Showing 1–73 of 73 results for all: kaniadakis statistics

The $κ$-generalised Distribution for Stock Returns

Authors: Samuel Forbes

Abstract: …-generalised distribution, originated in the context of statistical physics by Kaniadakis, is characterised by the $κ$-exponential function that is asymptotically exponential for small values and asymptotically power law for large values. This proves to be a useful property and makes it a good candidate distribution fo… ▽ More Empirical evidence shows stock returns are often heavy-tailed rather than normally distributed. The $κ$-generalised distribution, originated in the context of statistical physics by Kaniadakis, is characterised by the $κ$-exponential function that is asymptotically exponential for small values and asymptotically power law for large values. This proves to be a useful property and makes it a good candidate distribution for many types of quantities. In this paper we focus on fitting historic daily stock returns for the FTSE 100 and the top 100 Nasdaq stocks. Using a Monte-Carlo goodness of fit test there is evidence that the $κ$-generalised distribution is a good fit for a significant proportion of the 200 stock returns analysed. △ Less

Submitted 16 May, 2024; originally announced May 2024.

Relativistic Correction to Black Hole Entropy

Authors: Naman Kumar

Abstract: …ensemble and apply it to the cases of non-rotating BTZ and AdS-Schwarzschild black holes. This is realized by generalizing the equations obtained using Boltzmann-Gibbs(BG) statistics with its relativistic generalization, Kaniadakis statistics, or $κ$-… ▽ More In this paper, we study the relativistic correction to Bekenstein-Hawking entropy in the canonical ensemble and isothermal-isobaric ensemble and apply it to the cases of non-rotating BTZ and AdS-Schwarzschild black holes. This is realized by generalizing the equations obtained using Boltzmann-Gibbs(BG) statistics with its relativistic generalization, Kaniadakis statistics, or $κ$-statistics. The relativistic corrections are found to be logarithmic in nature and it is observed that their effect becomes appreciable in the high-temperature limit suggesting that the entropy corrections must include these relativistically corrected terms while taking the aforementioned limit. The non-relativistic corrections are recovered in the $κ\rightarrow 0$ limit. △ Less

Submitted 25 April, 2024; originally announced April 2024.

Journal ref: Gen Relativ Gravit 56, 47 (2024)

Cosmological FLRW phase transitions and micro-structure under Kaniadakis statistics

Authors: Joaquin Housset, Joel F. Saavedra, Francisco Tello-Ortiz

Abstract: This article is devoted to the study of the thermodynamics phase transitions and critical phenomena of an FLRW cosmological model under the so-called Kaniadakis's statistics. The equation of state is derived from the corrected Friedmann field equations and the thermodynamics unified first law. This reveals the exis… ▽ More This article is devoted to the study of the thermodynamics phase transitions and critical phenomena of an FLRW cosmological model under the so-called Kaniadakis's statistics. The equation of state is derived from the corrected Friedmann field equations and the thermodynamics unified first law. This reveals the existence of non-trivial critical points where a first-order phase transition takes place. The system behaves as an "inverted" van der Waals fluid in this concern. Interestingly, the numerical values of the critical exponents are the same as those of the van der Waals system. Besides, to obtain more insights into the thermodynamics description, the so-called Ruppeiner's geometry is studied through the normalized scalar curvature, disclosing this invariant zone where the system undergoes repulsive/attractive interactions. Near the critical point, this curvature provides again the same critical exponent and universal constant value as for van der Waals fluid. △ Less

Submitted 8 May, 2024; v1 submitted 9 December, 2023; originally announced December 2023.

Comments: 11 pages, 4 figures. Accepted version at Physics Letter B

Microscopic interpretation of generalized entropy

Authors: Shin'ichi Nojiri, Sergei D. Odintsov, Tanmoy Paul

Abstract: Generalized entropy, that has been recently proposed, puts all the known and apparently different entropies like The Tsallis, the Rényi, the Barrow, the Kaniadakis, the Sharma-Mittal and the loop quantum gravity entropy within a single umbrella. However, the microscopic origin of such generalized entropy as well as its relation to thermodynamic system(s) is… ▽ More Generalized entropy, that has been recently proposed, puts all the known and apparently different entropies like The Tsallis, the Rényi, the Barrow, the Kaniadakis, the Sharma-Mittal and the loop quantum gravity entropy within a single umbrella. However, the microscopic origin of such generalized entropy as well as its relation to thermodynamic system(s) is not clear. In the present work, we will provide a microscopic thermodynamic explanation of generalized entropy(ies) from canonical and grand-canonical ensembles. It turns out that in both the canonical and grand-canonical descriptions, the generalized entropies can be interpreted as the statistical ensemble average of a series of microscopic quantity(ies) given by various powers of $\left(-k\lnρ\right)^n$ (with $n$ being a positive integer and $ρ$ symbolizes the phase space density of the respective ensemble), along with a term representing the fluctuation of Hamiltonian and number of particles of the system under consideration (in case of canonical ensemble, the fluctuation on the particle number vanishes). △ Less

Submitted 7 November, 2023; originally announced November 2023.

Comments: PLB Accepted

criticalities, phase transitions and geometrothermodynamics of charged AdS black holes from Kaniadakis statistics

Authors: Giuseppe Gaetano Luciano, Emmanuel Saridakis

Abstract: …criticalities and van der Waals-like phase transitions. In this work we extend the study of these critical phenomena to Kaniadakis theory, which is a non-extensive generalization of the classical… ▽ More Boltzmann entropy-based thermodynamics of charged anti-de Sitter (AdS) black holes has been shown to exhibit physically interesting features, such as $P-V$ criticalities and van der Waals-like phase transitions. In this work we extend the study of these critical phenomena to Kaniadakis theory, which is a non-extensive generalization of the classical statistical mechanics incorporating relativity. By applying the typical framework of condensed-matter physics, we analyze the impact of Kaniadakis entropy onto the equation of state, the Gibbs free energy and the critical exponents of AdS black holes in the extended phase space. Additionally, we investigate the underlying micro-structure of black holes in Ruppeiner geometry, which reveals appreciable deviations of the nature of the particle interactions from the standard behavior. Our analysis opens up new perspectives on the understanding of black hole thermodynamics in a relativistic statistical framework, highlighting the role of non-extensive corrections in the AdS black holes/van der Waals fluids dual picture. △ Less

Submitted 24 August, 2023; originally announced August 2023.

Comments: 15 pages, 11 labelled figures, comments are welcome

Is Kaniadakis $κ$-generalized statistical mechanics general?

Authors: T. F. A. Alves, J. F. da Silva Neto, F. W. S. Lima, G. A. Alves, P. R. S. Carvalho

Abstract: …-generalized statistics, namely $κ$-generalized statistical field theory. In particular, we show, by computations through analytic and simulation results, that the $κ$-generalized Ising-like systems are not capable of describing the nonconventional critical properties of real imperfect crystals, \emph{e. g.} of mangani… ▽ More In this Letter we introduce some field-theoretic approach for computing the critical properties of systems undergoing continuous phase transitions governed by the $κ$-generalized statistics, namely $κ$-generalized statistical field theory. In particular, we show, by computations through analytic and simulation results, that the $κ$-generalized Ising-like systems are not capable of describing the nonconventional critical properties of real imperfect crystals, \emph{e. g.} of manganites, as some alternative generalized theory is, namely nonextensive statistical field theory, as shown recently in literature. Although $κ$-Ising-like systems do not depend on $κ$, we show that a few distinct systems do. Thus the $κ$-generalized statistical field theory is not general, \emph{i. e.} it fails to generalize Ising-like systems for describing the critical behavior of imperfect crystals, and must be discarded as one generalizing statistical mechanics. For the latter systems we present the physical interpretation of the theory by furnishing the general physical interpretation of the deformation $κ$-parameter. △ Less

Submitted 11 July, 2023; originally announced July 2023.

Comments: 19 pages, 2 figures

Journal ref: Phys. Lett. B 843, 138005 (2023)

Slow-roll inflation and growth of perturbations in Kaniadakis modification of Friedmann cosmology

Authors: Gaetano Lambiase, Giuseppe Gaetano Luciano, Ahmad Sheykhi

Abstract: Kaniadakis entropy is a one-parameter deformation of the classical Boltzmann-Gibbs-Shannon entropy, arising from a self-consistent relativistic… ▽ More Kaniadakis entropy is a one-parameter deformation of the classical Boltzmann-Gibbs-Shannon entropy, arising from a self-consistent relativistic statistical theory. Assuming a Kaniadakis-type generalization of the entropy associated with the apparent horizon of Friedmann-Robertson-Walker (FRW) Universe and using the gravity-thermodynamics conjecture, a new cosmological scenario is obtained based on the modified Friedmann equations. By employing such modified equations, we analyze the slow-roll inflation, driven by a scalar field with power-law potential, at the early stages of the Universe. We explore the phenomenological consistency of this model by computation of the scalar spectral index and tensor-to-scalar ratio. Comparison with the latest Planck data allows us to constrain Kaniadakis parameter to $κ\lesssim\mathcal{O}(10^{-12}\div10^{-11})$, which is discussed in relation to other observational bounds in the past literature. We also disclose the effects of Kaniadakis correction term on the growth of perturbations at the early stages of the Universe by employing the spherically symmetric collapse formalism in the linear regime of density perturbations. We find out that the profile of density contrast is non-trivially affected in this scenario. Interestingly enough, we observe that increasing Kaniadakis parameter $κ$ corresponds to a faster growth of perturbations in a Universe governed by the corrected Friedmann equations. Finally, we comment on the consistency of the primordial power spectrum for scalar perturbations with the best data-fit provided by Planck. △ Less

Submitted 8 November, 2023; v1 submitted 8 July, 2023; originally announced July 2023.

Comments: Discussion added on the primordial power spectrum for scalar perturbations

Journal ref: Eur.Phys.J.C 83 (2023) 10, 936

On the Kaniadakis distributions applied in statistical physics and natural sciences

Authors: Tatsuaki Wada, Antonio M. Scarfone

Abstract: …-deformed functions, some constitutive relations are generalized. We here show some applications of the Kaniadakis distributions based on the inverse hyperbolic sine function to some topics belonging to the realm of statistical physics and natural science. ▽ More Constitutive relations are fundamental and essential to characterize physical systems. By utilizing the $κ$-deformed functions, some constitutive relations are generalized. We here show some applications of the Kaniadakis distributions based on the inverse hyperbolic sine function to some topics belonging to the realm of statistical physics and natural science. △ Less

Submitted 24 January, 2023; v1 submitted 2 January, 2023; originally announced January 2023.

Comments: 14 pages, 2 figures, submitted to Entropy

Journal ref: Entropy 2023, 25(2), 292

Studies of transport coefficients in charged AdS$_{4}$ black holes on $κ$-deformed space

Authors: Fabiano F. Santos, Bruno G. da Costa, Ignacio S. Gomez

Abstract: …black hole for an Einstein-Maxwell model where the derivative quadrivector is replaced by a deformed version inspired in Kaniadakis statistics. Besides, we extract the transport coefficient know as electrical conductivity. ▽ More In this work, we study the effect of $κ$-deformed space on the thermodynamic quantities, this are find through the holographic renormalization that provide the free energy, which is fundamental to derive the another thermodynamic quantities. For this scenario we consider an charged AdS$_{4}$ black hole for an Einstein-Maxwell model where the derivative quadrivector is replaced by a deformed version inspired in Kaniadakis statistics. Besides, we extract the transport coefficient know as electrical conductivity. △ Less

Submitted 24 November, 2022; originally announced November 2022.

Comments: 5 pages, 8 figures

Implications of non-extensivity on Gamow theory

Authors: H. Moradpour, M. Javaherian, E. Namvar, A. H. Ziaie

Abstract: Relying on the quantum tunneling concept and the Maxwell-Boltzmann-Gibbs statistics, Gamow shows that the star burning process happens at temperatures comparable to a critical value, called the Gamow temperature ($\texttt{T}$) and less than the prediction of the classical framework. In order to highlight the role of the equipartition theorem in the Gamow arg… ▽ More Relying on the quantum tunneling concept and the Maxwell-Boltzmann-Gibbs statistics, Gamow shows that the star burning process happens at temperatures comparable to a critical value, called the Gamow temperature ($\texttt{T}$) and less than the prediction of the classical framework. In order to highlight the role of the equipartition theorem in the Gamow argument, a thermal length scale is defined and thereinafter, the effects of non-extensivity on the Gamow temperature have been investigated focusing on the Tsallis and Kaniadakis statistics. The results attest that while the Gamow temperature decreases in the framework of the Kaniadakis statistics, it can be bigger or smaller than $\texttt{T}$ when the Tsallis statistics is employed. △ Less

Submitted 3 May, 2022; originally announced May 2022.

Comments: 3 pages

New power-law tailed distributions emerging in $κ$-statistics

Authors: G. Kaniadakis

Abstract: …which imposes the generalization of Newton's classical mechanics into Einstein's special relativity, implies a generalization, or deformation, of the ordinary statistical mechanics. The exponential function, which defines the Boltzmann's factor, emerges properly deformed within this formalism. Starting from this, so-called $κ$-deformed exponenti… ▽ More Over the last two decades, it has been argued that the Lorentz transformation mechanism, which imposes the generalization of Newton's classical mechanics into Einstein's special relativity, implies a generalization, or deformation, of the ordinary statistical mechanics. The exponential function, which defines the Boltzmann's factor, emerges properly deformed within this formalism. Starting from this, so-called $κ$-deformed exponential function, we introduce new classes of statistical distributions emerging as the $κ$-deformed version of already known distribution as the Generalized Gamma, Weibull, Logistic which can be adopted in the analysis of statistical data that exhibit power-law tails. △ Less

Submitted 3 March, 2022; originally announced March 2022.

Journal ref: EuroPhysics Letters 133, 10002 (2021)

Generalized statistics: applications to data inverse problems with outlier-resistance

Authors: João V. T. de Lima, Sérgio Luiz E. F. da Silva, João M. de Araújo, Gilberto Corso, Gustavo Z. dos Santos Lima

Abstract: The conventional approach to data-driven inversion framework is based on Gaussian statistics that presents serious difficulties, especially in the presence of outliers in the measurements. In this work, we present maximum likelihood estimators associated with generalized Gaussian distributions in the context of Rényi, Tsallis and… ▽ More The conventional approach to data-driven inversion framework is based on Gaussian statistics that presents serious difficulties, especially in the presence of outliers in the measurements. In this work, we present maximum likelihood estimators associated with generalized Gaussian distributions in the context of Rényi, Tsallis and Kaniadakis statistics. In this regard, we analytically analyse the outlier-resistance of each proposal through the so-called influence function. In this way, we formulate inverse problems by constructing objective functions linked to the maximum likelihood estimators. To demonstrate the robustness of the generalized methodologies, we consider an important geophysical inverse problem with high noisy data with spikes. The results reveal that the best data inversion performance occurs when the entropic index from each generalized statistic is associated with objective functions proportional to the inverse of the error amplitude. We argue that in such a limit the three approaches are resistant to outliers and are also equivalent, which suggests a lower computational cost for the inversion process due to the reduction of numerical simulations to be performed and the fast convergence of the optimization process. △ Less

Submitted 28 January, 2022; originally announced January 2022.

Comments: 17 pages, 14 figures

How fundamental is entropy? From non-extensive statistics and black hole physics to the holographic dark universe

Authors: Shin'ichi Nojiri, Sergei D. Odintsov, Valerio Faraoni

Abstract: We propose a new entropy construct that generalizes the Tsallis, Rényi, Sharma-Mittal, Barrow, Kaniadakis, and Loop Quantum Gravity entropies and reduces to the Bekenstein-Hawking entropy in a certain limit. This proposal is applied to the Schwarzschild black hole and to spatially homogeneous and isotropic cosmology, where it is shown that it can potentially… ▽ More We propose a new entropy construct that generalizes the Tsallis, Rényi, Sharma-Mittal, Barrow, Kaniadakis, and Loop Quantum Gravity entropies and reduces to the Bekenstein-Hawking entropy in a certain limit. This proposal is applied to the Schwarzschild black hole and to spatially homogeneous and isotropic cosmology, where it is shown that it can potentially describe inflation and/or holographic dark energy. △ Less

Submitted 7 January, 2022; originally announced January 2022.

Comments: 13 pages, no figures

Kaniadakis holographic dark energy in Brans-Dicke cosmology

Authors: S. Ghaffari

Abstract: By using the holographic hypothesis and Kaniadakis generalized entropy, which is based on relativistic… ▽ More By using the holographic hypothesis and Kaniadakis generalized entropy, which is based on relativistic statistical theory and modified Boltzmann-Gibbs entrory, we build Kaniadakis holographic dark energy (DE) model in the Brans-Dicke framework. We drive cosmological parameters of Kaniadakis holographic DE model, with IR cutoff as the Hubble horizon, in order to investigate its cosmological consequences. Our study shows that, even in the absence of an interaction between the dark sectors of cosmos, the Kaniadakis holographic dark energy model with the Hubble radius as IR cutoff can explain the present accelerated phase of the universe expansion in the Brans-Dicke theory. The stability of the model, using the squared of sound speed, has been checked and it is found that the model is unstable in non-interacting case and can be stable for some range of model parameters within the interacting case. △ Less

Submitted 10 December, 2021; originally announced December 2021.

Comments: 7 pages, 6 figures

Observational constraints and dynamical analysis of Kaniadakis horizon-entropy cosmology

Authors: A. Hernández-Almada, Genly Leon, Juan Magaña, Miguel A. García-Aspeitia, V. Motta, Emmanuel N. Saridakis, Kuralay Yesmakhanova, Alfredo D. Millano

Abstract: We study the scenario of Kanadiakis horizon entropy cosmology which arises from the application of the gravity-thermodynamics conjecture using the Kaniadakis modified entropy. The resulting modified Friedmann equations contain extra terms that constitute an effective dark energy sector. We use data from Cosmic chronometers, Supernova Type Ia, HII galaxies, S… ▽ More We study the scenario of Kanadiakis horizon entropy cosmology which arises from the application of the gravity-thermodynamics conjecture using the Kaniadakis modified entropy. The resulting modified Friedmann equations contain extra terms that constitute an effective dark energy sector. We use data from Cosmic chronometers, Supernova Type Ia, HII galaxies, Strong lensing systems, and Baryon acoustic oscillations observations and we apply a Bayesian Markov Chain Monte Carlo analysis to construct the likelihood contours for the model parameters. We find that the Kaniadakis parameter is constrained around 0, namely, around the value where the standard Bekenstein-Hawking is recovered. Concerning the normalized Hubble parameter, we find $h=0.708^{+0.012}_{-0.011}$, a result that is independently verified by applying the $\mathbf{\mathbb{H}}0(z)$ diagnostic and, thus, we conclude that the scenario at hand can alleviate the $H_0$ tension problem. Regarding the transition redshift, the reconstruction of the cosmographic parameters gives $z_T=0.715^{+0.042}_{-0.041}$. Furthermore, we apply the AICc, BIC and DIC information criteria and we find that in most datasets the scenario is statistical equivalent to $Λ$CDM one. Moreover, we examine the Big Bang Nucleosynthesis (BBN) and we show that the scenario satisfies the corresponding requirements. Additionally, we perform a phase-space analysis, and we show that the Universe past attractor is the matter-dominated epoch, while at late times the Universe results in the dark-energy-dominated solution. Finally, we show that Kanadiakis horizon entropy cosmology accepts heteroclinic sequences, but it cannot exhibit bounce and turnaround solutions. △ Less

Submitted 18 March, 2022; v1 submitted 8 December, 2021; originally announced December 2021.

Comments: Version accepted in MNRAS

Entropic Gravity and Cosmology in Kaniadakis Statistics

Authors: N. Sadeghnezhad

Abstract: By using the Kaniadakis… ▽ More By using the Kaniadakis statistics, we discuss the modifications of Newtonian gravity and radial velocity profile in the light of Verlinde's formalism for gravitational entropy. After considering the implications of k-statistics on the gravitational potential, it is shown that an accelerated universe may be obtained by considering the Friedmann first equation in this non-extensive statistics. △ Less

Submitted 21 September, 2022; v1 submitted 26 November, 2021; originally announced November 2021.

An outlier-resistant $κ$-generalized approach for robust physical parameter estimation

Authors: Sérgio Luiz E. F. da Silva, R. Silva, Gustavo Z. dos Santos Lima, João M. de Araújo, Gilberto Corso

Abstract: …to mitigate the undesirable effects caused by outliers to generate reliable physical models. In this way, we formulate the inverse problems theory in the context of Kaniadakis statistical mechanics (or $κ$-statistics), in which the classical approach is a particular case. In this… ▽ More In this work we propose a robust methodology to mitigate the undesirable effects caused by outliers to generate reliable physical models. In this way, we formulate the inverse problems theory in the context of Kaniadakis statistical mechanics (or $κ$-statistics), in which the classical approach is a particular case. In this regard, the errors are assumed to be distributed according to a finite-variance $κ$-generalized Gaussian distribution. Based on the probabilistic maximum-likelihood method we derive a $κ$-objective function associated with the finite-variance $κ$-Gaussian distribution. To demonstrate our proposal's outlier-resistance, we analyze the robustness properties of the $κ$-objective function with help of the so-called influence function. In this regard, we discuss the role of the entropic index ($κ$) associated with the Kaniadakis $κ$-entropy in the effectiveness in inferring physical parameters by using strongly noisy data. In this way, we consider a classical geophysical data-inverse problem in two realistic circumstances, in which the first one refers to study the sensibility of our proposal to uncertainties in the input parameters, and the second is devoted to the inversion of a seismic data set contaminated by outliers. The results reveal an optimum $κ$-value at the limit $κ\rightarrow 2/3$, which is related to the best results. △ Less

Submitted 18 November, 2021; originally announced November 2021.

Comments: 37 pages, 23 figures

Kaniadakis holographic dark energy and cosmology

Authors: Niki Drepanou, Andreas Lymperis, Emmanuel N. Saridakis, Kuralay Yesmakhanova

Abstract: We construct a holographic dark energy scenario based on Kaniadakis entropy, which is a generalization of Boltzmann-Gibbs entropy that arises from relativistic statistical theory and is characterized by a single parameter $K$ which quantifies the deviations from standard expressions, and we use the future event horizon… ▽ More We construct a holographic dark energy scenario based on Kaniadakis entropy, which is a generalization of Boltzmann-Gibbs entropy that arises from relativistic statistical theory and is characterized by a single parameter $K$ which quantifies the deviations from standard expressions, and we use the future event horizon as the Infrared cutoff. We extract the differential equation that determines the evolution of the effective dark energy density parameter, and we provide analytical expressions for the corresponding equation-of-state and deceleration parameters. We show that the universe exhibits the standard thermal history, with the sequence of matter and dark-energy eras, while the transition to acceleration takes place at $z\approx0.6$. Concerning the dark-energy equation-of-state parameter we show that it can have a rich behavior, being quintessence-like, phantom-like, or experience the phantom-divide crossing in the past or in the future. Finally, in the far future dark energy dominates completely, and the asymptotic value of its equation of state depends on the values of the two model parameters. △ Less

Submitted 16 May, 2022; v1 submitted 19 September, 2021; originally announced September 2021.

Comments: 16 pages, 3 figures

Modified cosmology through Kaniadakis horizon entropy

Authors: Andreas Lymperis, Spyros Basilakos, Emmanuel N. Saridakis

Abstract: We apply the gravity-thermodynamics conjecture, namely the first law of thermodynamics on the Universe horizon, but using the generalized Kaniadakis entropy instead of the standard Bekenstein-Hawking one. The former is a one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy, arising from a coherent and self-consistent relativistic… ▽ More We apply the gravity-thermodynamics conjecture, namely the first law of thermodynamics on the Universe horizon, but using the generalized Kaniadakis entropy instead of the standard Bekenstein-Hawking one. The former is a one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy, arising from a coherent and self-consistent relativistic statistical theory. We obtain new modified cosmological scenarios, namely modified Friedmann equations, which contain new extra terms that constitute an effective dark energy sector depending on the single model Kaniadakis parameter $K$. We investigate the cosmological evolution, by extracting analytical expressions for the dark energy density and equation-of-state parameters and we show that the Universe exhibits the usual thermal history, with a transition redshift from deceleration to acceleration at around 0.6. Furthermore, depending on the value of $K$, the dark energy equation-of-state parameter deviates from $Λ$CDM cosmology at small redshifts, while lying always in the phantom regime, and at asymptotically large times the Universe always results in a dark-energy dominated, de Sitter phase. Finally, even in the case where we do not consider an explicit cosmological constant the resulting cosmology is very interesting and in agreement with the observed behavior. △ Less

Submitted 4 December, 2021; v1 submitted 27 August, 2021; originally announced August 2021.

Comments: 10 pages, 5 figures

Statistical approaches on the apparent horizon entropy and the generalized second law of thermodynamics

Authors: Everton M. C. Abreu, Jorge Ananias Neto

Abstract: In this work we have investigated the effects of three nongaussian entropies, namely, the modified Rényi entropy (MRE), the Sharma-Mittal entropy (SME) and the dual Kaniadakis entropy (DKE) in the investigation of the generalized second law (GSL) of thermodynamics violation. The GSL is an extension of the second law for black holes. Recently, it was conclude… ▽ More In this work we have investigated the effects of three nongaussian entropies, namely, the modified Rényi entropy (MRE), the Sharma-Mittal entropy (SME) and the dual Kaniadakis entropy (DKE) in the investigation of the generalized second law (GSL) of thermodynamics violation. The GSL is an extension of the second law for black holes. Recently, it was concluded that a total entropy is the sum of the entropy enclosed by the apparent horizon plus the entropy of the horizon itself when the apparent horizon is described by the Barrow entropy. It was assumed that the universe is filled with matter and dark energy fluids. Here, the apparent horizon will be described by MRE, SME, and then by DKE proposals. Since GSL holds for usual entropy, but it is conditionally violated in the extended entropies, this implies that the parameter of these entropies should be constrained in small values in order for the GSL to be satisfied. Hence, we have established conditions where the second law of thermodynamics can or cannot be obeyed considering these three statistical concepts just as it was made in Barrow's entropy. Considering the $ΛCDM$ cosmology we can observe that for MRE, SME and DKE, the GSL of thermodynamics is not obeyed for small redshift values. △ Less

Submitted 29 November, 2021; v1 submitted 10 July, 2021; originally announced July 2021.

Comments: Enlarged version, to appear in PLB

The k-statistics approach to epidemiology

Authors: Giorgio Kaniadakis, Mauro M. Baldi, Thomas S. Deisboeck, Giulia Grisolia, Dionissios T. Hristopulos, Antonio M. Scarfone, Amelia Sparavigna, Tatsuaki Wada, Umberto Lucia

Abstract: A great variety of complex physical, natural and artificial systems are governed by statistical distributions, which often follow a standard exponential function in the bulk, while their tail obeys the Pareto power law. The recently introduced $κ$-statistics framework predicts distribution functions with this feature.… ▽ More A great variety of complex physical, natural and artificial systems are governed by statistical distributions, which often follow a standard exponential function in the bulk, while their tail obeys the Pareto power law. The recently introduced $κ$-statistics framework predicts distribution functions with this feature. A growing number of applications in different fields of investigation are beginning to prove the relevance and effectiveness of $κ$-statistics in fitting empirical data. In this paper, we use $κ$-statistics to formulate a statistical approach for epidemiological analysis. We validate the theoretical results by fitting the derived $κ$-Weibull distributions with data from the plague pandemic of 1417 in Florence as well as data from the COVID-19 pandemic in China over the entire cycle that concludes in April 16, 2020. As further validation of the proposed approach we present a more systematic analysis of COVID-19 data from countries such as Germany, Italy, Spain and United Kingdom, obtaining very good agreement between theoretical predictions and empirical observations. For these countries we also study the entire first cycle of the pandemic which extends until the end of July 2020. The fact that both the data of the Florence plague and those of the Covid-19 pandemic are successfully described by the same theoretical model, even though the two events are caused by different diseases and they are separated by more than 600 years, is evidence that the $κ$-Weibull model has universal features. △ Less

Submitted 25 November, 2020; originally announced December 2020.

Comments: 15 pages, 1 table, 5 figures

Journal ref: Scientific Report (2020) 10:19949

Generalized entropies and corresponding holographic dark energy models

Authors: H. Moradpour, A. H. Ziaie, M. Kord Zangeneh

Abstract: Using Tsallis statistics and its relation with Boltzmann entropy, the Tsallis entropy content of black holes is achieved, a result in full agreement with a recent study (Phys. Lett. B 794, 24 (2019)). In addition, employing… ▽ More Using Tsallis statistics and its relation with Boltzmann entropy, the Tsallis entropy content of black holes is achieved, a result in full agreement with a recent study (Phys. Lett. B 794, 24 (2019)). In addition, employing Kaniadakis statistics and its relation with that of Tsallis, the Kaniadakis entropy of black holes is obtained. The Sharma-Mittal and Rényi entropy contents of black holes are also addressed by employing their relations with Tsallis entropy. Thereinafter, relying on the holographic dark energy hypothesis and the obtained entropies, two new holographic dark energy models are introduced and their implications on the dynamics of a flat FRW universe are studied when there is also a pressureless fluid in background. In our setup, the apparent horizon is considered as the IR cutoff, and there is not any mutual interaction between the cosmic fluids. The results indicate that the obtained cosmological models have $i$) notable powers to describe the cosmic evolution from the matter-dominated era to the current accelerating universe, and $ii$) suitable predictions for the universe age. △ Less

Submitted 12 May, 2020; originally announced May 2020.

Report number: 80,732

Journal ref: EPJC (2020)

Analysis on hadron spectra in heavy-ion collisions with a new non-extensive approach

Authors: Ke-Ming Shen

Abstract: …momentum spectra of identified charged hadrons stemming from high energy collisions at different beam energies are described by a new non-extensive distribution, the Kaniadakis $κ$-distribution, with respect to the constraints in non-extensive quantum statistics. All fittings are also compared with the Tsallis distribu… ▽ More The transverse momentum spectra of identified charged hadrons stemming from high energy collisions at different beam energies are described by a new non-extensive distribution, the Kaniadakis $κ$-distribution, with respect to the constraints in non-extensive quantum statistics. All fittings are also compared with the Tsallis distributions as well as the usual Boltzmann-Gibbs one. $χ^2/ndf$ is also used to test the fitting goodness of all functions. Our results show that these different non-extensive approaches can be well applied in high energy collisions rather than the classical one. The Kaniadakis statistics is typically better applied into such systems with both positive and negative particles considered. This provides an alternative non-extensive view to study high energy physics. Analysis on the fitting parameters are present as well. The similar relationships of all functions remind us of the further understanding of the non-extensivity. △ Less

Submitted 16 July, 2019; v1 submitted 2 July, 2019; originally announced July 2019.

Comments: 18 pages, 14 figures

Non-Gaussian thermostatistical considerations upon the Saha equation

Authors: Bráulio B. Soares, Edésio M. Barboza Jr., Everton M. C. Abreu, Jorge Ananias Neto

Abstract: …of the atoms in a gas ensemble. Saha equation can also consider the partitions functions for both states and its main application is in stellar astrophysics population statistics. This paper presents two non-Gaussian thermostatistical generalizations for the Saha equation: the first one towards the Tsallis nonextensive $q$-entropy and the other one is based… ▽ More The Saha equation provides the relation between two consecutive ionization state populations, like the Maxwell-Boltzmann velocity distribution of the atoms in a gas ensemble. Saha equation can also consider the partitions functions for both states and its main application is in stellar astrophysics population statistics. This paper presents two non-Gaussian thermostatistical generalizations for the Saha equation: the first one towards the Tsallis nonextensive $q$-entropy and the other one is based upon Kaniadakis $κ$-statistics. Both thermostatistical formalisms are very successful when used in several complex astrophysical statistical systems and we have demonstrated here that they work also in Saha's ionization distribution. We have obtained new chemical $q$-potentials and their respective graphical regions with a well defined boundary that separated the two symmetric intervals for the $q$-potentials. The asymptotic behavior of the $q$-potential was also discussed. Besides the proton-electron, we have also investigated the complex atoms and pair production ionization reactions. △ Less

Submitted 27 December, 2018; originally announced January 2019.

Comments: 15 pages. Preprint format

Loop Quantum Gravity Immirzi parameter and the Kaniadakis statistics

Authors: Everton M. C. Abreu, Jorge Ananias Neto, Albert C. R. Mendes, Rodrigo M. de Paula

Abstract: …surface can emerge depending on the thermostatistics theory previously chosen. Starting from the Boltzmann-Gibbs entropy, the Immirzi parameter can be reobtained. Using the Kaniadakis… ▽ More In this letter we have shown that a possible connection between the LQG Immirzi parameter and the area of a punctured surface can emerge depending on the thermostatistics theory previously chosen. Starting from the Boltzmann-Gibbs entropy, the Immirzi parameter can be reobtained. Using the Kaniadakis statistics, which is an important non-Gaussian statistics, we have derived a new relation between the Immirzi parameter, the kappa parameter and the area of a punctured surface. After that, we have compared our result with the Immirzi parameter previously obtained in the literature within the context of Tsallis' statistics. We have demonstrated in an exact way that the LQG Immirzi parameter can also be used to compare both Kaniadakis and Tsallis statics. △ Less

Submitted 31 July, 2018; originally announced August 2018.

Comments: 7 pages. Preprint format

Nonlinear Kinetics on Lattices based on the Kinetic Interaction Principle

Authors: Giorgio Kaniadakis, Dionissios T. Hristopulos

Abstract: …dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-P… ▽ More Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker-Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker-Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker-Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in [G. Kaniadakis, Physica A, 296, 405 (2001)], univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker-Planck equation in its most general form. △ Less

Submitted 2 June, 2018; originally announced June 2018.

Comments: 26 pages

MSC Class: 82

Journal ref: Entropy, 20(6):426, 2018

A new $κ$-deformed parametric model for the size distribution of wealth

Authors: Adams Vallejos, Ignacio Ormazabal, Felix A. Borotto, Hernan F. Astudillo

Abstract: …with a distributed saving parameter can be resolved as a mixture of Gamma distributions corresponding to particular subsets of agents. Here, we propose a new four-parameter statistical distribution which is a $κ$-deformation of the Generalized Gamma distribution with a power-law tail, based on the deformed exponential and logarithm functions introduced by… ▽ More It has been pointed out by Patriarca et al. (2005) that the power-law tailed equilibrium distribution in heterogeneous kinetic exchange models with a distributed saving parameter can be resolved as a mixture of Gamma distributions corresponding to particular subsets of agents. Here, we propose a new four-parameter statistical distribution which is a $κ$-deformation of the Generalized Gamma distribution with a power-law tail, based on the deformed exponential and logarithm functions introduced by Kaniadakis(2001). We found that this new distribution is also an extension to the $κ$-Generalized distribution proposed by Clementi et al. (2007), with an additional shape parameter $ν$, and properly reproduces the whole range of the distribution of wealth in such heterogeneous kinetic exchange models. We also provide various associated statistical measures and inequality measures. △ Less

Submitted 17 May, 2018; originally announced May 2018.

Tsallis and Kaniadakis statistics from a point of view of the holographic equipartition law

Authors: Everton M. C. Abreu, Jorge Ananias Neto, Albert C. R. Mendes, Alexander Bonilla

Abstract: In this work, we have illustrated the difference between both Tsallis and Kaniadakis entropies through cosmological models obtained from the formalism proposed by Padmanabhan, which is called holographic equipartition law. Similarly to the formalism proposed by Komatsu, we have obtained an extra driving constant term in the Friedmann equation if we deform th… ▽ More In this work, we have illustrated the difference between both Tsallis and Kaniadakis entropies through cosmological models obtained from the formalism proposed by Padmanabhan, which is called holographic equipartition law. Similarly to the formalism proposed by Komatsu, we have obtained an extra driving constant term in the Friedmann equation if we deform the Tsallis entropy by Kaniadakis' formalism. We have considered initially Tsallis entropy as the Black Hole (BH) area entropy. This constant term may lead the universe to be in an accelerated mode. On the other hand, if we start with the Kaniadakis entropy as the BH area entropy and then by modifying the Kappa expression by Tsallis' formalism, the same constant, which shows that the universe have an acceleration is obtained. In an opposite limit, no driving inflation term of the early universe was derived from both deformations. △ Less

Submitted 16 April, 2018; v1 submitted 17 November, 2017; originally announced November 2017.

Comments: 8 pages, preprint format. Final version to appear in Europhysics Letters

Tsallis and Kaniadakis statistics from the viewpoint of entropic gravity formalism

Authors: Everton M. C. Abreu, Jorge Ananias Neto, Edesio M. Barboza Jr., Rafael C. Nunes

Abstract: It has been shown in the literature that effective gravitational constants, which are derived from Verlinde's formalism, can be used to introduce the Tsallis and Kaniadakis… ▽ More It has been shown in the literature that effective gravitational constants, which are derived from Verlinde's formalism, can be used to introduce the Tsallis and Kaniadakis statistics. This method provides a simple alternative to the usual procedure normally used in these non-Gaussian statistics. We have applied our formalism in the Jeans mass criterion of stability and in the free fall time collapsing of a self-gravitating system where new results are obtained. A possible connection between our formalism and deviations of Newton's law of gravitation in a submillimeter range is made. △ Less

Submitted 18 January, 2017; originally announced January 2017.

Comments: To appear in IJMPA

Different Non-extensive Models for heavy-ion collisions

Authors: Keming Shen, Tamas S. Biro, Enke Wang

Abstract: …) spectra from heavy-ion collisions at intermediate momenta are described by non-extensive statistical models. Assuming a fixed relative variance of the temperature fluctuating event by event or alternatively a fixed mean multiplicity in a negative binomial distribution (NBD), two different linear relations emerge between the temperature, $T$, and the Tsalli… ▽ More The transverse momentum ($p_T$) spectra from heavy-ion collisions at intermediate momenta are described by non-extensive statistical models. Assuming a fixed relative variance of the temperature fluctuating event by event or alternatively a fixed mean multiplicity in a negative binomial distribution (NBD), two different linear relations emerge between the temperature, $T$, and the Tsallis parameter $q-1$. Our results qualitatively agree with that of G.~Wilk. Furthermore we revisit the "Soft+Hard" model, proposed recently by G.~G.~Barnaföldi \textit{et.al.}, by a $T$-independent average $p_T^2$ assumption. Finally we compare results with those predicted by another deformed distribution, using Kaniadakis' $κ$ parametrization. △ Less

Submitted 11 January, 2017; originally announced January 2017.

Comments: 12 pages, 3 figures

Jeans instability criterion from the viewpoint of non-gaussian statistics

Authors: Everton M. C. Abreu, Jorge Ananias Neto, Edesio M. Barboza Jr., Rafael C. Nunes

Abstract: In this Letter we have derived the Jeans length in the context of the Kaniadakis statistics. We have compared this result with the Jeans length obtained in the non-extensive Tsallis statistics and discussed the main differences between these two models. We have also obtained the… ▽ More In this Letter we have derived the Jeans length in the context of the Kaniadakis statistics. We have compared this result with the Jeans length obtained in the non-extensive Tsallis statistics and discussed the main differences between these two models. We have also obtained the kappa-sound velocity. Finally, we have applied the results obtained here to analyze an astrophysical system. △ Less

Submitted 29 February, 2016; originally announced March 2016.

Comments: 6 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1503.05874

Variational Approach and Deformed Derivatives

Authors: José Weberszpil, José Abdalla Helayël-Neto

Abstract: Recently, we have demonstrated that there exists a possible relationship between q-deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis' framework and the Kaniadakis' scenario, with a local form of fractional-derivative operators for fractal media, the so-called Hausdorff de… ▽ More Recently, we have demonstrated that there exists a possible relationship between q-deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis' framework and the Kaniadakis' scenario, with a local form of fractional-derivative operators for fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. Here, in this paper, we present an extension of the traditional calculus of variations for systems containing deformed-derivatives embedded into the Lagrangian and the Lagrangian densities for classical and field systems. The results extend the classical Euler-Lagrange equations and the Hamiltonian formalism. The resulting dynamical equations seem to be compatible with those found in the literature, specially with mass-dependent and with nonlinear equations for systems in classical and quantum mechanics. Examples are presented to illustrate applications of the formulation. Also, the conserved Nether current, are worked out. △ Less

Submitted 9 November, 2015; originally announced November 2015.

Comments: 13 pages. arXiv admin note: text overlap with arXiv:1502.07606

Journal ref: Phys A 450 (2016) 217-227

Probing the cosmological viability of non-gaussian statistics

Authors: Rafael C. Nunes, Edésio M. Barboza Jr., Everton M. C. Abreu, Jorge Ananias Neto

Abstract: …description takes into account the entropy and temperature intrinsic to the horizon of the universe due to the information holographically stored there through non-gaussian statistical theories proposed by Tsallis and Kaniadakis. The effect of these non-gaussian statistics in the… ▽ More Based on the relationship between thermodynamics and gravity we propose, with the aid of Verlinde's formalism, an alternative interpretation of the dynamical evolution of the Friedmann-Robertson-Walker Universe. This description takes into account the entropy and temperature intrinsic to the horizon of the universe due to the information holographically stored there through non-gaussian statistical theories proposed by Tsallis and Kaniadakis. The effect of these non-gaussian statistics in the cosmological context is change the strength of the gravitational constant. In this paper, we consider the $w$CDM model modified by the non-gaussian statistics and investigate the compatibility of these non-gaussian modification with the cosmological observations. In order to analyze in which extend the cosmological data constrain these non-extensive statistics, we use type Ia supernovae, baryon acoustic oscillations, Hubble expansion rate function and the linear growth of matter density perturbations data. △ Less

Submitted 25 May, 2016; v1 submitted 16 September, 2015; originally announced September 2015.

Comments: 14 pages, 1 figure. arXiv admin note: text overlap with arXiv:1503.05874, arXiv:1403.5706

Holographic considerations on non-gaussian statistics and gravothermal catastrophe

Authors: Everton M. C. Abreu, Jorge Ananias Neto, Edesio M. Barboza Jr., Rafael da C. Nunes

Abstract: In this paper we have derived the equipartition law of energy using Tsallis formalism and the Kaniadakis power law… ▽ More In this paper we have derived the equipartition law of energy using Tsallis formalism and the Kaniadakis power law statistics in order to obtain a modified gravitational constant. We have applied this result in the gravothermal collapse phenomenon. We have discussed the equivalence between Tsallis and the Kaniadakis statistics in the context of Verlinde entropic formalism. In the same way we have analyzed negative heat capacities in the light of gravothermal catastrophe. The relative deviations of the modified gravitational constants are derived. △ Less

Submitted 18 August, 2014; originally announced March 2015.

Comments: 15 pages. 4 figures. Preprint format. arXiv admin note: text overlap with arXiv:1204.2005

On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric

Authors: J. Weberszpil, Matheus Jatkoske Lazo, J. A. Helayël-Neto

Abstract: …decades, diverse formalisms have emerged that are adopted to approach complex systems. Amongst those, we may quote the q-calculus in Tsallis' version of Non-Extensive Statistics with its undeniable success whenever applied to a wide class of different systems;… ▽ More Over the recent decades, diverse formalisms have emerged that are adopted to approach complex systems. Amongst those, we may quote the q-calculus in Tsallis' version of Non-Extensive Statistics with its undeniable success whenever applied to a wide class of different systems; Kaniadakis' approach, based on the compatibility between relativity and thermodynamics; Fractional Calculus (FC), that deals with the dynamics of anomalous transport and other natural phenomena, and also some local versions of FC that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations. The question we might ask is whether or not there are common aspects that connect these alternative approaches. In this short communication, we discuss a possible relationship between q-deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis' framework and the Kaniadakis' scenario, with local form of fractional-derivative operators defined in fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. This connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics, such as porous or granular media. Possible connections with other alternative definitions of FC are also contempled. Insights on complexity connected to concepts like coarse-grained space-time and physics in general are pointed out. △ Less

Submitted 4 March, 2015; v1 submitted 26 February, 2015; originally announced February 2015.

Comments: 12 pages. Original Article

Journal ref: Physica A 436 (2015) 399-404

Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions

Authors: G. Kaniadakis

Abstract: …ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions which have been observed experimentally in many physical, natural and art… ▽ More We present the main features of the mathematical theory generated by the κ-deformed exponential function exp_κ(x)=(\sqrt{1+κ^2 x^2}+κx)^{1/κ}, with 0<κ<1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra we present the associated κ-differential and κ-integral calculus. Then we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics. △ Less

Submitted 25 September, 2013; originally announced September 2013.

Comments: Review paper, 22 pages, 5 figures

Journal ref: Entropy, 15(10) 3983-4010 (2013)

Comments on "A two-parameter generalization of Shannon-Khinchin Axioms and the uniqueness theorem"

Authors: Velimir M. Ilic, Edin H. Mulalic, Miomir S. Stankovic

Abstract: …class by fixing the incorrectness which occurs in the mentioned paper. Also, we consider a two-parameter class of entropies derived from the maxent principle proposed in [Kaniadakis, G. and Lissia, M. and Scarfone, AM, Physica A: Statistical Mechanics and its Applications, 340(1)]. We rederived this class by changing i… ▽ More Wada and Suyari proposed a two-parameter generalization of Shannon-Khinchin axioms (TGSK axioms) [T. Wada and H. Suyari, Physics Letters A, 368(3)]. We derive a new class of entropies which differs from Wada-Suyari's class by fixing the incorrectness which occurs in the mentioned paper. Also, we consider a two-parameter class of entropies derived from the maxent principle proposed in [Kaniadakis, G. and Lissia, M. and Scarfone, AM, Physica A: Statistical Mechanics and its Applications, 340(1)]. We rederived this class by changing initial condition, obtaining the same class as our class derived from TGSK axioms. △ Less

Submitted 29 January, 2013; originally announced January 2013.

Comments: 4 pages

Generalized information entropies depending only on the probability distribution

Authors: O. Obregón, A. Gil-Villegas

Abstract: …can be described by a superposition of several statistics, a "super statistics". We consider first, the Gamma, log-normal and $F$-distributions of $β$. It is assumed that they depend only on $p_l$, the probability associated with the microscopic configuration of the system. For each of the three $β-$distributi… ▽ More Systems with a long-term stationary state that possess as a spatio-temporally fluctuation quantity $β$ can be described by a superposition of several statistics, a "super statistics". We consider first, the Gamma, log-normal and $F$-distributions of $β$. It is assumed that they depend only on $p_l$, the probability associated with the microscopic configuration of the system. For each of the three $β-$distributions we calculate the Boltzmann factors and show that they coincide for small variance of the fluctuations. For the Gamma distribution it is possible to calculate the entropy in a closed form, depending on $p_l$, and to obtain then an equation relating $p_l$ with $βE_l$. We also propose, as other examples, new entropies close related with the Kaniadakis and two possible Sharma-Mittal entropies. The entropies presented in this work do not depend on a constant parameter $q$ but on $p_l$. For the $p_l$-Gamma distribution and its corresponding $B_{p_l}(E)$ Boltzmann factor and the associated entropy, we show the validity of the saddle-point approximation. We also briefly discuss the generalization of one of the four Khinchin axioms to get this proposed entropy. △ Less

Submitted 14 June, 2012; originally announced June 2012.

Comments: 13 pages, 3 figures

Non-Gaussian statistics, maxwellian derivation and stellar polytropes

Authors: E. P. Bento, J. R. P. Silva, R. Silva

Abstract: In this letter we discuss the Non-gaussian statistics considering two aspects. In the first, we show that the Maxwell's first derivation of the stationary distribution function for a dilute gas can be extended in the context of… ▽ More In this letter we discuss the Non-gaussian statistics considering two aspects. In the first, we show that the Maxwell's first derivation of the stationary distribution function for a dilute gas can be extended in the context of Kaniadakis statistics. The second one, by investigating the stellar system, we study the Kaniadakis analytical relation between the entropic parameter $κ$ and stellar polytrope index $n$. We compare also the Kaniadakis relation $n=n(κ)$ with $n=n(q)$ proposed in the Tsallis framework. △ Less

Submitted 7 May, 2013; v1 submitted 8 May, 2012; originally announced May 2012.

Comments: 10 pages, 1 figure

Kappa-deformed random-matrix theory based on Kaniadakis statistics

Authors: A. Y. Abul-Magd, M. Abdel-Mageed

Abstract: We present a possible extension of the random-matrix theory, which is widely used to describe spectral fluctuations of chaotic systems. By considering the Kaniadakis non-Gaussian… ▽ More We present a possible extension of the random-matrix theory, which is widely used to describe spectral fluctuations of chaotic systems. By considering the Kaniadakis non-Gaussian statistics, characterized by the index κ (Boltzmann-Gibbs entropy is recovered in the limit κ\rightarrow0), we propose the non-Gaussian deformations (κ \neq 0) of the conventional orthogonal and unitary ensembles of random matrices. The joint eigenvalue distributions for the κ-deformed ensembles are derived by applying the principle maximum entropy to Kaniadakis entropy. The resulting distribution functions are base invarient as they depend on the matrix elements in a trace form. Using these expressions, we introduce a new generalized form of the Wigner surmise valid for nearly-chaotic mixed systems, where a basis-independent description is still expected to hold. We motivate the necessity of such generalization by the need to describe the transition of the spacing distribution from chaos to order, at least in the initial stage. We show several examples about the use of the generalized Wigner surmise to the analysis of the results of a number of previous experiments and numerical experiments. Our results suggest the entropic index κ as a measure for deviation from the state of chaos. We also introduce a κ-deformed Porter-Thomas distribution of transition intensities, which fits the experimental data for mixed systems better than the commonly-used gamma-distribution. △ Less

Submitted 1 January, 2012; originally announced January 2012.

Comments: 18 pages, 8 figures

Journal ref: Modern Physics Letters B 26 (2012) 1250059

Group entropies, correlation laws and zeta functions

Authors: Piergiulio Tempesta

Abstract: …of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is w… ▽ More The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function. △ Less

Submitted 10 May, 2011; originally announced May 2011.

Comments: to appear in Physical Review E

Statistical field theories deformed within different calculi

Authors: A. I. Olemskoi, S. S. Borysov, I. A. Shuda

Abstract: Within framework of basic-deformed and finite-difference calculi, as well as deformation procedures proposed by Tsallis, Abe, and Kaniadakis to be generalized by Naudts, we develop field-theoretical schemes of… ▽ More Within framework of basic-deformed and finite-difference calculi, as well as deformation procedures proposed by Tsallis, Abe, and Kaniadakis to be generalized by Naudts, we develop field-theoretical schemes of statistically distributed fields. We construct a set of generating functionals and find their connection with corresponding correlators for basic-deformed, finite-difference, and Kaniadakis calculi. Moreover, we introduce pair of additive functionals, whose expansions into deformed series yield both Green functions and their irreducible proper vertices. We find as well formal equations, governing by the generating functionals of systems which possess a symmetry with respect to a field variation and are subjected to an arbitrary constrain. Finally, we generalize field-theoretical schemes inherent in concrete calculi in the Naudts spirit. △ Less

Submitted 15 April, 2010; originally announced April 2010.

Comments: 12 pages, 3 figures

Complete versus incomplete definitions of the deformed logarithmic and exponential functions

Authors: Thomas Oikonomou, G. Baris Bagci

Abstract: The recent generalizations of Boltzmann-Gibbs statistics mathematically relies on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed logarithmic/exponential map is a bijection from $\mathbb{R}^+/\mathbb{R}$ (set of positive real numbers/all real numbers) to… ▽ More The recent generalizations of Boltzmann-Gibbs statistics mathematically relies on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed logarithmic/exponential map is a bijection from $\mathbb{R}^+/\mathbb{R}$ (set of positive real numbers/all real numbers) to $\mathbb{R}/\mathbb{R}^+$, as their undeformed counterparts. We show that their inverse map exists only in some subsets of the aforementioned (co)domains. Furthermore, we present conditions which a generalized deformed function has to satisfy, so that the most important properties of the ordinary functions are preserved. The fulfillment of these conditions permits us to determine the validity interval of the deformation parameters. We finally apply our analysis to Tsallis, Kaniadakis, Abe and Borges-Roditi deformed functions. △ Less

Submitted 23 July, 2009; originally announced July 2009.

Comments: 12 pages, 4 figures

Maximum entropy principle and power-law tailed distributions

Authors: G. Kaniadakis

Abstract: In ordinary statistical mechanics the Boltzmann-Shannon entropy is related to the Maxwell-Bolzmann distribution $p_i$ by means of a twofold link. The first link is differential and is offered by the Jaynes Maximum Entropy Principle. The second link is algebraic and imposes that both the entropy and the distribution must be expressed in terms of the same func… ▽ More In ordinary statistical mechanics the Boltzmann-Shannon entropy is related to the Maxwell-Bolzmann distribution $p_i$ by means of a twofold link. The first link is differential and is offered by the Jaynes Maximum Entropy Principle. The second link is algebraic and imposes that both the entropy and the distribution must be expressed in terms of the same function in direct and inverse form. Indeed, the Maxwell-Boltzmann distribution $p_i$ is expressed in terms of the exponential function, while the Boltzmann-Shannon entropy is defined as the mean value of $-\ln(p_i)$. In generalized statistical mechanics the second link is customarily relaxed. Here we consider the question if and how is it possible to select generalized statistical theories in which the above mentioned twofold link between entropy and the distribution function continues to hold, such as in the case of ordinary statistical mechanics. Within this scenario, there emerge new couples of direct-inverse functions, i.e. generalized logarithms $Λ(x)$ and generalized exponentials $Λ^{-1}(x)$, defining coherent and self-consistent generalized statistical theories. Interestingly, all these theories preserve the main features of ordinary statistical mechanics, and predict distribution functions presenting power-law tails. Furthermore, the obtained generalized entropies are both thermodynamically and Lesche stable. △ Less

Submitted 26 May, 2009; v1 submitted 27 April, 2009; originally announced April 2009.

Comments: To appear in EPJB. 11 journal pages, 100 references

Journal ref: Eur. Phys. J. B 70, 3-13 (2009)

$κ$-exponential models from the geometrical viewpoint

Authors: Giovanni Pistone

Abstract: We discuss the use of Kaniadakis' $κ$-exponential in the construction of a statistical manifold modelled on Lebesgue spaces of real random variables. Some algebraic features of the deformed exponential models are considered. A chart is defined for each strictly positive densities; every other strictly positive dens… ▽ More We discuss the use of Kaniadakis' $κ$-exponential in the construction of a statistical manifold modelled on Lebesgue spaces of real random variables. Some algebraic features of the deformed exponential models are considered. A chart is defined for each strictly positive densities; every other strictly positive density in a suitable neighborhood of the reference probability is represented by the centered $\Kln$ likelihood △ Less

Submitted 11 March, 2009; originally announced March 2009.

Non-gaussian statistics and stellar rotational velocities of main sequence field stars

Authors: J. C. Carvalho, J. D. Jr. do Nascimento, R. Silva, J. R. De Medeiros

Abstract: …measurements. We show that the velocity distributions cannot be fitted by a maxwellian. On the other hand, an analysis based on both Tsallis and Kaniadakis power-law… ▽ More In this letter we study the observed distributions of rotational velocity in a sample of more than 16,000 nearby F and G dwarf stars, magnitude complete and presenting high precision $Vsin i$ measurements. We show that the velocity distributions cannot be fitted by a maxwellian. On the other hand, an analysis based on both Tsallis and Kaniadakis power-law statistics is by far the most appropriate statistics and give a very good fit. It is also shown that single and binary stars have similar rotational distributions. This is the first time, to our knowledge, that these two new statistics are tested for the rotation of such a large sample of stars, pointing solidly to a solution of the puzzling problem on the function governing the distribution of stellar rotational velocity △ Less

Submitted 4 March, 2009; originally announced March 2009.

Comments: 10 pages, 2 figures, accepted for publication in ApJ Letters

Journal ref: Astrophys.J.696:L48-L51,2009

Power-law statistics and stellar rotational velocities in the Pleiades

Authors: J. C. Carvalho, R. Silva, J. D. Jr. do Nascimento, J. R. De Medeiros

Abstract: In this paper we will show that, the non-gaussian statistics framework based on the Kaniadakis statistics is more appropriate to fit the observed distributions of projected rotational velocity measurements of stars in the Pleiades open cluster. To this end, we compare the results… ▽ More In this paper we will show that, the non-gaussian statistics framework based on the Kaniadakis statistics is more appropriate to fit the observed distributions of projected rotational velocity measurements of stars in the Pleiades open cluster. To this end, we compare the results from the $κ$ and $q$-distributions with the Maxwellian. △ Less

Submitted 4 March, 2009; originally announced March 2009.

Comments: 13 pages, 3 figures

Journal ref: Europhys. Lett.84:59001, 2008

Non-gaussian statistics and the relativistic nuclear equation of state

Authors: F. I. M. Pereira, R. Silva, J. S. Alcaniz

Abstract: We investigate possible effects of quantum power-law statistical mechanics on the relativistic nuclear equation of state in the context of the Walecka quantum hadrodynamics theory. By considering the Kaniadakis non-Gaussian statistics, characterized by the index $κ$ (Boltzmann-Gi… ▽ More We investigate possible effects of quantum power-law statistical mechanics on the relativistic nuclear equation of state in the context of the Walecka quantum hadrodynamics theory. By considering the Kaniadakis non-Gaussian statistics, characterized by the index $κ$ (Boltzmann-Gibbs entropy is recovered in the limit $κ\to 0$), we show that the scalar and vector meson fields become more intense due to the non-Gaussian statistical effects ($κ\neq 0$). From an analytical treatment, an upper bound on $κ$ ($κ< 1/4$) is found. We also show that as the parameter $κ$ increases the nucleon effective mass diminishes and the equation of state becomes stiffer. A possible connection between phase transitions in nuclear matter and the $κ$-parameter is largely discussed. △ Less

Submitted 13 February, 2009; originally announced February 2009.

Comments: 8 pages, 8 fifures, LaTeX

Journal ref: Nucl.Phys.A828:136-148,2009

Generalized information and entropy measures in physics

Authors: Christian Beck

Abstract: The formalism of statistical mechanics can be generalized by starting from more general measures of information than the Shannon entropy and maximizing those subject to suitable constraints. We discuss some of the most important examples of information measures that are useful for the description of complex systems. Examples treated are the Renyi entropy, Ts… ▽ More The formalism of statistical mechanics can be generalized by starting from more general measures of information than the Shannon entropy and maximizing those subject to suitable constraints. We discuss some of the most important examples of information measures that are useful for the description of complex systems. Examples treated are the Renyi entropy, Tsallis entropy, Abe entropy, Kaniadakis entropy, Sharma-Mittal entropies, and a few more. Important concepts such as the axiomatic foundations, composability and Lesche stability of information measures are briefly discussed. Potential applications in physics include complex systems with long-range interactions and metastable states, scattering processes in particle physics, hydrodynamic turbulence, defect turbulence, optical lattices, and quite generally driven nonequilibrium systems with fluctuations of temperature. △ Less

Submitted 14 February, 2009; v1 submitted 7 February, 2009; originally announced February 2009.

Comments: 23 pages, 6 figures. Short review article, to appear in Contemporary Physics. References extended

A k-generalized statistical mechanics approach to income analysis

Authors: F. Clementi, M. Gallegati, G. Kaniadakis

Abstract: This paper proposes a statistical mechanics approach to the analysis of income distribution and inequality. A new distribution function, having its roots in the framework of k-generalized… ▽ More This paper proposes a statistical mechanics approach to the analysis of income distribution and inequality. A new distribution function, having its roots in the framework of k-generalized statistics, is derived that is particularly suitable to describe the whole spectrum of incomes, from the low-middle income region up to the high-income Pareto power-law regime. Analytical expressions for the shape, moments and some other basic statistical properties are given. Furthermore, several well-known econometric tools for measuring inequality, which all exist in a closed form, are considered. A method for parameter estimation is also discussed. The model is shown to fit remarkably well the data on personal income for the United States, and the analysis of inequality performed in terms of its parameters reveals very powerful. △ Less

Submitted 23 February, 2009; v1 submitted 31 January, 2009; originally announced February 2009.

Comments: LaTeX2e; 15 pages with 1 figure; corrected typos

Journal ref: Journal of Statistical Mechanics: Theory and Experiment, 16 February 2009, start page: P02037

Conservative Force Fields in Non-Gaussian Statistics

Authors: J. M. Silva, R. Silva, J. A. S. Lima

Abstract: …framework of kinetic theory with basis on the Vlasov equation. Such a result is significant as a preliminary to the discussion on the role of long range interactions in the Kaniadakis thermostatistics and the underlying kinetic theory. ▽ More In this letter, we determine the $κ$-distribution function for a gas in the presence of an external field of force described by a potential U(${\bf r}$). In the case of a dilute gas, we show that the $κ$-power law distribution including the potential energy factor term can rigorously be deduced in the framework of kinetic theory with basis on the Vlasov equation. Such a result is significant as a preliminary to the discussion on the role of long range interactions in the Kaniadakis thermostatistics and the underlying kinetic theory. △ Less

Submitted 21 July, 2008; originally announced July 2008.

Comments: 10 pages, 1 figure

Multifractal spectrum of phase space related to generalized thermostatistics

Authors: A. I. Olemskoi, V. O. Kharchenko, V. N. Borisyuk

Abstract: …. Related thermostatistics is shown to be governed by the Tsallis formalism of the non-extensive statistics, where the non-additivity parameter is equal to ${\barτ}(q)\equiv 1/τ(q)>1$, and the multifractal function $τ(q)= qd_q-f(d_q)$ is the specific heat determined with multifractal parameter $q\in [1,\infty)$. In this way, the equipartition law is shown… ▽ More We consider a self-similar phase space with specific fractal dimension $d$ being distributed with spectrum function $f(d)$. Related thermostatistics is shown to be governed by the Tsallis formalism of the non-extensive statistics, where the non-additivity parameter is equal to ${\barτ}(q)\equiv 1/τ(q)>1$, and the multifractal function $τ(q)= qd_q-f(d_q)$ is the specific heat determined with multifractal parameter $q\in [1,\infty)$. In this way, the equipartition law is shown to take place. Optimization of the multifractal spectrum function $f(d)$ derives the relation between the statistical weight and the system complexity. It is shown the statistical weight exponent $τ(q)$ can be modeled by hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponentials to describe arbitrary multifractal phase space explicitly. The spectrum function $f(d)$ is proved to increase monotonically from minimum value $f=-1$ at $d=0$ to maximum one $f=1$ at $d=1$. At the same time, the number of monofractals increases with growth of the phase space volume at small dimensions $d$ and falls down in the limit $d\to 1$. △ Less

Submitted 23 October, 2007; v1 submitted 14 August, 2007; originally announced August 2007.

Comments: 16 pages, 7 figures

The property of kappa-deformed statistics for a relativistic gas in an electromagnetic field: kappa parameter and kappa-distribution

Authors: Guo Lina, Du Jiulin, Liu Zhipeng

Abstract: We investigate the physical property of the kappa parameter and the kappa-distribution in the kappa-deformed statistics, based on… ▽ More We investigate the physical property of the kappa parameter and the kappa-distribution in the kappa-deformed statistics, based on Kaniadakis entropy, for a relativistic gas in an electromagnetic field. We derive two relations for the relativistic gas in the framework of kappa-deformed statistics, which describe the physical situation represented by the relativistic kappa-distribution function, provide a reasonable connection between the parameter kappa, the temperature four-gradient and the four-vector potential gradient, and thus present for the case kappa different from zero a clearly physical meaning. It is shown that such a physical situation is a meta-equilibrium state of the system, but has a new physical characteristic. △ Less

Submitted 11 April, 2007; originally announced April 2007.

Comments: 10 pages, 15 references. Accepted by Phys.Lett.A

Journal ref: Phys. Lett. A 367(2007)431

The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

Authors: R. Silva

Abstract: …theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G.… ▽ More We have investigated the proof of the $H$ theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phy. Rev. E {\bf 66}, 056125, 2002; {\it ibid.} {\bf 72}, 036108, 2005]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a $κ$ power law generalization of the exponential Juttner distribution, e.g., $f(x,p)\propto (\sqrt{1+ κ^2θ^2}+κθ)^{1/κ}\equiv\exp_κθ$, with $θ=α(x)+β_μp^μ$, where $α(x)$ is a scalar, $β_μ$ is a four-vector, and $p^μ$ is the four-momentum. As a simple example, we calculate the relativistic $κ$ power law for a dilute charged gas under the action of an electromagnetic field $F^{μν}$. All standard results are readly recovered in the particular limit $κ\to 0$. △ Less

Submitted 29 November, 2006; originally announced November 2006.

Comments: 7 pages; to be published in EPJB

Journal ref: Eur. Phys. J. B 54, 499 (2006)

k-Generalized Statistics in Personal Income Distribution

Authors: F. Clementi, M. Gallegati, G. Kaniadakis

Abstract: …, proposed in Ref. [G. Kaniadakis, Physica A \textbf{296}, 405 (2001)], the survival function $P_{>}(x)=\exp_κ(-βx^α)$, where $x\in\mathbf{R}^{+}$, $α,β>0$, and $κ\in[0,1)$, is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter… ▽ More Starting from the generalized exponential function $\exp_κ(x)=(\sqrt{1+κ^{2}x^{2}}+κx)^{1/κ}$, with $\exp_{0}(x)=\exp(x)$, proposed in Ref. [G. Kaniadakis, Physica A \textbf{296}, 405 (2001)], the survival function $P_{>}(x)=\exp_κ(-βx^α)$, where $x\in\mathbf{R}^{+}$, $α,β>0$, and $κ\in[0,1)$, is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function $P_{>}^{0}(x)=\exp(-βx^α)$\textemdash to which reduces as $κ$ approaches zero\textemdash behaving in very different way in the $x\to0$ and $x\to\infty$ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law $P_{>}(x)\sim(2βκ)^{-1/κ}x^{-α/κ}$. This makes the $κ$-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range. △ Less

Submitted 1 February, 2007; v1 submitted 31 July, 2006; originally announced July 2006.

Comments: Latex2e v1.6; 14 pages with 12 figures; for inclusion in the APFA5 Proceedings

Journal ref: The European Physical Journal B, Vol: 57, Issue: 2, May II, 2007, pp: 187-193

Multifractal spectrum of the phase space related to generalized thermostatistics

Authors: A. I. Olemskoi, V. O. Kharchenko

Abstract: We consider the set of monofractals within a multifractal related to the phase space being the support of a generalized thermostatistics. The statistical weight exponent $τ(q)$ is shown to can be modeled by the hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponentials whose using allows on… ▽ More We consider the set of monofractals within a multifractal related to the phase space being the support of a generalized thermostatistics. The statistical weight exponent $τ(q)$ is shown to can be modeled by the hyperbolic tangent deformed in accordance with both Tsallis and Kaniadakis exponentials whose using allows one to describe explicitly arbitrary multifractal phase space. The spectrum function $f(d)$, determining the specific number of monofractals with reduced dimension $d$, is proved to increases monotonically from minimum value $f=-1$ at $d=0$ to maximum $f=1$ at $d=1$. The number of monofractals is shown to increase with growth of the phase space volume at small dimensions $d$ and falls down in the limit $d\to 1$. △ Less

Submitted 25 February, 2006; v1 submitted 23 February, 2006; originally announced February 2006.

Comments: 10 pages, 6 figures

Statistical descriptions of nonlinear systems at the onset of chaos

Authors: Massmimo Coraddu, Marcello Lissia, Roberto Tonelli

Abstract: …ensemble entropy shows an asymptotic linear growth with rate K. The rate K matches the logarithm of the corresponding asymptotic sensitivity to initial conditions λ. The statistical formalism and the equality K=λcan be extended to weakly chaotic systems by suitable and corresponding generalizations of the logarithm and of the entropy. Using the logistic map… ▽ More Ensemble of initial conditions for nonlinear maps can be described in terms of entropy. This ensemble entropy shows an asymptotic linear growth with rate K. The rate K matches the logarithm of the corresponding asymptotic sensitivity to initial conditions λ. The statistical formalism and the equality K=λcan be extended to weakly chaotic systems by suitable and corresponding generalizations of the logarithm and of the entropy. Using the logistic map as a test case we consider a wide class of deformed statistical description which includes Tsallis, Abe and Kaniadakis proposals. The physical criterion of finite-entropy growth K strongly restricts the suitable entropies. We study how large is the region in parameter space where the generalized description is useful. △ Less

Submitted 30 November, 2005; originally announced November 2005.

Comments: RevTeX preprint 10 pages, 4 figures, to appear in The European Physical Journal B

Journal ref: Physica A365:252-257,2006

Two generalizations of the Boltzmann equation

Authors: T. S. Biro, G. Kaniadakis

Abstract: We connect two different generalizations of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint. Direct transfor… ▽ More We connect two different generalizations of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint. Direct transformation formulas between key functions of the two approaches are given. △ Less

Submitted 27 September, 2005; originally announced September 2005.

Comments: Talk given at the 3rd NEXT-Sigma-Phi Conference, Crete, Aug.2005, revtex, 10 page, 2 figs

Journal ref: Eur.Phys.J. B50 (2006) 3

Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics

Authors: G. Kaniadakis, M. Lissia, A. M. Scarfone

Abstract: A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributi… ▽ More A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed. △ Less

Submitted 13 April, 2005; v1 submitted 26 September, 2004; originally announced September 2004.

Comments: Version to appear in PRE: about 20% shorter, references updated, 13 PRE pages, 3 figures

Journal ref: Phys.Rev. E71 (2005) 046128

Deformed logarithms and entropies

Authors: G. Kaniadakis, M. Lissia, A. M. Scarfone

Abstract: By solving a differential-functional equation inposed by the MaxEnt principle we obtain a class of two-parameter deformed logarithms and construct the corresponding two-parameter generalized trace-form entropies. Generalized distributions follow from these generalized entropies in the same fashion as the Gaussian distribution follows from the Shannon entropy, which is a special limiting case of… ▽ More By solving a differential-functional equation inposed by the MaxEnt principle we obtain a class of two-parameter deformed logarithms and construct the corresponding two-parameter generalized trace-form entropies. Generalized distributions follow from these generalized entropies in the same fashion as the Gaussian distribution follows from the Shannon entropy, which is a special limiting case of the family. We determine the region of parameters where the deformed logarithm conserves the most important properties of the logarithm, and show that important existing generalizations of the entropy are included as special cases in this two-parameter class. △ Less

Submitted 16 February, 2004; originally announced February 2004.

Comments: Presented at next2003, Second Sardinian International Conference on News and Expectations in Thermostatistics, Villasimius (Cagliari) Italy, 21st-28th September 2003. In press Physica A (2004). Elsevier LaTeX macros, 11 pages, 1 figure

Journal ref: Physica A340:41-49,2004

Lesche Stability of $κ$-Entropy

Authors: G. Kaniadakis, A. M. Scarfone

Abstract: …for the Shannon entropy [B. Lesche, J. Stat. Phys. 27, 419 (1982)], represents a fundamental test, for its experimental robustness, for systems obeying the Maxwell-Boltzmann statistical mechanics. Of course, this stability condition must be satisfied by any entropic functional candidate to generate non-conventional… ▽ More The Lesche stability condition for the Shannon entropy [B. Lesche, J. Stat. Phys. 27, 419 (1982)], represents a fundamental test, for its experimental robustness, for systems obeying the Maxwell-Boltzmann statistical mechanics. Of course, this stability condition must be satisfied by any entropic functional candidate to generate non-conventional statistical mechanics. In the present effort we show that the $κ$-entropy, recently introduced in literature [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002)], satisfies the Lesche stability condition. △ Less

Submitted 11 February, 2004; v1 submitted 30 October, 2003; originally announced October 2003.

Comments: Presented at next2003, Second Sardinian International Conference on News and Expectations in Thermostatistics, Villasimius (Cagliari) Italy, 21st-28th September 2003. In press Physica A (2004). Elsevier LaTeX macros, 10 pages, minor changes

Journal ref: Physica A 340 (2004) 102-109

Statistical mechanics and thermodynamics of complex systems

Authors: V. Garcia-Morales, J. Pellicer

Abstract: An unified thermodynamical framework based in the use of a generalized Massieu-Planck thermodynamic potential is proposed and a new formulation of Boltzmann-Gibbs Statistical Mechanics is established. Under this philosophy a generalization of (classical) Boltzmann-Gibbs thermostatistics is suggested and connected to recent nonextensive… ▽ More An unified thermodynamical framework based in the use of a generalized Massieu-Planck thermodynamic potential is proposed and a new formulation of Boltzmann-Gibbs Statistical Mechanics is established. Under this philosophy a generalization of (classical) Boltzmann-Gibbs thermostatistics is suggested and connected to recent nonextensive statistics formulations. This is accomplished by defining a convenient squeezing function which restricts among the collections of Boltzmann-Gibbs configurations of the complete equilibrium closure. The formalism embodies Beck-Cohen superstatistics and a direct connection with the nonlinear kinetic theory due to Kaniadakis is provided, being the treatment presented fully consistent with it. As an example Tsallis nonextensive statistics is completely rebuilt into our formulation adding new insights (zeroth law of thermodynamics, non ad hoc definition of the mean value of a physical quantity,...). We relate all the formal development to physical and measurable quantities and suggest a way to establish the relevant statistics of any system based on determinations of temperature. △ Less

Submitted 6 July, 2003; v1 submitted 10 April, 2003; originally announced April 2003.

Comments: 23 pages, no figures, extensively revised (added comment on zeroth law of thermodynamics)

MSC Class: 82B03; 82C03

Kappa-deformed Statistics and the Formation of a Quark-Gluon Plasma

Authors: A. M. Teweldeberhan, H. G. Miller, R. Tegen

Abstract: The effect of the non-extensive form of statistical mechanics proposed by Tsallis on the formation of a quark-gluon plasma (QGP) has been recently investigated in ref. \cite{1}. The results show that for small deviations ($\approx 10%$) from Boltzmann-Gibbs (BG) statistics in the QGP phase, the critical temperature for… ▽ More The effect of the non-extensive form of statistical mechanics proposed by Tsallis on the formation of a quark-gluon plasma (QGP) has been recently investigated in ref. \cite{1}. The results show that for small deviations ($\approx 10%$) from Boltzmann-Gibbs (BG) statistics in the QGP phase, the critical temperature for the formation of a QGP does not change substantially for a large variation of the chemical potential. In the present effort we use the extensive $κ$-deformed statistical mechanics constructed by Kaniadakis to represent the constituents of the QGP and compare the results with ref. [1]. △ Less

Submitted 3 March, 2003; originally announced March 2003.

Comments: 2 Figures

Journal ref: Int.J.Mod.Phys. E12 (2003) 669-673

Kinetical Foundations of Non Conventional Statistics

Authors: G. Kaniadakis, P. Quarati, A. M. Scarfone

Abstract: After considering the kinetical interaction principle (KIP) introduced in ref. Physica A {\bf296}, 405 (2001), we study in the Boltzmann picture, the evolution equation and the H-theorem for non extensive systems. The $q$-kinetics and the $κ$-kinetics are studied in detail starting from the most general non linear Boltzmann equation compatible with the KIP. After considering the kinetical interaction principle (KIP) introduced in ref. Physica A {\bf296}, 405 (2001), we study in the Boltzmann picture, the evolution equation and the H-theorem for non extensive systems. The $q$-kinetics and the $κ$-kinetics are studied in detail starting from the most general non linear Boltzmann equation compatible with the KIP. △ Less

Submitted 3 October, 2001; originally announced October 2001.

Comments: 11 pages, no figures. Contribution paper to the proseedings of the International School and Workshop on Nonextensive Thermodynamics and Physical Applications, NEXT 2001, 23-30 May 2001, Cagliari Sardinia, Italy (Physica A)

Journal ref: Physica A 305, 76 (2002)

Non Linear Kinetics underlying Generalized Statistics

Authors: G. Kaniadakis

Abstract: …Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand,… ▽ More The purpose of the present effort is threefold. Firstly, it is shown that there exists a principle, that we call Kinetical Interaction Principle (KIP), underlying the non linear kinetics in particle systems, independently on the picture (Kramers, Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand, to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and, on the other hand, suggests how we can introduce naturally new distributions. Finally, as a working example of the approach to the non linear kinetics here presented, a new non extensive statistics is constructed and studied starting from a one-parameter deformation of the exponential function holding the relation $f(-x)f(x)=1$. △ Less

Submitted 17 May, 2001; v1 submitted 22 March, 2001; originally announced March 2001.

Comments: 31 pages, no figures

Journal ref: Physica A 296, 405 (2001)

Microscopic dynamics underlying the anomalous diffusion

Authors: G. Kaniadakis, G. Lapenta

Abstract: The time dependent Tsallis statistical distribution describing anomalous diffusion is usually obtained in the literature as the solution of a non-linear Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347 (1995)]. The scope of the present paper is twofold. Firstly we show that this distribution can be obtained also as solution of… ▽ More The time dependent Tsallis statistical distribution describing anomalous diffusion is usually obtained in the literature as the solution of a non-linear Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347 (1995)]. The scope of the present paper is twofold. Firstly we show that this distribution can be obtained also as solution of the non-linear porous media equation. Secondly we prove that the time dependent Tsallis distribution can be obtained also as solution of a linear FP equation [G. Kaniadakis and P. Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the velocity, that describes a generalized Brownian motion. This linear FP equation is shown to arise from a microscopic dynamics governed by a standard Langevin equation in presence of multiplicative noise. △ Less

Submitted 19 July, 2000; originally announced July 2000.

Comments: 4 pag. - no figures. To appear on Phys. Rev. E 62, September 2000

Journal ref: Phys.Rev.E 62, 3246 (2000).

Nonextensive statistical effects in nuclear physics problems

Authors: G. Kaniadakis, A. Lavagno, M. Lissia, P. Quarati

Abstract: Recent progresses in statistical mechanics indicate the Tsallis nonextensive thermostatistics as the natural generalization of the standard classical and quantum… ▽ More Recent progresses in statistical mechanics indicate the Tsallis nonextensive thermostatistics as the natural generalization of the standard classical and quantum statistics, when memory effects and long-range forces are not negligible. In this framework, weakly nonextensive statistical deviations can strongly reduce the puzzling discrepancies between experimental data and theoretical previsions for solar neutrinos and for pion transverse-momentum correlations in Pb-Pb high-energy nuclear collisions. △ Less

Submitted 12 December, 1998; originally announced December 1998.

Comments: 8 pages, LaTeX with lamuphys.sty and bibnorm.sty, to appear in the proceedings of ``VII Convegno su Problemi di Fisica Nucleare Teorica'', Cortona (Italy), 19-21 October 1998

Report number: INFNCA-TH9817

Kinetic model for q-deformed bosons and fermions

Authors: G. Kaniadakis, A. Lavagno, P. Quarati

Abstract: …-oscillators equilibrium statistics. ▽ More We have studied the kinetics of $q$-deformed bosons and fermions, within a semiclassical approach. This investigation is realized by introducing a generalized exclusion-inclusion principle, intrinsically connected with the quantum $q$-algebra by means of the creation and annihilation operators matrix elements. In this framework, we have derived a non-linear Fokker-Planck equation for $q$-deformed bosons and fermions which can be seen as a time evolution equation, appropriate to consider non-equilibrium or near-equilibrium systems in a semiclassical approximation. The steady state of this equation reproduces in a simple mode the $q$-oscillators equilibrium statistics. △ Less

Submitted 22 January, 1997; originally announced January 1997.

Comments: revtex, 4 pages, to appear in Phys. Lett. A

Non-Extensive Statistics and Solar Neutrinos

Authors: G. Kaniadakis, A. Lavagno, P. Quarati

Abstract: …on the light and heavy ions (or, equally, because of anomalous diffusion of solar core constituents of light mass and of normal diffusion of heavy ions), the equilibrium statistical distribution that these particles must obey, is that of generalized Boltzmann-Gibbs… ▽ More In this paper we will show that, because of the long-range microscopic memory of the random force, acting in the solar core, mainly on the electrons and the protons than on the light and heavy ions (or, equally, because of anomalous diffusion of solar core constituents of light mass and of normal diffusion of heavy ions), the equilibrium statistical distribution that these particles must obey, is that of generalized Boltzmann-Gibbs statistics (or the Tsallis non-extensive statistics), the distribution differing very slightly from the usual Maxwellian distribution. Due to the high-energy depleted tail of the distribution, the nuclear rates are reduced and, using earlier results on the standard solar model neutrino fluxes, calculated by Clayton and collaborators, we can evaluate fluxes in good agreement with the experimental data. While proton distribution is only very slightly different from Maxwellian there is a little more difference with electron distribution. We can define one central electron temperature as a few percent higher than the ion central temperature nearly equal to the standard solar model temperature. The difference is related to the different reductions with respect to the standard solar model values needed for $B$ and $CNO$ neutrinos and for $Be$ neutrinos. △ Less

Submitted 17 January, 1997; originally announced January 1997.

Comments: RevTeX, 12 pages

Journal ref: Astrophys.Space Sci.258:145-162,1998

Photofission and Quasi-Deuteron-Nuclear State as Mixing of Bosons and Fermions

Authors: G. Kaniadakis, A. Lavagno, P. Quarati

Abstract: The empirical-phenomenological quasi-deuteron photofission description is theoretically justified within the semiclassical, intermediate statistics model. The transmutational fermion (nucleon) - boson (quasi-deuteron) potential plays an essential role in the present context and is expressed in terms of thermodynamical and of microscopical quantities, analogo… ▽ More The empirical-phenomenological quasi-deuteron photofission description is theoretically justified within the semiclassical, intermediate statistics model. The transmutational fermion (nucleon) - boson (quasi-deuteron) potential plays an essential role in the present context and is expressed in terms of thermodynamical and of microscopical quantities, analogous to those commonly used in the superfluid nuclear model. △ Less

Submitted 20 March, 1996; originally announced March 1996.

Comments: 7 pages, RevTex, to appear in Zeit. f. Phys. A

Generalized Fractional Statistics

Authors: G. Kaniadakis, A. Lavagno, P. Quarati

Abstract: We link, by means of a semiclassical approach, the fractional statistics of particles obeying the Haldane exclusion principle to the Tsallis statistics and derive a generalized quantum entropy and its associated statistics. We link, by means of a semiclassical approach, the fractional statistics of particles obeying the Haldane exclusion principle to the Tsallis statistics and derive a generalized quantum entropy and its associated statistics. △ Less

Submitted 19 March, 1996; originally announced March 1996.

Comments: 4 pages, RevTex

Generalized Statistics and Solar Neutrinos

Authors: G. Kaniadakis, A. Lavagno, P. Quarati

Abstract: The generalized Tsallis statistics produces a distribution function appropriate to describe the interior solar plasma, thought as a stellar polytrope, showing a tail depleted respect to the Maxwell-Boltzmann distribution and reduces to zero at energies greater than about $20 \, k_{_B} T$. The Tsallis statistics can the… ▽ More The generalized Tsallis statistics produces a distribution function appropriate to describe the interior solar plasma, thought as a stellar polytrope, showing a tail depleted respect to the Maxwell-Boltzmann distribution and reduces to zero at energies greater than about $20 \, k_{_B} T$. The Tsallis statistics can theoretically support the distribution suggested in the past by Clayton and collaborators, which shows also a depleted tail, to explain the solar neutrino counting rate. △ Less

Submitted 20 March, 1996; originally announced March 1996.

Comments: 4 pages, RevTex

Journal ref: Phys.Lett. B369 (1996) 308-312

Kinetic Approach to Fractional Exclusion Statistics

Authors: G. Kaniadakis, A. Lavagno, P. Quarati

Abstract: We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion… ▽ More We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed [Phys. Rev. E49, 5103 (1994)] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular cases. The thermodynamic properties of the statistical systems obeying the generalized exclusion-inclusion principle are discussed. △ Less

Submitted 22 July, 1995; originally announced July 1995.

Comments: 6 pages, REVTEX

Journal ref: Nucl.Phys.B466:527-537,1996