Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells

A comparative analysis of the Dirichlet and Neumann boundary conditions (BCs) of the one-dimensional (1D) quantum well extracts similarities and differences of the Rényi R(α) as well as Tsallis T(α) entropies between these two geometries. It is shown, in particular, that for either BC the dependences of the Rényi position components on the parameter α are the same for all orbitals but the lowest Neumann one, for which the corresponding functional R is not influenced by the variation of α. Lower limit α _TH of the semi-infinite range of the dimensionless Rényi/Tsallis coefficient where momentum entropies exist crucially depends on the position BC and is equal to 1/4 for the Dirichlet requirement and 1/2 for the Neumann one. At α approaching this critical value, the corresponding momentum functionals do diverge. The gap between the thresholds α _TH of the two BCs causes different behavior of the Rényi uncertainty relations as functions of α. For both configurations, the lowest energy level at α = 1/2 does saturate either type of the entropic inequality, thus confirming an earlier surmise about it. It is also conjectured that the threshold α _TH of ½ is characteristic of any 1D non-Dirichlet system. Other properties are discussed and analyzed from the mathematical and physical points of view.     

Quantum vibrational partition function in the non-extensive Tsallis framework

The quantum vibrational partition function has been obtained in the Tsallis statistics framework for the entropic index, q, between 1 and 2. The effect of non-extensivity on the population of states and thermodynamic properties have been studied and compared with their corresponding values obtained in the Boltzmann-Gibbs (BG) statistics. Our results show that the non-extensive partition function of harmonic oscillator at any temperature is larger than its corresponding values for an extensive system and that their differences increase with temperature and entropic index. Also, the number of accessible states increases with q but, compared to the BG statistics, the occupation number decreases for low energy levels while the population of the higher energy levels increases. The internal energy and heat capacity have also been obtained for the non-extensive harmonic oscillator system. Results indicate that the heat capacity is greater than its corresponding value in the extensive (BG) system at low temperatures but that this trend is reversed at higher temperatures.     

Approach on Tsallis statistical interpretation of hydrogen-atom by adopting the generalized radial distribution function

This paper revisits the statistical interpretation of the hydrogen atom within the framework of Tsallis Statistical Mechanics in the Canonical Ensemble. The convergence of the partition function does not exhibit for all the temperatures, while the well-known T → T′ transformation method of Tsallis Statistics fails, since non-monotonicity is observed between the ordinary temperature, T, and the auxiliary one, T′. Here we re-examine the inconsistency of T → T′ transformation method, in the case where the partition function converges for all the temperatures, by considering the generalized radial distribution function. We find that both the transformation method inconsistency and the partition function divergence can be recovered for all the temperatures, if the hydrogen atom is restricted within a critical radius R _c  ≤ 4.832 bohr, while Tsallis entropic index values are given by . An erratum to this article can be found at     

Hierarchical Polygamy Inequality for Entanglement of Tsallis q-Entropy

In this paper, we study the polygamy inequality of quantum entanglement in terms of Tsallis q-entropy.We first give a lower bound of Tsallis q-entropy entanglement of assistance(TOA) in the 2  d systems. The relationships between Tsallis q-entropy entanglement(TEE) and TOA are also given. Furthermore, we prove TOA follows a hierarchical polygamy inequality in a 2  2  2~(N-2) systems.     

Quasi-additivity of Tsallis entropies and correlated subsystems

We use Beck's quasi-additivity of Tsallis entropies for n   independent subsystems to show that like the case of n=2$n=2$, the entropic index q$q$ approaches 1 by increasing system size. Then, we will generalize that concept to correlated subsystems to find that in the case of correlated subsystems, when system size increases, q$q$ also approaches a value corresponding to the additive case.     

Uncertainty and Certainty Relations for Successive Projective Measurements of a Qubit in Terms of Tsallis' Entropies

We study uncertainty and certainty relations for two successive measurements of two-dimensional observables.Uncertainties in successive measurement are considered within the following two scenarios.In the first scenario,the second measurement is performed on the quantum state generated after the first measurement with completely erased information.In the second scenario,the second measurement is performed on the post-first-measurement state conditioned on the actual measurement outcome.Induced quantum uncertainties are characterized by means of the Tsallis entropies.For two successive projective measurement of a qubit,we obtain minimal and maximal values of related entropic measures of induced uncertainties.Some conclusions found in the second scenario are extended to arbitrary finite dimensionality.In particular,a connection with mutual unbiasedness is emphasized.     

A Two-Parameter Fractional Tsallis Decision Tree

Decision trees are decision support data mining tools that create, as the name suggests, a tree-like model. The classical C4.5 decision tree, based on the Shannon entropy, is a simple algorithm to calculate the gain ratio and then split the attributes based on this entropy measure. Tsallis and Renyi entropies (instead of Shannon) can be employed to generate a decision tree with better results. In practice, the entropic index parameter of these entropies is tuned to outperform the classical decision trees. However, this process is carried out by testing a range of values for a given database, which is time-consuming and unfeasible for massive data. This paper introduces a decision tree based on a two-parameter fractional Tsallis entropy. We propose a constructionist approach to the representation of databases as complex networks that enable us an efficient computation of the parameters of this entropy using the box-covering algorithm and renormalization of the complex network. The experimental results support the conclusion that the two-parameter fractional Tsallis entropy is a more sensitive measure than parametric Renyi, Tsallis, and Gini index precedents for a decision tree classifier.     

Kaniadakis Entropy Leads to Particle–Hole Symmetric Distribution

We discuss generalized exponentials, whose inverse functions are at the core of generalized entropy formulas, with respect to particle–hole (KMS) symmetry. The latter is fundamental in field theory; so, possible statistical generalizations of the Boltzmann formula-based thermal field theory have to take this property into account. We demonstrate that Kaniadakis’ approach is KMS ready and discuss possible further generalizations.     

Stochastic and Self-Organisation Patterns in a 17-Year PM10 Time Series in Athens,Greece

This paper utilises statistical and entropy methods for the investigation of a 17-year PM₁₀ time series recorded from five stations in Athens, Greece, in order to delineate existing stochastic and self-organisation trends. Stochastic patterns are analysed via lumping and sliding, in windows of various lengths. Decreasing trends are found between Windows 1 and 3500–4000, for all stations. Self-organisation is studied through Boltzmann and Tsallis entropy via sliding and symbolic dynamics in selected parts. Several values are below −2 (Boltzmann entropy) and 1.18 (Tsallis entropy) over the Boltzmann constant. A published method is utilised to locate areas for which the PM₁₀ system is out of stochastic behaviour and, simultaneously, exhibits critical self-organised tendencies. Sixty-six two-month windows are found for various dates. From these, nine are common to at least three different stations. Combining previous publications, two areas are non-stochastic and exhibit, simultaneously, fractal, long-memory and self-organisation patterns through a combination of 15 different fractal and SOC analysis techniques. In these areas, block-entropy (range 0.650–2.924) is significantly lower compared to the remaining areas of non-stochastic but self-organisation trends. It is the first time to utilise entropy analysis for PM₁₀ series and, importantly, in combination with results from previously published fractal methods.Data Set License: license under which the dataset is made available (CC0, CC-BY, CC-BY-SA, CC-BY-NC, etc.)     

Non-Extensive Statistical Analysis of Acoustic Emissions: The Variability of Entropic Index q during Loading of Brittle Materials Until Fracture

Non-extensive statistical mechanics (NESM), introduced by Tsallis based on the principle of non-additive entropy, is a generalisation of the Boltzmann–Gibbs statistics. NESM has been shown to provide the necessary theoretical and analytical implementation for studying complex systems such as the fracture mechanisms and crack evolution processes that occur in mechanically loaded specimens of brittle materials. In the current work, acoustic emission (AE) data recorded when marble and cement mortar specimens were subjected to three distinct loading protocols until fracture, are discussed in the context of NESM. The NESM analysis showed that the cumulative distribution functions of the AE interevent times (i.e., the time interval between successive AE hits) follow a q-exponential function. For each examined specimen, the corresponding Tsallis entropic q-indices and the parameters β_q and τq were calculated. The entropic index q shows a systematic behaviour strongly related to the various stages of the implemented loading protocols for all the examined specimens. Results seem to support the idea of using the entropic index q as a potential pre-failure indicator for the impending catastrophic fracture of the mechanically loaded specimens.