles

Plasma oscillations and spectral index in non-extensive statistics

In this paper, we have discussed the dependency of the plasma oscillations on the numbers of degrees of freedom (dimensionality) involved in the spectral indices (q or κ) of non-extensive statistical mechanics. Generally, the spectral indices have an inherent dimensional dependency, which may be ignored when a reduced velocity distribution function (VDF) has been derived from the higher dimensional ones, e.g., when studying the longitudinal oscillations or the related topics. In such cases, if we ignore the equipartition of energy, the higher dimensional dependency of the original VDF may be still included in the spectral index of our solutions. On the other hand, in many studies of the plasma oscillations in non-extensive statistics, the ordinary formalism of canonical probability distribution (the old version) had been used instead of the escort probability distribution (the modern version). The escort formalism has some advantages, as opposed to the ordinary version, which may provide more appropriate solutions for the plasma oscillations. In this study, we have re-examined this problem by utilizing the modern version of kappa distribution formalism labeled with an invariant kappa index as of zero dimensionality spectral index, κ0, which is independent of the dimensionality, the degrees of freedom, or the numbers of particles. The related dispersion relation and Landau damping rate of plasma oscillations (Langmuir waves) have been analytically derived and studied by using the Vlasov–Poisson equations for the weakly damped modes. The physical and thermodynamics features of the problem have been considered in terms of the variation of the invariant spectral index κ0.

Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions.

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function , Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended to the complex plane z and conjectured that all nontrivial zeros are in the axis. The nonadditive entropy , where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function . It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as , which recover the number x for . The q-prime numbers are then defined as the q-natural numbers , where n is a prime number We show that, for any value of q, infinitely many q-prime numbers exist; for they diverge for increasing prime number, whereas they converge for ; the standard prime numbers are recovered for . For , we generalize the function as follows: (). We show that this function appears to diverge at , . Also, we alternatively define, for , and , which, for , generically satisfy , in variance with the case, where of course .

Thermodynamic Definitions of Temperature and Kappa and Introduction of the Entropy Defect.

This paper develops explicit and consistent definitions of the independent thermodynamic properties of temperature and the kappa index within the framework of nonextensive statistical mechanics and shows their connection with the formalism of kappa distributions. By defining the “entropy defect” in the composition of a system, we show how the nonextensive entropy of systems with correlations differs from the sum of the entropies of their constituents of these systems. A system is composed extensively when its elementary subsystems are independent, interacting with no correlations; this leads to an extensive system entropy, which is simply the sum of the subsystem entropies. In contrast, a system is composed nonextensively when its elementary subsystems are connected through long-range interactions that produce correlations. This leads to an entropy defect that quantifies the missing entropy, analogous to the mass defect that quantifies the mass (energy) associated with assembling subatomic particles. We develop thermodynamic definitions of kappa and temperature that connect with the corresponding kinetic definitions originated from kappa distributions. Finally, we show that the entropy of a system, composed by a number of subsystems with correlations, is determined using both discrete and continuous descriptions, and find: (i) the resulted entropic form expressed in terms of thermodynamic parameters; (ii) an optimal relationship between kappa and temperature; and (iii) the correlation coefficient to be inversely proportional to the temperature logarithm.

Theoretical self-consistency in nonextensive statistical mechanics with parameter transformation

Self-consistency in nonextensive statistical mechanics is studied as a recourse to parameter transformation, where different nonextensive parameters are presented for various theoretical branches. The unification between the first and third choices of the average definition and that between the normal and escort distributions are both examined. The problem of parameter inversion in the generalized H theorem is also investigated. The inconsistency between the statistical ensemble pressure and molecular dynamics pressure can be eliminated. This work also verifies the equivalence of physical temperature and gravitational temperature in nonextensive statistical mechanics. In these parameter transformations, the Tsallis entropy form is observed to remain invariant.

Exploring the nonextensive thermodynamics of partially ionized gas in magnetic field.

Contrary to classical thermodynamics, which deals with systems in thermal equilibrium, partially ionized gases generally do not reach thermal equilibrium. Nonextensive statistical mechanics has helped extend classical thermodynamics to nonequilibrium ionized gas. However, the fundamental question on whether the statistics of non-Maxwellian electrons satisfy the laws of thermodynamics has not been resolved. Here, we verify the thermodynamic laws of reversible and adiabatic processes for a magnetically expanding ionized gas. Together with the experimental evidence of the non-Maxwellian electron distribution, the κ distribution, which measures the thermal equilibrium states, shows the Tsallis entropy to be nearly constant and the polytropic index to be close to adiabatic values along a divergent magnetic field. These results verify that the collisionless magnetic expansion of a nonequilibrium plasma is reversible and adiabatic, and an isentropic process is the origin of the high-energy tail of the energy distribution far downstream.

On the distribution function of suprathermal particles at collisionless shocks

The departure of particle distributions from the Maxwellian is commonly observed in space plasmas. These non-Maxwellian distributions which are typical for plasmas that are not in thermal equilibrium, can be modeled with κ-distribution. Kinetic simulations of quasi-parallel collisionless shocks show that proton distribution is a composite of thermal, suprathermal, and non-thermal parts. By using particle-in-cell shock simulations, we show that κ-distribution adequately fits thermal and suprathermal parts together, as a single continuous distribution in early proton spectra. We derive suprathermal proton distribution directly from the generalized entropy of non-extensive statistical mechanics, and show that thermal and suprathermal populations are both naturally embedded in κ-distribution. We find that the index κ of the distribution increases with the distance from the shock, following the decrease in suprathermal part. The non-equilibrium plasma distribution which is continuously being enriched with suprathermal particles at the reforming shock barrier, reaches the thermal equilibrium in the far downstream. The suprathermal part completely fades there, and the shape of proton distribution becomes a Maxwellian from which directly emerges a power-law.

The ‘wavelet’ entropic index q of non-extensive statistical mechanics and superstatistics

Generalized entropies developed for non-extensive statistical mechanics are derived from the Boltzmann-Gibbs-Shannon entropy by a real number q that is a parameter based on q-calculus; where q is called ‘the entropic index’ and determines the degree of non-extensivity of a system in the interval between 1 and 3. In a very recent study, we introduced a new calculation method of the entropic index q of non-extensive statistical mechanics. In this study, we show the mathematical proof of this calculation method of the entropic index. Firstly, we propose that the number of degrees of freedom, n is proportional to the inverse of the wavelet scale index,n≡1iscale, where iscale is a wavelet based parameter called wavelet scale index that quantitatively measures the non-periodicity of a signal in the interval between 0 and 1. Then, by applying this proposition to the superstatistics approach, we derive the equation that expresses the relationship between the entropic index and the wavelet scale index, q=1+2iscale. Therefore, we name this q-index as the ‘wavelet’ entropic index. Lastly, we calculate the Abe entropy, Landsberg-Vedral entropy and q-dualities of the Tsallis entropy of the Logistic Map and Hennon Map using the ‘wavelet’ entropic index, and based on our results, compare and discuss these generalized entropies.

Experimental investigations of pressure stimulated currents from stressed sandstone used as precursors to rock fracture

Precursor information of rock fracture is a prerequisite for rock dynamic disaster (RDD) prediction. To investigate the precursor information of rock fracture based on pressure stimulated current (PSC), progressive loading (PL) and step-like loading (SLL) experiments on sandstone samples were conducted. The response characteristics of PSCs from sandstone samples being actively loaded in the laboratory were analyzed and potential precursors to rock fracture based on PSCs are discussed. The results show that weak currents are generated instantaneously when a load is applied on sandstone samples and the PSCs correspond well to stress variations. In the PL experiments, the PSC responses are different in the different stages of deformation. In the compaction stage, the PSC first increases linearly and rapidly as the stress becomes greater but then decreases slowly a few seconds later; In the elastic stage, the current increases slowly and linearly with stress but it begins to rise with an increasing rate when the sample enters the plastic deformation stage, reaching the peak value of PSC the instant the final fracture occurs. In the SLL experiments, when stress is increased from a lower to a higher stress step, the PSCs increase rapidly and reach peak values the moment the stress stops increasing. In addition, the peak current increases exponentially with stress rate with a negative exponent. During stress maintenance, the PSCs decay slowly before finally reaching a stable value, and the PSC decaying process obeys the non-extensive statistical mechanics . The accumulated charge increases linearly with the relative stress, but the charge increasing rate are different in the three deformation stages. The variation rate of accumulated charge in plastic deformation stage is approximately eleven times as that in elastic deformation stage. The variation of current has a precursor response to rock fracture: the increase of PSC with an accelerative rate can be used as a precursor of rock failure during a progressive loading, the increasingly intensive abrupt drops during the PSCs decay are precursors for rock fracture during creep, and the rapid increase of accumulated charge variation rate can be used as a precursor of rock fracture both under progressive and step-like loadings. These research results are expected to provide new ideas and methods for RDD prediction.

Innovation for measuring the distribution function with nonextensive single electric probe

In the field of plasma diagnosis, the measurement of the distribution function is significant because the distribution function is the basis for the use of plasma kinetic theory and it is the prerequisite for analyzing many physical phenomena, such as Landau damping (wave-particle resonance phenomenon) and ion sheath. Theoretical analysis and a large number of experiments have proved that plasma components do not obey Boltzmann–Gibbs statistics and can be well described by nonextensive statistical mechanics. The field of nonextensive electric probe has also made great progress, and the invention of the nonextensive single electric probes has developed and strengthened the power of plasma diagnostics. The nonextensive electric probe can not only measure the electron nonextensive parameter of plasma that cannot be measured by traditional probes but can also measure more accurate plasma parameters that can also be measured by traditional probes, such as Te, Φp, ne, Φf, and αqFe. However, diagnosing the plasma distribution function by the nonextensive electric probe has not been thoroughly and systematically analyzed and discussed. Here, we show the measurement of the plasma distribution function with a nonextensive single electric probe. This work expands the diagnostic capabilities of the nonextensive single electric probe. We utilize the nonextensive single electric probe theory to analyze the experimental data points of the I–V curve, measure the plasma electron distribution function fvx, and display the distribution curve (figure f-vx), and we also measure the plasma parameters of qFe, Te, Φp, ne, Φf, αqFe, etc. The proposed method provides a new approach to the diagnosis of the plasma distribution function and contributes to a more accurate and comprehensive grasp of plasma, which creates better conditions for us to take advantage of plasma. These initial results illustrate the potential of the nonextensive electric probe in the field of plasma diagnosis and, more generally, in accelerating the progress of fusion-energy science and helping to understand complex physical systems.

Non-extensive statistical mechanics for acoustic emission in disordered media: Entropy, size effect, and self-organization

The present work explores acoustic emission phenomenon of concrete like disordered material within the framework of non-extensive statistical mechanics. The aim of the study is threefold. At first, we re-derive the non-extensive distribution function using the power-function and exponential-function ansatzes. We assert that the power-function ansatz based distribution model, compared to exponential-function based model, is superior due to its ability to recover the exponential distribution in the limit when the tail index q→1 and due to its higher sensitivity for long-tail. The ability of recovering exponential distribution in the limit q→1 preserves the essence of Tsallis non-extensive statistical mechanics formulation and retains the pertinent meaning of entropic index q of the distribution. Second, we study the size effect on the various parameters of the distribution models and discover the size-independence of the entropic index. Thirdly, the self-organization phenomenon often observed in the complex dynamic systems is commented. The existence of criticality near failure for quasi-static loading of concrete beams appear speculative and the criticality might exist in midway of damage progress.

Anisotropic Kappa Distributions. I. Formulation Based on Particle Correlations

Abstract We develop the theoretical basis for the connection of the variety of anisotropic distributions with the statistical correlations among particles’ velocity components. By examining the most common anisotropic distribution function, we derive the correlation coefficient among particle energies, show how this correlation is connected to the effective dimensionality of the velocity distribution, and derive the connection between anisotropy and adiabatic polytropic index. Having established the importance of the correlation among particles in the formulation of anisotropic kappa distributions, we generalize these distributions within the framework of nonextensive statistical mechanics and based on the types of homogeneous or heterogeneous correlations among the particles’ velocity components. The formulation of the developed generalized distributions mediates the main two types of anisotropic kappa distributions that consider either (a) equal correlations, or (b) zero correlations, among different velocity components. Finally, the developed anisotropic kappa distributions are expressed in terms of the energy and pitch angle in arbitrary reference frames.

Nonextensive statistical mechanics for robust physical parameter estimation: the role of entropic index

The problem of inferencing parameters of complex systems from measured data has been extensively studied based on the inverse problem theory. Classically, an inverse problem is formulated as an optimisation task in which the objective function is based on the maximisation of Boltzmann–Gibbs entropy under appropriate constraints. In this classical framework, errors are assumed to follow a Gaussian-behaviour. However, in many situations, the errors are non-Gaussian and therefore the classical approach is predestined to failure. To mitigate the inverse-problem sensitivity to non-Gaussian errors, we have considered in this study a q-generalised objective function within the context of nonextensive statistical mechanics. In this regard, the errors are assumed to follow the q-Gaussian distribution, which arises from the maximisation of the nonadditive Tsallis entropy. We study the robustness properties of the q-objective function, which is a generalisation of the classical objective function, through the so-called influence function. In this work, we analyse and discuss from an analytical and numerical perspective, the role of the entropic index (q) of Tsallis entropy on the effectiveness and robustness in the inference of physical parameters from with strongly noisy-data. In this perspective, we show that exists an optimum value for the entropic index at the limit $$q \rightarrow 3$$ , which implies in $$\delta _q = 3 - q \approx 0$$ in the q-objective function. To validate our proposal, we consider a classic geophysical inverse problem by employing a realistic geological model and a data set contaminated by spikes.

Non-Extensive Statistical Analysis of Acoustic Emissions: The Variability of Entropic Index q during Loading of Brittle Materials until Fracture.

Non-extensive statistical mechanics (NESM), introduced by Tsallis based on the principle of non-additive entropy, is a generalisation of the Boltzmann–Gibbs statistics. NESM has been shown to provide the necessary theoretical and analytical implementation for studying complex systems such as the fracture mechanisms and crack evolution processes that occur in mechanically loaded specimens of brittle materials. In the current work, acoustic emission (AE) data recorded when marble and cement mortar specimens were subjected to three distinct loading protocols until fracture, are discussed in the context of NESM. The NESM analysis showed that the cumulative distribution functions of the AE interevent times (i.e., the time interval between successive AE hits) follow a q-exponential function. For each examined specimen, the corresponding Tsallis entropic q-indices and the parameters βq and were calculated. The entropic index shows a systematic behaviour strongly related to the various stages of the implemented loading protocols for all the examined specimens. Results seem to support the idea of using the entropic index as a potential pre-failure indicator for the impending catastrophic fracture of the mechanically loaded specimens.

A new structure entropy of complex networks based on nonextensive statistical mechanics and similarity of nodes.

Entropy has been widely measured the complexity of complex networks. At present, many measures about entropies were defined based on the directed connection of nodes. The similarity of nodes can better represent the relationship among all nodes in complex networks. In the definition of similarity of nodes, the importance of a node in the network depends not only on the degree of the node itself, but also on the extent of dependence of neighboring nodes on the node. In this paper, we proposed a new structure entropy based on nonextensive statistical mechanics and similarity of nodes. In the proposed method, the similarity of nodes and the betweenness of nodes are both quantified. The proposed method considers the extent of dependence between neighbouring nodes. For some complex networks, the proposed structure entropy can distinguish complexity of that while other entropies can not be. Meanwhile, we construct five ER random networks and small-world networks and some real-world complex networks such as the US Air Lines networks, the GD'01-GD Proceedings Self-Citing networks, the Science Theory networks, the Centrality Literature networks and the Yeast networks are measured by the proposed method. The results illustrated our method for quantifying the complexity of complex networks is effective.

A new structural entropy measurement of networks based on the nonextensive statistical mechanics and hub repulsion.

The structure properties of complex networks are an open issue. As the most important parameter to describe the structural properties of the complex network, the structure entropy has attracted much attention. Recently, the researchers note that hub repulsion plays an role in structural entropy. In this paper, the repulsion between nodes in complex networks is simulated when calculating the structure entropy of the complex network. Coulomb's law is used to quantitatively express the repulsive force between two nodes of the complex network, and a new structural entropy based on the Tsallis nonextensive statistical mechanics is proposed. The new structure entropy synthesizes the influence of repulsive force and betweenness. We study several construction networks and some real complex networks, the results show that the proposed structure entropy can describe the structural properties of complex networks more reasonably. In particular, the new structural entropy has better discrimination in describing the complexity of the irregular network. Because in the irregular network, the difference of the new structure entropy is larger than that of degree structure entropy, betweenness structure entropy and Zhang's structure entropy. It shows that the new method has better discrimination for irregular networks, and experiments on Graph, Centrality literature, US Aire lines and Yeast networks confirm this conclusion.

Модифицированный в рамках негауссовской каппа-статистики интегральный критерий устойчивости Чандрасекхара для равновесного сферического облака протозвезды

Within the framework of the non-extensive statistical mechanics of Kanyadakis, a generalization of the integral stability theorem of Chandrasekhar for the spherically symmetric distribution of matter and black radiation in an exoplanetary cloud in a state of gravitational equilibrium is obtained. For this purpose, the elements of deformed thermodynamics for an ideal gas, deformed canonical Gibbs distribution, as well as the effective gravitational constant, calculated in the formalisms of Kanyadakis and Verlinde, are used. In this, the deformation parameter κ (kappa) measures the so-called degree of nonextensiveness of the cloud system. In addition, the modified thermodynamic properties of blackbody radiation, in particular, the analogue of Stefan's law for radiation energy and generalized expressions for the entropy, heat capacity and radiation pressure, are discussed in the context of κ -statistics. The presented method of combining the indicated anomalous physical processes provides an alternative to the classical procedure of Chandrasekhar's derivation of the well-known integral theorems for gas configurations in gravitational equilibrium, and restores all standard expressions in the limit κ → 0. The results obtained will be able, according to the author, to explain some astrophysical problems of stellar-planetary cosmogony, associated, in particular, with modeling the processes of joint formation and evolution of a protosun and an exoplanetary cloud from a single nebula.

Non-Extensive Statistical Analysis of Acoustic Emissions Recorded in Marble and Cement Mortar Specimens Under Mechanical Load Until Fracture.

Non-extensive statistical mechanics (NESM), which is a generalization of the traditional Boltzmann-Gibbs statistics, constitutes a theoretical and analytical tool for investigating the irreversible damage evolution processes and fracture mechanisms occurring when materials are subjected to mechanical loading. In this study, NESM is used for the analysis of the acoustic emission (AE) events recorded when marble and cement mortar specimens were subjected to mechanical loading until fracture. In total, AE data originating from four distinct loading protocols are presented. The cumulative distribution of inter-event times (time interval between two consecutive AE events) and the inter-event distances (three-dimensional Euclidian distance between the centers of successive AE events) were examined under the above concept and it was found that NESM is suitable to detect criticality under the terms of mechanical status of a material. This was conducted by evaluating the fitting results of the q-exponential function and the corresponding q-indices of Tsallis entropy and , along with the parameters and . Results support that for AE data recorded from marble and cement mortar specimens of this work, which is in good agreement with the conjecture previously found in seismological data and AE data recorded from Basalt specimens.

The principle of maximum entropy explains the cores observed in the mass distribution of dwarf galaxies

Cold dark matter (CDM) simulations predict a central cusp in the mass distribution of galaxies. This prediction is in stark contrast with observations of dwarf galaxies that show a central plateau or “core” in their density distribution. The proposed solutions to this core-cusp problem can be classified into two types. One invokes feedback mechanisms produced by the baryonic component of the galaxies and the other assumes that the properties of the dark matter particle depart from the CDM hypothesis. Here we propose an alternative yet complementary explanation. We argue that cores are unavoidable in the self-gravitating systems of maximum entropy that result from non-extensive statistical mechanics. Their structure follows from the Tsallis entropy, which is attributed to systems with long-range interactions. Strikingly, the mass density profiles predicted by such thermodynamic equilibrium match the observed cores without any adjustment or tuning. Thus, the principle of maximum Tsallis entropy explains the presence of cores in dwarf galaxies.

Non-extensive processes associated with heating of the Galactic disc

We analyse the mechanisms ruling galactic disc heating through the dynamics of space velocities U, V and W, extracted from the Geneva-Copenhagen catalogue. To do this, we use a model based on non-extensive statistical mechanics, where we derive the probability distribution functions that quantify the non-Gaussian effects. Furthermore, we find that the deviation q –1 at a given stellar age follows a non-random behaviour. As a result, the q-index behaviour indicates that the vertical component W, perpendicular to the Galactic plane, does not “heat up” at random, which is in disagreement with previous works that attributed the evolution of W to randomness. Finally, our results bring a new perspective to this matter and open the way for studying Galactic kinematic components through the eyes of more robust statistical models that consider non-Gaussian effects.The authors would like to dedicate this paper to all the victims of the COVID-19 pandemic around the world.

Long-range correlation studies in deep earthquakes global series

In the present paper we have conducted studies on seismological properties using worldwide data of deep earthquakes (depth larger than 70 km), considering events with magnitude greater than or equal to 4.5. This work is an extension of a previous study using a similar approach, for shallow events. We have addressed the problem under the perspective of complex networks, using a time window model to build the networks for deep earthquakes. These networks present scale-free and small-world features, contributing to strengthen the use of the time window model to construct epicenters networks. The results for deep events were analyzed using Nonextensive Statistical Mechanics and they corroborate with those found for the shallow ones, since the connectivity distribution for deep earthquakes also follows a q-exponential distribution and the scaling behavior is present. Our findings reinforce the idea of long-range correlations between earthquakes and the criticality of the seismological system.