Real and spurious contributions for the Shannon, Rényi and Tsallis entropies

A two-parameter probability distribution is constructed by dilatation (or contraction) of the escort probability distribution. This transformation involves a physical probability distribution P associated with the system under study and an almost arbitrary reference probability distribution P^′. In contrast to the Shannon and Rényi entropies, the Tsallis entropy does not decompose as the sum of the physical contribution due to P and the reference or spurious part owing to P^′. For solving this problem, a slight modification to the relation between Tsallis and Rényi entropies must be introduced. The procedure in this paper gives rise to a nonconventional one-parameter Shannon entropy and to two-parameter Rényi and Tsallis entropies associated with P. It also contributes to clarify the meaning and role of the escort probabilities set.     

Quasi-stationary states of the NRT nonlinear Schrödinger equation

With regards to the nonlinear Schrödinger equation recently advanced by Nobre, Rego-Monteiro, and Tsallis (NRT), based on Tsallis q$q$-thermo-statistical formalism, we investigate the existence and properties of its quasi-stationary solutions, which have the time and space dependences “separated” in a q$q$-deformed fashion. One recovers the normal factorization into purely spatial and purely temporal factors, corresponding to the standard, linear Schrödinger equation, when the deformation vanishes (q=1)$(q=1)$. We discuss various specific examples of exact, quasi-stationary solutions of the NRT equation. In particular, we obtain a quasi-stationary solution for the Moshinsky model, providing the first example of an exact solution of the NRT equation for a system of interacting particles.     

Nonextensivity and entropy of astrophysical sources

We study the X-ray intensities of 142 light curves of cataclysmic variables, galaxies, pulsars, supernova remnants and other X-ray sources present in the public data collected by the instrument All Sky Monitor on board the satellite Rossi X-ray Timing Explorer. We show that the X-ray light curves coming from astrophysical systems obey Tsallis’s q$q$-Gaussian distribution as probability density. This fact strongly suggests that these astrophysical systems behave in a non-extensive manner. Furthermore, the q$q$ entropic indices for these systems were obtained and they provide an indication of the nonextensivity degree of each of these astrophysical systems. The q$q$-value increases for systems if the Tsallis entropy decreases.     

On Entropic Uncertainty Relations for Measurements of Energy and Its “Complement”

Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.     

Sufficient Conditions of the Same State Order Induced by Coherence

In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and l₁ norm of coherence, and give the sufficient conditions of the same state order induced by above coherence measures. First, we show that the above measures give the same ordering for single-qubit states in some special cases. Second, we consider some special states in a d-dimensional quantum system. We show that the above measures generate the same ordering for these special states. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find amplitude damping channel changes the coherence-induced ordering even though for single-qubit states with fixed mixedness.     

The Tsallis-complexity of a semiclassical time-evolution

An investigation is undertaken of semiclassical time-evolutions and their classical limit with the intent of getting insights into the classical–quantum frontier. We deal with a system that represents the interaction between matter and a given field, and our main research tool is the so-called q$q$-complexity quantifier, for which two different versions are considered. The probability distribution associated with the time-evolution process is determined by recourse to the Bandt–Pompe symbolic technique [C. Bandt, B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88 (2002) 174102:1–174102:4]. The most salient details of the quantum–classical transition turn out to be described not only well, but also in a better fashion than that of previous literature.     

The role of propagating stress waves on a geophysical scale: Evidence in terms of nonextensivity

Laboratory experiments have shown that, during a fracture, the breaking of a bond launches a propagating stress wave which may trigger the breaking of other bonds. We examine here the possibility that the same holds on a geophysical scale. Based on a nonextensive approach, we examine whether the transient stresses of seismic waves from a major earthquake (EQ) can trigger a considerably distant significant EQ. We use three different analytical approaches: (i) a recently introduced fragment-asperity interaction model for EQ dynamics based on nonextensive Tsallis statistics; (ii) the Hurst exponent; (iii) organization in terms of Fisher information. We find that the triggered seismicity displays higher nonextensivity, increased persistent behavior, and a higher level of organization. Using the same approaches, we further elucidate the link between the associated precursory kHz electromagnetic (EM) activity and the last stage of the impending EQ generation. We examine whether the statistics of regional seismicity could be a macroscopic reflection of physical processes in the EQ source, as would be expected by the fractal nature of fracture and faulting.     

Self-consistency in non-extensive thermodynamics of highly excited hadronic states

The self-consistency of a thermodynamical theory for hadronic systems based on the non-extensive statistics is investigated. We show that it is possible to obtain a self-consistent theory according to the asymptotic bootstrap principle if the mass spectrum and the energy density increase q-exponentially. A direct consequence is the existence of a limiting effective temperature for the hadronic system. We show that this result is in agreement with experiments.     

Information Shift Dynamics Described by Tsallis q = 3 Entropy on a Compact Phase Space

Recent mathematical investigations have shown that under very general conditions, exponential mixing implies the Bernoulli property. As a concrete example of statistical mechanics that are exponentially mixing we consider the Bernoulli shift dynamics by Chebyshev maps of arbitrary order N≥2, which maximizes Tsallis q=3 entropy rather than the ordinary q=1 Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd N. We show that the value of the fine structure constant αel=1/137 is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for N=3.     

On the fractal dimension of orbits compatible with Tsallis statistics

In a previous paper [A. Carati, Physica A 348 (2005) 110-120] it was shown how, for a dynamical system, the probability distribution function of sojourn-times in phase-space, defined in terms of the dynamical orbits (up to a given observation time), induces unambiguously a statistical ensemble in phase-space. In the present paper, the p.d.f. of the sojourn-times corresponding to a Tsallis ensemble is obtained (this, by the way, requires the solution of a problem of a general character, disregarded in paper [A. Carati, Physica A 348 (2005) 110-120]). In particular some qualitative properties, such as the fractal dimension, of the dynamical orbits compatible with the Tsallis ensembles are indicated.