Some New Results of Residual and Past Entropy Measures

In this paper, two new generalized entropies have been introduced with their respective properties. The results of these entropies have been verified for the exponential and weighted exponential distributions. These two entropies produce the results in the form of simple entropy, generalized entropy, residual entropy, cumulative entropy and mixtures of all these entropies. Some characteristics of residual & past entropy have been derived and special cases have also been obtained. These cases indicate that new generalized entropies are more comprehensive and useful. The main advantage of this study is to derive different types of generalization of entropies using the different parameter values of α and β.

Generalized entropies, density of states, and non-extensivity

The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. Independent or weakly interacting variables render the number of configurations scale exponentially with the number of variables, making the Boltzmann–Gibbs–Shannon entropy extensive. In systems with strongly interacting variables, or with variables driven by history-dependent dynamics, this is no longer true. Here we show that contrary to the generally held belief, not only strong correlations or history-dependence, but skewed-enough distribution of visiting probabilities, that is, first-order statistics, also play a role in determining the relation between configuration space size and system size, or, equivalently, the extensive form of generalized entropy. We present a macroscopic formalism describing this interplay between first-order statistics, higher-order statistics, and configuration space growth. We demonstrate that knowing any two strongly restricts the possibilities of the third. We believe that this unified macroscopic picture of emergent degrees of freedom constraining mechanisms provides a step towards finding order in the zoo of strongly interacting complex systems.

Generalized entropies and corresponding holographic dark energy models

Using Tsallis statistics and its relation with Boltzmann entropy, the Tsallis entropy content of black holes is achieved, a result in full agreement with a recent study (Mejrhit and Ennadifi in Phys Lett B 794:24, 2019). In addition, employing Kaniadakis statistics and its relation with that of Tsallis, the Kaniadakis entropy of black holes is obtained. The Sharma-Mittal and Rényi entropy contents of black holes are also addressed by employing their relations with Tsallis entropy. Thereinafter, relying on the holographic dark energy hypothesis and the obtained entropies, two new holographic dark energy models are introduced and their implications on the dynamics of a flat FRW universe are studied when there is also a pressureless fluid in background. In our setup, the apparent horizon is considered as the IR cutoff, and there is not any mutual interaction between the cosmic fluids. The results indicate that the obtained cosmological models have (i) notable powers to describe the cosmic evolution from the matter-dominated era to the current accelerating universe, and (ii) suitable predictions for the universe age.

The Python’s Lunch: geometric obstructions to decoding Hawking radiation

According to Harlow and Hayden [arXiv:1301.4504] the task of distilling information out of Hawking radiation appears to be computationally hard despite the fact that the quantum state of the black hole and its radiation is relatively un-complex. We trace this computational difficulty to a geometric obstruction in the Einstein-Rosen bridge connecting the black hole and its radiation. Inspired by tensor network models, we conjecture a precise formula relating the computational hardness of distilling information to geometric properties of the wormhole — specifically to the exponential of the difference in generalized entropies between the two non-minimal quantum extremal surfaces that constitute the obstruction. Due to its shape, we call this obstruction the ‘Python’s Lunch’, in analogy to the reptile’s postprandial bulge.

From Complexity to Information Geometry and Beyond

Complex Systems are ubiquitous in nature and man-made systems. In natural sciences, in social and economic models and in mathematical constructions are studied and analyzed, are applied in practical problems but without a clear and universal definition of "complexity", let alone classification and quantification. Following the "three-level scheme" of physical theories, observations/experiments, phenomenology, microscopic interactions, we need, starting from the experience of observation to establish appropriate phenomenological parameters and concepts, and in conjunction with a possible knowledge of the nature of microscopic structures to deepen our understanding of a particular system which we "understand as complex". Information Geometry seems to be a useful phenomenological framework, which using generalized entropies, provides some classification and quantification tools. But we need the next level, microscopic structure and interactions of the parts of complex systems. A useful direction is the conceptual niche of hyper-networks and super graphs, where a strong involvement of algebra offers concrete techniques. We believe that appropriate algebraic structures may systematize our approach to microscopic structures of complex systems, and help associate the information geometric phenomenology with concrete properties. In this paper after a short discussion of the problem of "definition of complexity", we introduce our information geometric quantities derived from generalized entropies. Then we present our results of application of information geometry for classification of complex systems. Finally we present our ideas for an abstract algebraic approach which may offer a framework for the microscopic study of complex systems.

Generalized Entropies, Variance and Applications

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy.

A functional equation related to generalized entropies and the modular group

We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies. These transformations generate the modular group, and this fact plays a crucial role in solving the system. The method suggests a more general relation between conditional probabilities and arithmetic.

Entropic Analysis of the Quantum Oscillator with a Minimal Length

The well-known Heisenberg–Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [ X , P ] = i ℏ ( 1 + β P 2 ) implies the existence of a minimal length proportional to β . The Bialynicki-Birula–Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β . Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.