Classical Entropy Research Articles

Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution

We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

Fast hybrid dimensionality reduction method for classification based on feature selection and grouped feature extraction

Dimensionality reduction is one basic and critical technology for data mining, especially in current “big data” era. As two different types of methods, feature selection and feature extraction each have their pros and cons. In this paper, we combine multi-strategy feature selection and grouped feature extraction and propose a novel fast hybrid dimension reduction method, incorporating their advantages of removing irrelevant and redundant information. Firstly, the intrinsic dimensionality of the data set is estimated by the maximum likelihood estimation method. Fisher Score and Information Gain based feature selection are used as multi-strategy methods to remove irrelevant features. With the redundancy among the selected features as clustering criterion, they are grouped into a certain amount of clusters. In every cluster, Principal Component Analysis (PCA) based feature extraction is carried out to remove redundant information. Four classical classifiers and representation entropy are used to evaluate the classification performance and information loss of the reduced set. The runtime results of different methods show that the proposed hybrid method is consistently much faster than the other three in almost all of the sets used. Meanwhile, the proposed method shows competitive classification performance, which has no significant difference basically compared with the other methods. The proposed method reduces the dimensionality of the raw data fast and it has excellent efficiency and competitive classification performance compared with the contrastive methods.

The fundamental connections between classical Hamiltonian mechanics, quantum mechanics and information entropy

We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.

Logarithmic terms in entropy of Schwarzschild black holes in higher loops

The Bekenstein-Hawking (BH) entropy is expected to be modified by certain correction terms in the quantum loop expansion. As is well known the logarithmic terms in the entropy of black holes appear as a one-loop addition to the classical BH entropy. In this note we study the further modifications of the logarithmic terms in the entropy of the Schwarzschild black holes due to higher quantum loops: up to three loops in a general renormalisable theory of gauge fields, scalars and fermions and two loops in quantum gravity. For a large class of field multiplets (including that of the Standard Model) that include graviton and for a certain range in the values of the couplings these modifications manifest themselves in cooling down the black holes at later stages of evaporation and, respectively, in increasing the life time of the black hole. If this picture persists to even higher loops, then the small black holes formed in the early stages of the cosmic evolution do not evaporate completely by now as is predicted in the standard picture. Instead, their long-lived (Planckian mass) remnants are present in abundance in today's Universe.

Optimal uniform continuity bound for conditional entropy of classical–quantum states

In this short note, I show how a recent result of Alhejji and Smith (A tight uniform continuity bound for equivocation, 2019. arXiv:1909.00787v1) regarding an optimal uniform continuity bound for classical conditional entropy leads to an optimal uniform continuity bound for quantum conditional entropy of classical–quantum states. The bound is optimal in the sense that there always exists a pair of classical–quantum states saturating the bound, and so, no further improvements are possible. An immediate application is a uniform continuity bound for the entanglement of formation that improves upon the one previously given by Winter (Commun Math Phys 347(1):291–313, 2016. arXiv:1507.07775). Two intriguing open questions are raised regarding other possible uniform continuity bounds for conditional entropy: one about quantum–classical states and another about fully quantum bipartite states.

Modified Entropy Estimator Using a Haar Matrix to Fit the Weibull Growth Model

It is well known that the Generalized Maximum Entropy method can be used to fit linear regression models, especially as they are not restricted by the conditions to be verified as are other classical methods. Therefore, in this paper, a new method for estimating the parameters of the four-parameter Weibull growth model was proposed using the Generalized Maximum Entropy function by fitting data based on the Haar matrix which was used in the wavelet method. The suggested and classical entropy estimators for Weibull growth model parameters were compared using simulation and the preference for the suggested method estimator was shown. The Modified Generalized Maximum Entropy estimator was applied to the real data representing annual Iraqi oil production for the period 2010–2017. Iraqi crude oil production for the year 2022 was predicted and appeared as 4.4 million bb/day.

Kinetic Correlation Functionals from the Entropic Regularization of the Strictly Correlated Electrons Problem.

In this work, we study the entropic regularization of the strictly correlated electrons formalism, discussing the implications for density functional theory and establishing a link with earlier works on quantum kinetic energy and classical entropy. We carry out a very preliminary investigation (using simplified models) on the use of the solution of the entropic regularized problem to build approximations for the kinetic correlation functional at large coupling strengths. We also analyze lower and upper bounds to the Hohenberg–Kohn functional using the entropic regularized strictly correlated electrons problem.

Analyzing causal structures using Tsallis entropies

Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realisable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional) independence between classical random variables and apply these to causal structures. We find that the number of independent constraints needed to characterise the causal structure is prohibitively high such that the computations required for the standard entropy vector method cannot be employed even for small causal structures. Instead, without solving the whole problem, we find new Tsallis entropic constraints for the triangle causal structure by generalising known Shannon constraints. Our results reveal new mathematical properties of classical and quantum Tsallis entropies and highlight difficulties of using Tsallis entropies for analysing causal structures.

Topological unified [formula omitted]-entropy

In the recent decade, different extensions of Shannon entropy have been introduced. One of them which generalizes many classical entropies is unified (r,s)-entropy. The properties of topological entropy and Shannon entropy are similar.In this paper, we extend the concept of unified (r,s)-entropy to the topological dynamical system by using Bowen’s definition of separated and spanning sets. We call this notion topological Unified (r,s)-entropy. Then we find the value of topological unified (r,s)-entropy when X is a noncompact metric space and get the value of the topological unified (r,s)-entropy for X∗ when X∗ is one point-compactification of X.

Entropies related to integral operators

Abstract We consider classical entropies associated with several continuous distributions of probabilities. Explicit expressions and properties of them are presented.

Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. A set of basic geometric and quantum mechanics and probability concepts together with methods of computation are developed in general covariant form for curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideas relating the Grigory Perelman's entropy for geometric flows and associated statistical thermodynamic systems to the quantum von Neumann entropy, classical and quantum relative and conditional entropy, mutual information etc. We formulate the concept of the entanglement entropy of quantum geometric information flows and study properties and inequalities for quantum, thermodynamic and geometric entropies characterising such systems.

A Novel Breast Cancer Detection Technology Using an Advanced Transfer Maximal Entropy Clustering Algorithm

The initial diagnosis of breast cancer involves analyzing the relevant examination report of the patient to determine whether the tumor is benign or malignant. Unsupervised clustering algorithms can be used with this type of problem. In a cluster analysis of a patient's examination data, the clustering results and the preliminary diagnosis results are obtained. However, due to the high cost of detection, medical datasets often have a small sample size or lack information. The traditional clustering technique usually has poor clustering effects in such scenarios. To solve this problem, this paper proposes an advanced transfer learning mechanism based on the classic maximum entropy clustering algorithm and proposes an advanced transfer maximal entropy clustering (AT-MEC) algorithm. A simulation experiment using the Wisconsin Breast Cancer Dataset is performed. This paper verifies that the proposed AT-MEC algorithm has a better clustering effect than other clustering algorithms in the Wisconsin Breast Cancer Dataset.

New quantum phase of the Universe before inflation and its cosmological and dark energy implications

The physical history of the Universe is completed by including the quantum Planckian and trans-Planckian phase before inflation in the Standard Model of the Universe in agreement with observations. In the absence of a complete quantum theory of gravity, we start from quantum physics and its foundational milestone. The universal classical-quantum (or wave-particle) duality, which we extend to gravity and the Planck domain. As a consequence, classical, quantum Planckian and super-Planckian regimes are covered, and the usual quantum domain as well. A new quantum precursor phase of the Universe appears beyond the Planck scale [Formula: see text]: [Formula: see text]; the known classical/semiclassical Universe being in the range: [Formula: see text]. We extend in this way the de Sitter Universe to the quantum domain: classical-quantum de Sitter duality. As a result: (i) The classical and quantum dual de Sitter temperatures and entropies are naturally included, and the different (classical, semiclassical, quantum Planckian and trans-Planckian) de Sitter regimes characterized in a precise and unifying way. (ii) We apply it to relevant cosmological examples as the CMB, inflation and dark energy. This allows us to find in a simple and consistent way. (iii) Full quantum inflationary spectra and their CMB observables, including in particular the classical known inflation spectra and the quantum corrections to them. (iv) A whole unifying picture for the Universe epochs and their quantum precursors emerges with the cosmological constant as the vacuum energy, entropy and temperature of the Universe, clarifying the so-called cosmological constant problem which once more in its rich history needed to be revised.

Positive entropy actions of countable groups factor onto Bernoulli shifts

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy), then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory.

Correlation Property of Multipartite Quantum Image

Inspired on the classical Bayesian method also called mutual information, by generalizing the classical case to quantum approach to study the multipartite correlation in quantum image by using the flexible representation of quantum images (FRQI) method. Some important entanglement measures are calculated including von Neumann entropy, conditional entropy, joint entropy, discord and mutual information. We take two typical images of size 2 × 2 pixels and 8 × 8 pixels and study their entanglement measures from different classical and quantum point of view. The quantum image of size 2 × 2 is studied under the change of color for one pixel, but the quantum image of 8 × 8 is treated under translation transform. We find that the classical joint entropy remains invariant under the change of color for one pixel but the quantum entropy is sensitive to such change. It is shown that the total correlation IT could arrive to the double amount of the classical joint entropy. The discord in the classical case is zero, but it varies much in the quantum case.

Measures of Entropy to Characterize Fatigue Damage in Metallic Materials.

This paper presents the entropic damage indicators for metallic material fatigue processes obtained from three associated energy dissipation sources. Since its inception, reliability engineering has employed statistical and probabilistic models to assess the reliability and integrity of components and systems. To supplement the traditional techniques, an empirically-based approach, called physics of failure (PoF), has recently become popular. The prerequisite for a PoF analysis is an understanding of the mechanics of the failure process. Entropy, the measure of disorder and uncertainty, introduced from the second law of thermodynamics, has emerged as a fundamental and promising metric to characterize all mechanistic degradation phenomena and their interactions. Entropy has already been used as a fundamental and scale-independent metric to predict damage and failure. In this paper, three entropic-based metrics are examined and demonstrated for application to fatigue damage. We collected experimental data on energy dissipations associated with fatigue damage, in the forms of mechanical, thermal, and acoustic emission (AE) energies, and estimated and correlated the corresponding entropy generations with the observed fatigue damages in metallic materials. Three entropic theorems—thermodynamics, information, and statistical mechanics—support approaches used to estimate the entropic-based fatigue damage. Classical thermodynamic entropy provided a reasonably constant level of entropic endurance to fatigue failure. Jeffreys divergence in statistical mechanics and AE information entropy also correlated well with fatigue damage. Finally, an extension of the relationship between thermodynamic entropy and Jeffreys divergence from molecular-scale to macro-scale applications in fatigue failure resulted in an empirically-based pseudo-Boltzmann constant equivalent to the Boltzmann constant.

Fault Diagnosis of Train Plug Door Based on a Hybrid Criterion for IMFs Selection and Fractional Wavelet Package Energy Entropy

The train plug door is the only way for passengers getting on and off. Its failures will make train operation ineffective. Taking the developed digital signal processing technologies into consideration, a data-driven diagnosis method for train plug doors is proposed based on sound recognition. First, a novel preprocessing method based on empirical mode decomposition and hybrid intrinsic mode functions (IMFs) selection criterion is proposed. The selected significant IMFs are used to reconstruct the signals. Inspired by the idea of fractional calculus, novel entropy named fractional wavelet package decomposition energy entropy (FWPDE) is proposed. Finally, multi-class support vector machine is used for classification and validation. Experimental results indicate that the proposed preprocessing method is of great significance to extract effective FWPDE features. In addition, FWPDE is more powerful in comparison with the classical wavelet package decomposition energy entropy. The identification accuracy using the proposed method reaches 96.28%, which demonstrates its effectiveness and superiority.

Entangled four-dimensional multicomponent topological states from photonic crystal defects

Recently, there has been a drive towards the realization of topological phases beyond conventional electronic materials, including phases defined in more than three dimensions. We propose a versatile and experimentally realistic approach of realizing a large variety of multi-component topological phases in 2D photonic crystals with quasi-periodically modulated defects. With a length scale introduced by a background resonator lattice, the defects are found to host various effective orbitals of $s$, $p$ and $d$-type symmetries, thus providing a monolithic platform for realizing multi-component topological states without requiring separate internal degrees of freedom in the physical setup. Notably, by coupling the defect modulations diagonally, we report the novel realization of an ``entangled'' 4D QH phase which cannot be factorized into two copies of 2D QH phases, each described by the 1st Chern number. The structure of this non-factorizability can be quantified by a classical entanglement entropy inspired by quantum information theory. In another embodiment, we present 4D p-orbital nodal lines in a nonsymmorphic photonic lattice, hosting boundary states with an exotic manifold. Our simple and versatile approach holds the promise of novel topological optoelectronic and photonic applications such as one-way optical fibers.

Double-Granule Conditional-Entropies Based on Three-Level Granular Structures.

Rough set theory is an important approach for data mining, and it refers to Shannon’s information measures for uncertainty measurements. The existing local conditional-entropies have both the second-order feature and application limitation. By improvements of hierarchical granulation, this paper establishes double-granule conditional-entropies based on three-level granular structures (i.e., micro-bottom, meso-middle, macro-top), and then investigates the relevant properties. In terms of the decision table and its decision classification, double-granule conditional-entropies are proposed at micro-bottom by the dual condition-granule system. By virtue of successive granular summation integrations, they hierarchically evolve to meso-middle and macro-top, to respectively have part and complete condition-granulations. Then, the new measures acquire their number distribution, calculation algorithm, three bounds, and granulation non-monotonicity at three corresponding levels. Finally, the hierarchical constructions and achieved properties are effectively verified by decision table examples and data set experiments. Double-granule conditional-entropies carry the second-order characteristic and hierarchical granulation to deepen both the classical entropy system and local conditional-entropies, and thus they become novel uncertainty measures for information processing and knowledge reasoning.

Quantum operator entropies under unitary evolution

For a quantum state undergoing unitary Schrödinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator [R. S. Ingarden, Quantum information theory, Rep. Math. Phys. 10, 43 (1976)RMHPBE0034-487710.1016/0034-4877(76)90005-7]. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system composed of a set of highly interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.