Exploring the Neighborhood of q-Exponentials

Lima, Henrique Santos; Tsallis, Constantino

doi:10.3390/e22121402

Entropy

2020

DOI: 10.3390/e22121402

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Exploring the Neighborhood of q-Exponentials

Henrique Santos Lima

Constantino Tsallis

Abstract: The q-exponential form eqx≡[1+(1−q)x]1/(1−q)(e1x=ex) is obtained by optimizing the nonadditive entropy Sq≡k1−∑ipiqq−1 (with S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann–Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms. We first discuss departures from q… Show more

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Towards a physical interpretation of the deformation parametrization in nonextensive statistics

Silveira

Benetti

2021

Eur. Phys. J. Plus

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We explore new ways to interpret the role played by the deformation q-parameter in nonextensive statistics. A generalized polytropic γ -index is deduced with basis on the Tsallis distribution. In the limit q → 1/3, it is shown that the γ -index decreases hyperbolically with the increase of the concentration n of charged particles in a nonextensive gas. An equation of state of a nonextensive system is derived following the generalized polytropic index. In the limit q → 1/3, it is found that a constant and uniform isotropic pressure develops throughout a nonextensive gas of charged particles in the absence of electric and magnetic fields, and in a stationary state of equilibrium of the system. The usual reduction of the Tsallis to Kappa distributions is examined with basis on their corresponding equations of state. It is shown that such a procedure leads to a general proof of the relationship between the qparameter and spectral κ-index, and between the T -Maxwellian and -Kappa temperatures. A generalization of the number of degrees of freedom in a nonextensive gas is provided. It is suggested that a nonextensive polytropic process might characterize a system that shall be something between a monoatomic gas with more than three translational degrees of freedom, and a diatomic gas with less than three translational plus two rotational degrees of freedom. Moreover, it is proved that the restriction of the Tsallis to Kappa distributions requires that the bulk concentration n in a nonextensive gas does not exceed 50% of the concentration n 0 on its boundary, thereby characterizing a low-density system.

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Towards a physical interpretation of the deformation parametrization in nonextensive statistics

Silveira

Benetti

2021

Eur. Phys. J. Plus

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Entropy

Tsallis

2022

Encyclopedia

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The concept of entropy constitutes, together with energy, a cornerstone of contemporary physics and related areas. It was originally introduced by Clausius in 1865 along abstract lines focusing on thermodynamical irreversibility of macroscopic physical processes. In the next decade, Boltzmann made the genius connection—further developed by Gibbs—of the entropy with the microscopic world, which led to the formulation of a new and impressively successful physical theory, thereafter named statistical mechanics. The extension to quantum mechanical systems was formalized by von Neumann in 1927, and the connections with the theory of communications and, more widely, with the theory of information were respectively introduced by Shannon in 1948 and Jaynes in 1957. Since then, over fifty new entropic functionals emerged in the scientific and technological literature. The most popular among them are the additive Renyi one introduced in 1961, and the nonadditive one introduced in 1988 as a basis for the generalization of the Boltzmann–Gibbs and related equilibrium and nonequilibrium theories, focusing on natural, artificial and social complex systems. Along such lines, theoretical, experimental, observational and computational efforts, and their connections to nonlinear dynamical systems and the theory of probabilities, are currently under progress. Illustrative applications, in physics and elsewhere, of these recent developments are briefly described in the present synopsis.

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Exploring the Neighborhood of q-Exponentials

Cited by 2 publications

References 30 publications

Towards a physical interpretation of the deformation parametrization in nonextensive statistics

Towards a physical interpretation of the deformation parametrization in nonextensive statistics

Entropy

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