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A Utility-Based Approach to Information Theory
Published in Craig Friedman, Sven Sandow, Utility-Based Learning from Data, 2016
The Tsallis entropy (see Tsallis (1988)) provides a theoretical framework for a nonextensive thermodynamics, i.e., for a thermodynamics for which the entropy of a system made up of two independent subsystems is not simply the sum of the entropies of the subsystems. Examples for physical application of such a theory might be astronomical self-gravitating systems (see, for example, Plastino and Plastino (1999)).
Bivariate Cumulative Tsallis Past Entropy: Properties and Applications
Published in American Journal of Mathematical and Management Sciences, 2023
David Chris Raju, S. M. Sunoj, G. Rajesh
There have been several attempts to generalize Shannon’s entropy based on modifying underlying axioms, introducing additional parameters, different physical conditions, new functional forms, etc. Among these, an important generalized measure is due to Tsallis (1988), introduced in the context of equilibrium thermodynamics. If a system is out of equilibrium or its component states depend strongly on one another, then Tsallis entropy or non-additive entropy is found to be a better measure of uncertainty than (1). Various applications of Tsallis entropy include fluctuations of the magnetic field in the solar wind, signs of breast cancer in mammograms, cold atoms in optical lattices, MRI scanning, and so on (see Tsallis, 1988; Cartwright, 2014; Raju et al., 2020). For a continuous random variable X, the Tsallis entropy is defined as, Similar to (2), Sati and Gupta (2015) introduced a cumulative form of (4) and later extended it to the bivariate case by Sati and Singh (2017). However, Rajesh and Sunoj (2019) introduced an alternative form to the cumulative Tsallis entropy of Sati and Gupta (2015), given by
The nonadditive entropy for the ground state of helium-like ions using Hellmann potential
Published in Molecular Physics, 2020
I. Nasser, Mostafa Zeama, Afaf Abdel-Hady
For a long time, the extensive and additive properties of the entropy were the corner stones for the study and calculation of the behaviour of many physical phenomena [1]. But, about 3 decades ago, a nonadditive entropy, known as the ‘Tsallis entropy ’ was first proposed by Tsallis himself [2]. Accordingly, the concept that related with nonadditive statistical mechanics has found applications in a variety of disciplines including physics, fluids, informatics, and linguistics, among others [3,4]. We refer to Ref. [5] for recent, more historical details, and applications. The Tsallis entropy (TSE) is claimed to be useful in cases where there are strong correlations between the different microstates in a system. is defined as the q-generalised entropy, in the form: where and are the Tsallis index and the electron states’ density, respectively, and is the volume element. The entropy satisfies the following property: if and are two probabilistically independent systems, then for the combined system satisfies the pseudoadditivity relation:
Influence of dust-neutral collisional frequency and nonextensivity on dynamic motion of dust-acoustic waves
Published in Waves in Random and Complex Media, 2021
It has been examined that for the systems occurring in long-term communications, prolong-time series, section of the entity fractal of the associating space-time, is not irresistible throughout the conventional Boltzmann-Gibbs statistics [28,29]. The additive or momentous formalism is the major source for this disruption. In order to improve the statistical features of the systems involving long-range correlations, Tsallis [30] monotonously elongated BG thermodynamics by postulating the theory of entropy to nonextensive influences. The entropy of of two independent systems P and Q is given by , where q gives the measure of its interaction and it frames the basis for the generalized Tsallis entropy that is correlated with the main dynamics of the composition. The comprehensive classifications of physical structures which are related to Tsallis entropy are observed in plasma dynamics [31,32], Hamiltonian structures [33] with long-term communications, and gravitational systems [34]. Moreover, an investigation done by Lima et al. [31] reported that in astrophysical and plasma environments, the nonextensive parameter (q) is important for systems with long-term communications. Furthermore, Lima et al. [31] and Liu et al. [35] acknowledged that plasma observational specification leads to non-Maxwellian velocity profile. Therefore, it is significant to follow that the Tsallis q-entropy and the advanced conventional statistics were promoted with appreciable progress for the diversified plasma environments [36–42]. Considering non-Maxwellian distributed electrons, Gill et al. [43,44] investigated nonlinear solitary waves and double layers in non-Maxwellian plasmas. Hafez et al. [45,46] studied nonlinear solitary waves with q-nonextensive velocity distributed electrons. Furthermore, Aly et al. [47,48] studied the stability analysis for Zakharov–Kuznetsov and three-dimensional modified KdV-ZK equations for nonlinear features in plasmas.