A Brief Review of Generalized Entropies

A measure D [ t 1 , t 2 ] for the amount of dynamical evolution exhibited by a quantum system during a time interval [ t 1 , t 2 ] is defined in terms of how distinguishable from each other are, on average, the states of the system at different times. We investigate some properties of the measure D showing that, for increasing values of the interval’s duration, the measure quickly reaches an asymptotic value given by the linear entropy of the energy distribution associated with the system’s (pure) quantum state. This leads to the formulation of an entropic variational problem characterizing the quantum states that exhibit the largest amount of dynamical evolution under energy constraints given by the expectation value of the energy.

Section: Entropic Variational Approach To Quantum States Exhibiting Mmentioning

confidence: 99%

Entropic Characterization of Quantum States with Maximal Evolution under Given Energy Constraints

Majtey

Valdés-Hernández

Maglione

et al. 2019

“…Here, we focus our attention on the Rényi entropy by [37] and its twin sister: the q-entropy by [38]. For discussions and applications of these two entropy measures, we refer the reader to the survey of [39], and the references therein.…”

Section: Measures Of Entropymentioning

confidence: 99%

Statistical Inference of the Half-Logistic Inverse Rayleigh Distribution

Almarashi

Badr

Elgarhy

et al. 2020

The inverse Rayleigh distribution finds applications in many lifetime studies, but has not enough overall flexibility to model lifetime phenomena where moderately right-skewed or near symmetrical data are observed. This paper proposes a solution by introducing a new two-parameter extension of this distribution through the use of the half-logistic transformation. The first contribution is theoretical: we provide a comprehensive account of its mathematical properties, specifically stochastic ordering results, a general linear representation for the exponentiated probability density function, raw/inverted moments, incomplete moments, skewness, kurtosis, and entropy measures. Evidences show that the related model can accommodate the treatment of lifetime data with different right-skewed features, so far beyond the possibility of the former inverse Rayleigh model. We illustrate this aspect by exploring the statistical inference of the new model. Five classical different methods for the estimation of the model parameters are employed, with a simulation study comparing the numerical behavior of the different estimates. The estimation of entropy measures is also discussed numerically. Finally, two practical data sets are used as application to attest of the usefulness of the new model, with favorable goodness-of-fit results in comparison to three recent extended inverse Rayleigh models.Entropy 2020, 22, 449 2 of 24 respectively, where α is a scale parameter. As notable features, the IR distribution has tractable and simple probability functions, is unimodal and right-skewed, and possesses a hazard rate function with a singular curvature: it increases at a certain value, then decreases until attain a kind of stabilization. The pioneer studies are [2] which presents some properties of the maximum likelihood estimator of α, and [3] which provides closed-form expressions for the (standard) mean, harmonic mean, geometric mean, mode and median of the IR distribution. Also, among the amount of works investigating the statistical aspects of the IR distribution, the reader can be referred to [4][5][6][7][8][9][10][11][12].In the recent years, several extensions of the IR distribution were developed, using different mathematical techniques, often at the basis of general families of distributions. Among them, there are the beta IR (BIR) distribution by [13], transmuted IR (TIR) distribution by [14], modified IR (MIR) distribution by [15], transmuted modified IR (TMIR) distribution by [16], transmuted exponentiated IR (TEIR) distribution by [17], Kumaraswamy exponentiated IR (KEIR) distribution by [18], weighted IR (WIR) distribution by [19], odd Fréchet IR (OFIR) distribution by [20], type II Topp-Leone IR (TIITLIR) distribution by [21], type II Topp-Leone generalized IR (TIITLGIR) distribution by [22] and exponentiated IR (EIR) distribution by [23].However, to the best of our knowledge, the use of the half-logistic transformation to extend the IR distribution remains unexplored, despite recent success in this regard. This half-log...

“…In the limit q → 1 the standard Boltzmann-Gibbs entropy, S 1 = −k σ ln(σ/σ 0 )d N x, is recovered. The power-law entropy S q constitutes a distinguished member among the family of generalized entropies that are currently being intensively studied [40][41][42]. An essential ingredient of the S q thermostatistical formalism is given by the q-exponential function.…”

Section: S Q Entropies Q-exponentials and Q-gaussiansmentioning

confidence: 99%

“…Inserting (42) in (38), and following a procedure similar to the one adopted in our previous discussion of the solutions of the nonlinear Fokker-Planck equation, one can show that (41) satisfies the evolution Equation (38) if the parameters A and β evolve according to…”

Section: A Two-dimensional System With Particles Interacting Through mentioning

confidence: 99%

Nonlinear Fokker–Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions

Plastino

Wedemann

2020

Nonlinear Fokker–Planck equations (NLFPEs) constitute useful effective descriptions of some interacting many-body systems. Important instances of these nonlinear evolution equations are closely related to the thermostatistics based on the S q power-law entropic functionals. Most applications of the connection between the NLFPE and the S q entropies have focused on systems interacting through short-range forces. In the present contribution we re-visit the NLFPE approach to interacting systems in order to clarify the role played by the range of the interactions, and to explore the possibility of developing similar treatments for systems with long-range interactions, such as those corresponding to Newtonian gravitation. In particular, we consider a system of particles interacting via forces following the inverse square law and performing overdamped motion, that is described by a density obeying an integro-differential evolution equation that admits exact time-dependent solutions of the q-Gaussian form. These q-Gaussian solutions, which constitute a signature of S q -thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker–Planck equation.