A Shannon–Tsallis transformation

Via a first-order linear differential equation, we determine a general link between two different solutions of the MaxEnt variational problem, namely, the ones that correspond to using either Shannon’s or Tsallis’ entropies in the concomitant variational problem. It is shown that the two variations lead to equivalent solutions that have different appearances but contain the same information. These solutions are linked by our transformation. However, the so-called collision entropy (Tsallis’ one with q=2$q=2$) does not have a Shannon counterpart.     

Correlation and entanglement of a three-level atom inside a dissipative cavity

We discuss the correlation and entanglement of a three-level atom with a single-mode quantized field in a coherent state inside a phase-damped cavity. We analyze the influence of dissipation on the quantum and classical entropy. It has been shown that the quantum, classical and nonextensive entropy are sensitive to any change in the initial state setting of the atom and the quantized field. The relation between the long lived entanglement and dissipation is observed. On the other hand, a short disentanglement can be generated through special values of the atomic motion parameter.     

Information and complexity measures in the interface of a metal and a superconductor

Fisher information, Shannon information entropy and Statistical Complexity are calculated for the interface of a normal metal and a superconductor, as a function of the temperature for several materials. The order parameter $\mathrm{\Psi }(\mathbf{r})$ derived from the Ginzburg–Landau theory is used as an input together with experimental values of critical transition temperature $T_c$ and the superconducting coherence length ${\xi }_0$. Analytical expressions are obtained for information and complexity measures. Thus $T_c$ is directly related in a simple way with disorder and complexity. An analytical relation is found of the Fisher Information with the energy profile of superconductivity i.e. the ratio of surface free energy and the bulk free energy. We verify that a simple relation holds between Shannon and Fisher information i.e. a decomposition of a global information quantity (Shannon) in terms of two local ones (Fisher information), previously derived and verified for atoms and molecules by Liu et al. Finally, we find analytical expressions for generalized information measures like the Tsallis entropy and Fisher information. We conclude that the proper value of the non-extensivity parameter $q?1$, in agreement with previous work using a different model, where $q?1.005$.     

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.     

Generalized ICM for image segmentation based on Tsallis statistics

In this paper, the iterated conditional modes optimization method of a Markov random field technique for image segmentation is generalized based on Tsallis statistics. It is observed that, for some q$q$ entropic index values the new algorithm performs better segmentation than the classical one. The proposed algorithm also does not have a local minimum problem and reaches a global minimum energy point although the number of iterations remains the same as ICM. Based on the findings of the new algorithm, it can be expressed that the new technique can be used for the image segmentation processes in which the objects are Gaussian or nearly Gaussian distributed.     

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.     

Distributivity and deformation of the reals from Tsallis entropy

We propose a one-parameter family R_q of deformations of the reals, which is motivated by the generalized additivity of the Tsallis entropy. We introduce a generalized multiplication which is distributive with respect to the generalized addition of the Tsallis entropy. These operations establish a one-parameter family of field isomorphisms τ_q between R and R_q through which an absolute value on R_q is introduced. This turns out to be a quasisymmetric map, whose metric and measure-theoretical implications are pointed out.     

Shannon and Fisher entropies for a hydrogen atom under soft spherical confinement

We calculate Shannon and Fisher entropies in the position and momentum space, and some complexity measures for a variationally described hydrogen atom confined in soft and hard spherical boxes of varying dimension $r_c$ and selected values of strength $U_0$. We include calculations for a free particle trapped in impenetrable boxes. It is found that the Shannon entropy $S_r$ becomes negative for small cavity radii and large values of $U_0$, due to the highly localized nature of the particle. For soft confinement and small cavity dimensions, the entropies change very rapidly over short radial intervals.     

On generalized entropy measures and pathways

Product probability property, known in the literature as statistical independence, is examined first. Then generalized entropies are introduced, all of which give generalizations to Shannon entropy. It is shown that the nature of the recursivity postulate automatically determines the logarithmic functional form for Shannon entropy. Due to the logarithmic nature, Shannon entropy naturally gives rise to additivity, when applied to situations having product probability property. It is argued that the natural process is non-additivity, important, for example, in statistical mechanics [C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988) 479-487; E.G.D. Cohen, Boltzmann and Einstein: statistics and dynamics—an unsolved problem, Pramana 64 (2005) 635-643.], even in product probability property situations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function. Generalized entropies are introduced and some of their properties are examined. Situations are examined where a generalized entropy of order α leads to pathway models, exponential and power law behavior and related differential equations. Connection of this entropy to Kerridge's measure of “inaccuracy” is also explored.     

Non-extensive statistical entropy,quantum groups and quantum entanglement

In this work, we explore a new connection between quantum groups and Tsallis entropy through the energy spectrum of a Hamiltonian with S_Uq(2)$S{U}_q\left(2\right)$ symmetry. Identifying the deformation parameter of the entropy with the parameter of deformation of the associated quantum group, we deduce Tsallis entropy for states related to such a system with S_Uq(2)$S{U}_q\left(2\right)$ symmetry and conducted an investigation of quantum entanglement.     

Generalized entropies from first principles

A derivation of power law canonical distributions from first principle statistical mechanics, including the exponential distribution as a particular case is presented. It is shown that these distributions arise naturally, and that the heat capacity of the heat bath is the condition that determines its type. As a consequence, a physical interpretation for the parameter q of the generalized entropy is given.     

Dynamic statistical information theory

In recent years we extended Shannon static statistical information theory to dynamic processes and established a Shannon dynamic statistical information theory, whose core is the evolution law of dynamic entropy and dynamic information. We also proposed a corresponding Boltzmman dynamic statistical information theory. Based on the fact that the state variable evolution equation of respective dynamic systems, i.e. Fokker-Planck equation and Liouville diffusion equation can be regarded as their information symbol evolution equation, we derived the nonlinear evolution equations of Shannon dynamic entropy density and dynamic information density and the nonlinear evolution equations of Boltzmann dynamic entropy density and dynamic information density, that describe respectively the evolution law of dynamic entropy and dynamic information. The evolution equations of these two kinds of dynamic entropies and dynamic informations show in unison that the time rate of change of dynamic entropy densities is caused by their drift, diffusion and production in state variable space inside the systems and coordinate space in the transmission processes; and that the time rate of change of dynamic information densities originates from their drift, diffusion and dissipation in state variable space inside the systems and coordinate space in the transmission processes. Entropy and information have been combined with the state and its law of motion of the systems. Furthermore we presented the formulas of two kinds of entropy production rates and information dissipation rates, the expressions of two kinds of drift information flows and diffusion information flows. We proved that two kinds of information dissipation rates (or the decrease rates of the total information) were equal to their corresponding entropy production rates (or the increase rates of the total entropy) in the same dynamic system. We obtained the formulas of two kinds of dynamic mutual informations and dynamic channel capacities reflecting the dynamic dissipation characteristics in the transmission processes, which change into their maximum—the present static mutual information and static channel capacity under the limit case where the proportion of channel length to information transmission rate approaches to zero. All these unified and rigorous theoretical formulas and results are derived from the evolution equations of dynamic information and dynamic entropy without adding any extra assumption. In this review, we give an overview on the above main ideas, methods and results, and discuss the similarity and difference between two kinds of dynamic statistical information theories.