Nonextensive Statistical Mechanics Research Articles

A discrete probabilistic model yielding multidimensional q − Gaussians in the thermodynamic limit for specific parameter values

We show that the N → ∞ limiting probability distributions (with N being the number of random variables to be summed) of a particular case belonging to a family of d − dimensional scale-invariant probabilistic models based on Leibniz-like (d + 1) − dimensional hyperpyramids (introduced in Rodríguez and Tsallis, J. Math. Phys 2012) are given, after an appropriate change o variables, by d − dimensional q − Gaussian distributions ( fq,β(x)∝eq−βxTΣ−1x , with x taking values in a subset of Rd , q and β real parameters, Σ a positive definite matrix, and eqx=[1+1(1−q)x]+11−q , with [u]+=max{u,0} ), for the particular parameter values q = 3 and β < 0. Some features of these distributions, which arise in the context of Nonextensive Statistical Mechanics, as well as a connection between Dirichlet distributions and q − Gaussian distributions, which generalizes the one given for dimension 1 by Rodríguez et al, are also exposed.

Thermal leptogenesis in nonextensive cosmology

Thermal leptogenesis is a mechanism that explains the observed asymmetry between matter and antimatter in the early universe. In this study, we review the impact of nonextensive Tsallis statistical mechanics on the early universe and study its effect on thermal leptogenesis. The study has found that the use of nonextensive statistical mechanics can affect the production of baryon asymmetry in thermal leptogenesis by modifying the equilibrium abundance of particles, decay, and washout parameters. Also, we show that nonextensive statistical mechanics potentially reduce the required right-handed neutrino mass scale.

Assessing structural integrity of non-homogeneous systems by means of Acoustic Emissions and Non-Extensive Statistical Mechanics

Taking advantage of the interevent time intervals, namely the time intervals between two successive acoustic events recorded during mechanical loading of structural elements, it is attempted to detect indices warning about upcoming failure. The innovative aspect of the study is that the analysis is implemented in the frame of Non-Extensive Statistical Mechanics, a discipline founded on a class of entropies violating the additivity principle, which is the cornerstone of the Boltzmann-Gibbs Statistical Mechanics. The specimens used for the experiments were marble blocks simulating either fragmented and restored epistyles or mutually interconnected intact epistyles of the Parthenon Temple on the Acropolis of Athens. The specimens consisted of three mate­rials (marble, metallic connectors and cementitious pastes) exhibiting, thus, a strongly non-homogeneous nature. The entropic index, i.e., the parameter quantifying the degree of non-additivity, was used for the analysis of the ex­pe­rimental data. The results were considered in juxtaposition to the re­spective ones from experimental protocols with specimens of macroscopically homo­geneous nature. It was concluded that the temporal evolution of the entropic index provides very good insight into the level of damage ac­cumu­lated in the loaded structure, independently of whether the structure is homogeneous or not, providing an interesting pre-failure indicator.

Two-parameter deformed quantum mechanics based on Fibonacci calculus and Debye crystal model of two-parameter deformed quantum statistics

Starting on the basis of Fibonacci calculus and Fibonacci oscillator algebra, we introduce the main properties to develop a new formalism for the two-parameter (q1,q2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({q}_{1},{q}_{2})$$\\end{document}-deformed quantum mechanics, where q1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${q}_{1}$$\\end{document} and q2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${q}_{2}$$\\end{document} are real positive independent deformation parameters. As applications of such a two-parameter deformed formalism, we investigate the behavior of a quantum particle in some different physical phenomena covering the free particle and the inverse-harmonic potential case. The effect of two deformation parameters on the wave functions for these applications is studied. Another application is carried out onto the quantum statistics of lattice oscillations through a model of the (q1,q2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({q}_{1},{q}_{2})$$\\end{document}-deformed phonon gas, and it is shown that the high- and low-temperature behavior of the model specific heat differs notably from the classical theories for the interval 0<(q1,q2)<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<({q}_{1},{q}_{2})<\\infty $$\\end{document}. We also construct a two-parameter deformed non-extensive entropy based on some elements of the Fibonacci calculus and discuss its possible connection with the Tsallis entropy in non-extensive statistical mechanics. Finally, other possible application areas of the present two-parameter (q1,q2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({q}_{1},{q}_{2})$$\\end{document}-deformed construction on quantum mechanics are discussed.

Identifying the Occurrence Time of the Destructive Kahramanmaraş-Gazientep Earthquake of Magnitude M7.8 in Turkey on 6 February 2023

Here, we employ natural time analysis of seismicity together with non-extensive statistical mechanics aiming at shortening the occurrence time window of the Kahramanmaraş-Gazientep M7.8 earthquake. The results obtained are in the positive direction pointing to the fact that after 3 February 2023 at 11:05:58 UTC, a strong earthquake was imminent. Natural time analysis also reveals a minimum fluctuation of the order parameter of seismicity almost three and a half months before the M7.8 earthquake, pointing to the initiation of seismic electrical activity. Moreover, before this earthquake occurrence, the detrended fluctuation analysis of the earthquake magnitude time-series reveals random behavior. Finally, when applying earthquake nowcasting, we find average earthquake potential score values which are compatible with those previously observed before strong (M≥7.1) earthquakes. The results obtained may improve our understanding of the physics of crustal phenomena that lead to strong earthquakes.

Open Problems within Nonextensive Statistical Mechanics.

Nonextensive statistical mechanics has developed into an important framework for modeling the thermodynamics of complex systems and the information of complex signals. To mark the 80th birthday of the field's founder, Constantino Tsallis, a review of open problems that can stimulate future research is provided. Over the thirty-year development of NSM, a variety of criticisms have been published ranging from questions about the justification for generalizing the entropy function to the interpretation of the generalizing parameter q. While these criticisms have been addressed in the past and the breadth of applications has demonstrated the utility of the NSM methodologies, this review provides insights into how the field can continue to improve the understanding and application of complex system models. The review starts by grounding q-statistics within scale-shape distributions and then frames a series of open problems for investigation. The open problems include using the degrees of freedom to quantify the difference between entropy and its generalization, clarifying the physical interpretation of the parameter q, improving the definition of the generalized product using multidimensional analysis, defining a generalized Fourier transform applicable to signal processing applications, and re-examining the normalization of nonextensive entropy. This review concludes with a proposal that the shape parameter is a candidate for defining the statistical complexity of a system.

A SPIN-1 ISING MODEL INVESTIGATION OF THE MAGNETIC SYSTEM IS CARRIED OUT WITHIN THE CONTEXT OF GENERALIZED STATISTICAL MECHANICS

In this study magnetization has been investigated with the help of Ising model in the frame of non-extensive statistical mechanics where a behavior of interacting elementary moments ensemble is taken into consideration. To examine the physical systems with three states and two order parameters, researchers employ the spin-1 single lattice Ising model or three-state systems. Within this model, various thermodynamic characteristics of phenomena like ferromagnetism in binary alloys, liquid mixtures, liquid-crystal mixtures, freezing, magnetic order, phase transformations, semi-stable and unstable states, ordered and disordered transitions have been investigated for three distinct forms associated with q &lt; 1, q = 1, and q &gt; 1. In this context, q represents the non-extensivity parameter of Tsallis statistics.

First-Principle Validation of Fourier's Law: One-Dimensional Classical Inertial Heisenberg Model.

The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size L is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures, Th and Tl (Th>Tl), respectively. These particles at the extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators (i=2,⋯,L-1) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier's law for the heat flux is verified numerically, with the thermal conductivity becoming independent of the lattice size in the limit L→∞, scaling with the temperature, as κ(T)∼T-2.25, where T=(Th+Tl)/2. Moreover, the thermal conductance, σ(L,T)≡κ(T)/L, is well-fitted by a function, which is typical of nonextensive statistical mechanics, according to σ(L,T)=Aexpq(-Bxη), where A and B are constants, x=L0.475T, q=2.28±0.04, and η=2.88±0.04.

Statistical mechanical characterization of billiard systems

Area-preserving maps play an important role in diverse fields as they are widely used for modeling complex systems. In addition, these maps provide rich observations by presenting stable orbits and chaotic behavior separately or together in the phase space depending on the control parameter. In recent years, several studies on these maps, drawing inspiration from the phase space dynamics, have shown that nonextensive statistical mechanics provides appropriate instruments to characterize these systems. In this study, we perform a rigorous numerical analysis to delve into the statistical mechanical properties of a billiard system. Our primary goal is to confirm the presence of a q-Gaussian distribution, with an estimated q value of approximately 1.935. We accomplish this by examining the probability distribution of the cumulative sum of system iterates, focusing specifically on initial conditions within the stability islands. Our findings align seamlessly with the latest research in this field. Furthermore, we show that a multi-component probability distribution containing both Gaussian and q-Gaussians describes the entire system for some parameter regions where the phase space consists of stability islands together with the chaotic sea.

When may a system be referred to as complex?—an entropic perspective

Defining complexity is hard and far from unique—like defining beauty, intelligence, creativity, and many other such abstract concepts. In contrast, describing concrete complex systems is a sensibly simpler task. We focus here on such an issue from the perspective of entropic functionals, either additive or nonadditive. Indeed, for the systems currently referred to as simple, the statistical mechanics and associated (additive) entropy is that of Boltzmann–Gibbs, formulated 150 years ago. This formalism constitutes a pillar of contemporary theoretical physics and is typically grounded on strong chaos, mixing, ergodicity, and similar hypotheses, which typically emerge for systems with short-range space–time generic correlations. It fails, however, for the so-called complex systems, where generic long-range space–time correlations prevail, typically grounded on weak chaos. Many such nontrivial systems are satisfactorily handled within a generalization of the Boltzmann–Gibbs theory, namely, nonextensive statistical mechanics, introduced in 1988 and grounded on nonadditive entropies. Illustrations are presented in terms of D-dimensional simplexes such as nodes (D = 0), bonds (D = 1), plaquettes (D = 2), polyhedra (D = 3, …), and higher-order ones. A regularly updated bibliography is available at http://tsallis.cat.cbpf.br/biblio.htm.

Extensive and nonextensive statistical mechanics: Exp and log distribution functions

The dominance of Boltzmann–Gibbs distribution (BG) in statistical physics appears to endow an impression that this would be the only statistical approach available. In fact, a large class of statistical approaches already exists. This is generalizing BG statistics referring to various types of nonextensivity, nonadditivity, nonequilibrium, nonlinearity, etc. For instance, [Formula: see text] and [Formula: see text] functions play crucial roles in both extensive and nonextensive domains of statistical mechanics. Emerging in a physical system, BG statistics defines extensive entropy which relates the number of microstates to thermodynamic quantities or macroscopic states. In this regard, the Boltzmann distribution, extensive statistics, refers to a well-defined probability distribution, while the various types of nonextensive statistics categorically violate the fourth Shannon–Khinchen additivity axiom. The [Formula: see text] and [Formula: see text] distribution functions in BG, Tsallis, and generic statistics are systematically compared. We focus on the mathematical properties of both distribution functions and conclude that their compatibility exclusively depends on the nonextensive parameters, i.e. the mathematical properties of both distribution functions depend on the nonextensive parameters either that of Tsallis- or that of the generic-type of nonextensivity. We also conclude that the statistical nature of the underlying ensemble should be taken into consideration when applying the statistical approach.

Nonextensive realizations in interacting ion channels: Implications for mechano-electrical transducer mechanisms

We propose a theoretical model to investigate the thermodynamics of single and coupled two-state ion channels, associated with mechanoelectrical transduction (MET) and hair cell biophysics. The modeling was based on the Tsallis nonextensive statistical mechanics. The choice for a nonextensive framework in modeling ion channels is encouraged on the fact that we take into account the presence of interactions or long-range correlations in the dynamics of single and coupled ion channels. However, the basic assumptions that support Boltzmann–Gibbs statistics, traditionally used to model ion channel dynamics, state that the system is formed by independent or weakly interacting elements. Despite being well studied in many biological systems, the literature has not yet addressed the study of both entropy and mutual information related to isolated or physically interacting pairs of MET channels. Inspired by hair cell biophysics, we show how the presence of nonextensivity, or subadditivity and superadditivity modulates the nonextensive entropy and mutual information as functions of stereocilia displacements. We also observe that the magnitude of the interaction between the two channels, given by a nonextensive parameter, influences the amplitude of the nonextensive joint entropy and mutual information as functions of the hair cell displacements. Finally, we show how nonextensivity regulates the current versus displacement curve for a single and a pair of interacting two-state channels. The present findings shed light on the thermodynamic process involved in the molecular mechanisms of the auditory system.

Loewner time conversion for q-generalized stochastic dynamics

Generalized statistical mechanics based on q-Gaussian has been demonstrated to be an effective theoretical framework for the analysis of non-equilibrium systems. Since q-generalized (non-extensive) statistical mechanics reduces the nonlinearity in the system into deformed entropy and probability distributions, we introduce an alternative method based on the time conversion method using the Loewner equation by investigating the statistical physical properties of one-dimensional stochastic dynamics described by the Langevin equation with multiplicative noise. We demonstrate that a randomized time transformation using Loewner time enables the conversion of the multiplicative Langevin dynamics into an equilibrium system obeying a conventional microcanonical ensemble. For the equilibrium Langevin system after the Loewner time conversion, the fluctuation-dissipation relation and path integral fluctuation theorem were discussed to derive the response function under a nonlinear perturbation and an extended Jarzynski equality. The present results suggest the efficacy of the introducing randomized time for analyzing non-equilibrium systems, and indicate a novel connection between q-generalized (non-extensive) and Boltzmann–Gibbs statistical mechanics.

Chromatin structure and context-dependent sequence features control prime editing efficiency.

Prime editing (PE) is a highly versatile CRISPR-Cas9 genome editing technique. The current constructs, however, have variable efficiency and may require laborious experimental optimization. This study presents statistical models for learning the salient epigenomic and sequence features of target sites modulating the editing efficiency and provides guidelines for designing optimal PEs. We found that both regional constitutive heterochromatin and local nucleosome occlusion of target sites impede editing, while position-specific G/C nucleotides in the primer-binding site (PBS) and reverse transcription (RT) template regions of PE guide RNA (pegRNA) yield high editing efficiency, especially for short PBS designs. The presence of G/C nucleotides was most critical immediately 5' to the protospacer adjacent motif (PAM) site for all designs. The effects of different last templated nucleotides were quantified and observed to depend on the length of both PBS and RT templates. Our models found AGG to be the preferred PAM and detected a guanine nucleotide four bases downstream of the PAM to facilitate editing, suggesting a hitherto-unrecognized interaction with Cas9. A neural network interpretation method based on nonextensive statistical mechanics further revealed multi-nucleotide preferences, indicating dependency among several bases across pegRNA. Our work clarifies previous conflicting observations and uncovers context-dependent features important for optimizing PE designs.

Natural time analysis together with non-extensive statistical mechanics shorten the time window of the impending 2011 Tohoku [formula omitted]9 earthquake in Japan

Applying natural time analysis (NTA) to the Japanese seismic data, we have found that the system enters the critical stage upon the occurrence of M=4.2-5.0 earthquakes from 08:36 to 13:14 local time (LT) on 10 March 2011, i.e., almost one day before the M9 Tohoku earthquake on 11 March 2011. In addition, here, we find that just after this period the entropy change ΔS under time reversal of NTA along with the Tsallis entropy of non-extensive statistical mechanics (NESM) show distinct simultaneous changes. This simultaneous appearance enables the shortening of the time window of the impending mainshock to several hours. Furthermore, upon the occurrence of the M7.3 foreshock at 11:45 LT on 9 March 2011, the following fact emerged: The Tsallis entropy of NESM exhibited a scaling behavior with a characteristic exponent 1/3 that conforms to Lifshitz–Slyozov–Wagner theory for phase transitions as had been also observed in NTA for the fluctuations of the entropy change ΔS under time reversal upon a M7.8 earthquake occurrence in Japan on 22 December 2010. The latter obeyed the form A(t−t0)c where c is approximately equal to 1/3 and the pre-factors A are proportional to the scale i (number of events) while t0 is almost 0.2 days after this M7.8 earthquake.

Characteristic Function of the Tsallis q-Gaussian and Its Applications in Measurement and Metrology

The Tsallis q-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the q-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, q-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent q-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent q-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.

Senses along Which the Entropy Sq Is Unique.

The Boltzmann-Gibbs-von Neumann-Shannon additive entropy SBG=-k∑ipilnpi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the nonextensive statistical mechanics-as currently referred to-grounded on the nonadditive entropy Sq=k1-∑ipiqq-1 as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. Sq plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity-plectics, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely In what senses is entropy Sq unique? The present effort is dedicated to a-surely non exhaustive-mathematical answer to this basic question.

Medical Applications of Nonadditive Entropies.

The Boltzmann-Gibbs additive entropy SBG=-k∑ipilnpi and associated statistical mechanics were generalized in 1988 into nonadditive entropy Sq=k1-∑ipiqq-1 and nonextensive statistical mechanics, respectively. Since then, a plethora of medical applications have emerged. In the present review, we illustrate them by briefly presenting image and signal processings, tissue radiation responses, and modeling of disease kinetics, such as for the COVID-19 pandemic.

Non-Extensive Statistical Mechanics in Acoustic Emissions: Detection of Upcoming Fracture in Rock Materials

Acoustic emission (AE), recorded during uniaxial compressive loading with constantly increasing stress and stepped stress increments until the fracture of prismatic marble specimens, were analyzed in terms of non-extensive statistical mechanics (NESM). Initially introduced by Tsallis, NESM has proven to be an autonomous robust theoretical framework for studying fracture mechanisms and damage evolution processes during fracture experiments in specimens made of brittle materials. In the current work, the time intervals of the recorded AE data are analyzed in terms of NESM. For each examined specimen, the corresponding q entropic indices and the βq parameters were calculated, and their variability in terms of the stress applied were studied. Furthermore, a possible linear relationship between the entropic index q and the parameter βq was examined, and it was investigated whether the observed deviation from monotonicity between q and βq may signal increased accumulation of damage, eventually leading to the final fracture of the specimens. Through this work, the emergence of an additional pre-failure indicator (i.e., the deviation from monotonicity between q and βq) alongside well-established ones can provide further insight regarding the underlying crack development mechanisms and damage accumulation processes during the fracture of rock materials.

Nonextensive effects on QCD chiral phase diagram and baryon-number fluctuations within Polyakov-Nambu-Jona-Lasinio model* *Supported by the National Natural Science Foundation of China (12005192) and the Project funded by China Postdoctoral Science Foundation (2020M672255, 2020TQ0287)

In this paper, a version of the Polyakov-Nambu-Jona-Lasinio (PNJL) model based on nonextensive statistical mechanics is presented. This new statistics summarizes all possible factors that violate the assumptions of the Boltzmann-Gibbs (BG) statistics to a dimensionless nonextensivity parameter q. Thus, when q tends to 1, it returns to the BG case. Within the nonextensive PNJL model, we found that as q increases, the location of the critical end point (CEP) exhibits non-monotonic behavior. That is, for , CEP moves in the direction of lower temperature and larger quark chemical potential. However, for , CEP turns to move in the direction of lower temperature and lower quark chemical potential. In addition, we studied the moments of the net-baryon number distribution, that is, variance (), skewness (S), and kurtosis (κ). Our results are generally consistent with the latest experimental data reported, especially for , when q is set to .