Non-Gaussian statistics, Maxwellian derivation and stellar polytropes

Bento, E. P.; Silva, J.R.P.; Silva, R.

doi:10.1016/j.physa.2012.10.022

Cited by 21 publications

(18 citation statements)

References 39 publications

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“…On the other hand specific applications to physical systems have been considered, e.g. the cosmic rays [3], relativistic [37] and classical [38] plasmas in presence of external electromagnetic fields, the relaxation in relativistic plasmas under wave-particle interactions [39,40], anomalous diffusion [41,42], nonlinear kinetics [43][44][45], kinetics of interacting atoms and photons [46], particle kinetics in the presence of temperature gradients [47,48], particle systems in external conservative force fields [49], stellar distributions in astrophysics [50][51][52][53], quark-gluon plasma formation [54], quantum hadrodynamics models [55], the fracture propagation [56], etc. Other applications concern dynamical systems at the edge of chaos [57][58][59], fractal systems [60], field theories [61], the random matrix theory [62][63][64], the error theory [65], the game theory [66], the theory of complex networks [67], the information theory [68], etc.…”

Section: Introductionmentioning

confidence: 99%

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

Kaniadakis

2013

Entropy

107

114

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We present the main features of the mathematical theory generated by the κ-deformed exponential function exp κ (x) = (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.

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Section: Introductionmentioning

confidence: 99%

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

Kaniadakis

2013

Entropy

107

114

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“…], quark-gluon plasma [13], kinetic models describing a gas of interacting atoms and photons [14] and financial models [15]. The kinetic foundations for the κ-statistics lead to a velocity distribution for free particles given by [16] …”

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confidence: 99%

Jeans instability criterion from the viewpoint of Kaniadakis' statistics

Abreu¹,

Neto²,

Barboza³

et al. 2016

EPL

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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 κ-sound velocity. Finally, we have applied the results obtained here to analyze an astrophysical system. Keywords: Jeans' criterion; non-gaussian statisticsThe dynamical stability of a self-gravitating system usually can be described by the Jeans criterion of gravitational instability. The so-called Jeans's length [1] is given bywhere k B is the Boltzmann constant, T is the temperature, µ is the mean molecular weight, m H is the atomic mass of hydrogen, G is the gravitational constant and ρ 0 is the equilibrium mass density. The critical value λ J , Eq. (1), is derived by considering a small perturbation in a set of four equations that are the equation of continuity, the Euler's equation, the Poisson's equation and the equation of state of an ideal gas. The Jeans length establishes that if the wave length λ of density fluctuation is greater than λ J then the density will grow with time in an exponential form and the system will become gravitationally unstable. For more details see Jiulin in ref.[2]. Tsallis [3] has proposed an important extension of the Boltzman-Gibbs (BG) statistical theory. In a brief and technical terminology, this model is also currently referred to as a nonextensive (NE) statistical mechanics.Tsallis thermostatistics formalism defines a nonadditive entropy aswhere p i is the probability of the system to be in a microstate, W is the total number of configurations and q, known in the current literature as being the Tsallis * Electronic address: evertonabreu@ufrrj.br † Electronic address: jorge@fisica.ufjf.br ‡ Electronic address: edesiobarboza@uern.br § Electronic address: nunes@ecm.ub.edu parameter or NE parameter, is a real parameter which quantifies the degree of nonextensivity. The definition of entropy in Tsallis statistics carries the usual properties of positivity, equiprobability, concavity and irreversibility and it also has motivated the study of multifractals systems. It is important to stress that Tsallis thermostatistics formalism contains the BG statistics as a particular case in the limit q → 1 where the usual additivity of entropy is recovered. Plastino and Lima [4] have derived a NE equipartition law of energy. It has been shown that the kinetic foundations of Tsallis' NE statistics leads to a velocity distribution for free particles given by [5] where B q is a normalization constant. Then, the expectation value of v 2 , for each degree of freedom, is given byThe equipartition theorem is then obtained by using thatand we will arrive atThe range of q is 0 ≤ q < 5/3. For q = 5/3 (critical value) the expression of the equipartition law of energy, Eq. (6), diverges. It is also easy to observe that for q = 1, the classical equipartition theorem for each microscopic degrees of freedom can be recovered.

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“…Since Boltzmann distribution yields unphysical results, a generalized entropy, the Tsallis entropy, was used in [6] instead of Boltzmann entropy. The use of Kaniadakis entropy had been recently proposed too, in [7]. Here we will see that these two entropic measures are related in polytropic solutions, and that the result given in [7] can be easily obtained from [6].…”

Section: Introductionmentioning

confidence: 56%

“…The use of Kaniadakis entropy had been recently proposed too, in [7]. Here we will see that these two entropic measures are related in polytropic solutions, and that the result given in [7] can be easily obtained from [6]. After, following the same approach used for polytropes, we will see how the generalized logarithms and exponential functions, used in these two nonextensive methods, are linked.…”

Section: Introductionmentioning

confidence: 77%

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Tsallis and Kaniadakis Entropic Measures in Polytropic, Logarithmic and Exponential Functions

Sparavigna¹

2015

ijSciences

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Among nonextensive statistical approaches, those proposed by Constantino Tsallis and Giorgio Kaniadakis had been involved in the study of several physical phenomena. Here, we will discuss the case of the polytropic solutions of self-gravitating fluid spheres used in astrophysics as approximate stellar models. We will see how, in this problem, Tsallis and Kaniadakis entropic measures are related. After, following the same approach, we will find the links between generalized logarithms and exponential functions of the abovementioned entropic measures.

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Non-Gaussian statistics, Maxwellian derivation and stellar polytropes

Cited by 21 publications

References 39 publications

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

Jeans instability criterion from the viewpoint of Kaniadakis' statistics

Tsallis and Kaniadakis Entropic Measures in Polytropic, Logarithmic and Exponential Functions

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