Definition Of Entropy Research Articles

A Note on Localization of Entropy of Doubly Stochastic Operators

In this paper, we present a local approach to the entropy of doubly stochastic operators. The definition of the local entropy follows some steps, motivated by Downarowicz and Frej (Ergod Theory Dyn Syst 25(2):455–481, 2005). It results in a local approach to the entropy of doubly stochastic operators with finite numbers of ergodic measures as well as Koopman operators.

Entanglement entropy, relative entropy and duality

A definition for the entanglement entropy in both Abelian and non-Abelian gauge theories has been given in the literature, based on an extended Hilbert space construction. The result can be expressed as a sum of two terms, a classical term and a quantum term. It has been argued that only the quantum term is extractable through the processes of quantum distillation and dilution. Here we consider gauge theories in the continuum limit and argue that quite generically, the classical piece is dominated by modes with very high momentum, of order the cut-off, in the direction normal to the entangling surface. As a result, we find that the classical term does not contribute to the relative entropy or the mutual information, in the continuum limit, for states which only carry a finite amount of energy above the ground state. We extend these considerations for p-form theories, and also discuss some aspects pertaining to electric-magnetic duality.

A microcanonical entropy correcting finite-size effects in small systems

In a recent paper (Franzosi (2018 Physica A 494 302)), we have suggested to use the surface entropy, namely the logarithm of the area of a hypersurface of constant energy in the phase space, as an expression for the thermodynamic microcanonical entropy, in place of the standard definition usually known as Boltzmann entropy. In the present manuscript, we have tested the surface entropy on the Fermi–Pasta–Ulam model for which we have computed the caloric equations that derive from both the Boltzmann entropy and the surface entropy. The results achieved clearly show that in the case of the Boltzmann entropy there is a strong dependence of the caloric equation from the system size, whereas in the case of the surface entropy there is no such dependence. We infer that the issues that one encounters when the Boltzmann entropy is used in the statistical description of small systems could be a clue to a deeper defect of this entropy that derives from its basic definition. Furthermore, we show that the surface entropy is well founded from a mathematical point of view, and we show that it is the only admissible entropy definition, for an isolated and finite system with a given energy, which is consistent with the postulate of equal a priori probability.

A Matrix Method of Basic Belief Assignment's Negation in Dempster–Shafer Theory

Negation is a new perspective to represent knowledge. The negation of probability distribution has been proposed, and it has a lot of interesting properties, which can reach a maximum entropy. Because of the defects of the classical probability theory in the expression of uncertainty, the basic belief assignment (BBA) in the Dempster–Shafer theory (D–S theory) are widely used in decision theory. Thus, negation provides a new perspective for D–S theory to measure fuzziness. In this paper, a new definition of negation of BBA is presented. In the proposed negation, BBAs are represented as vectors, and negation is realized by matrix operators. This method has a good interpretation of the matrix operators and has the merit of simplifying the problem. With several different definitions of entropy to determinate the uncertainty of BBA, the proposed negation of BBA can reach a maximum belief entropy when the entropies satisfy a certain property.

Intelligent Classification Method for Grid-Monitoring Alarm Messages Based on Information Theory

Alarm messages for grid monitoring are an important way to supervise the operation of power grids. Since the use of alarm messages is increasing exponentially due to the continuous expansion of the scale of power grids, a processing method for alarm messages based on statistics is proposed in this study. Entropy theory in information theory is introduced into the calculation of information value in power-grid alarming. By means of multiple entropy definitions, an evaluation index system for information value is constructed. Based on the analytic hierarchy process (AHP), various alarm-message entropies are used as indices to comprehensively assess the information value and level of each alarm message. Finally, an example is given to illustrate the effectiveness and practicality of the proposed method. This study provides a new idea for the intelligent classification of alarm messages.

Quantum Features of Macroscopic Fields: Entropy and Dynamics.

Macroscopic fields such as electromagnetic, magnetohydrodynamic, acoustic or gravitational waves are usually described by classical wave equations with possible additional damping terms and coherent sources. The aim of this paper is to develop a complete macroscopic formalism including random/thermal sources, dissipation and random scattering of waves by environment. The proposed reduced state of the field combines averaged field with the two-point correlation function called single-particle density matrix. The evolution equation for the reduced state of the field is obtained by reduction of the generalized quasi-free dynamical semigroups describing irreversible evolution of bosonic quantum field and the definition of entropy for the reduced state of the field follows from the von Neumann entropy of quantum field states. The presented formalism can be applied, for example, to superradiance phenomena and allows unifying the Mueller and Jones calculi in polarization optics.

Characterizing with Shannon 1948 Entropy Leading to a Modeling and Testing Method for SOFC and SOEC High Temperature Heat Exchangers

A description of the reversible behavior of BoP-heat exchangers for SOFC and SOEC (r-SOC) is studied in this paper. Because considerations based on thermodynamics entropy could not offer a satisfactory explanation, a more fundamental definition of entropy based on the Shannon 1948 entropy is applied. A Shannon Isentropic Heat Exchanger shows insight in the formulation coming from Kays & London (1964) for compact heat exchangers. Starting from the laminar heat exchange model, it is enlarged towards heat exchangers with enhanced heat transfer due to micro-turbulences. A mathematical and graphical approach is discussed. A testing methodology is developed that offers the advantages to identify and quantify the performance quality accurately, and enables to discover links with the causes of performance degradation. This approach delivers a clear different view on the thermodynamics in a high effectiveness heat exchanger, which can be characterized as a result of “thinking out of the box”.

Exploring Biased Probability Using Loaded Dice: An Active Learning Exercise with Analogy to Entropic and Energetic Determinants of Equilibria in Chemical Systems

The equilibrium position of a chemical system is based on two distinct considerations, namely, the entropy and enthalpy differences between the states. Loaded dice provide a physical system for exploring the probability of observing different interconverting states on the basis of two distinct factors: the number of sides of the die (entropy) and the asymmetric distribution of mass in the die (gravitational potential energy). Although potential energy analogies such as pushing a rock up a hill have often been used for teaching differences in energy in chemistry, and dice have likewise often been used to explain entropy, such analogies can be unsatisfactory when entropic and enthalpic considerations favor different states. We present an activity that allows students to explore how potential energy and entropy together contribute to observed probability and then draw parallels to chemical systems. Students collect statistics for observed rolls of loaded dice, rationalize the different patterns as distinct states, and determine the difference in entropy between the states. These observations are then related to chemical reactions and to the definition of entropy. The effect of varying the loaded mass within the dice is related to the role enthalpy plays in determining equilibrium. This analogy and associated active learning exercises are applicable to a wide variety of college chemistry and physics courses; provide a hands-on approach to introducing concepts such as probability, entropy, and energy differences; and can be used to draw parallels to chemical reactions, enthalpy, and Gibbs free energy.

Valid Incremental Attribute Reduction Algorithm Based on Attribute Generalization for an Incomplete Information System

Attribute reduction, also known as feature selection, is a vital application of rough set theory in areas such as machine learning and data mining. With several information systems constantly and dynamically changing in reality, the method of continuing the incremental attribute reduction for these dynamic information systems is the focus of this research. In an incomplete information system, the increasing form of attribute sets is an important form of dynamic change. In this paper, the definition of conditional entropy is first introduced in the incomplete information system, and for the circumstances of the dynamic change of the attribute sets, two types of incremental mechanisms of the matrix and non-matrix forms based on conditional entropy are subsequently proposed. In addition, on the basis of the two incremental mechanisms, the incremental attribute reduction algorithm is given when the attribute set increases dynamically. Finally, the experimental results of the UCI (University of California Irvine) datasets verify that the two proposed incremental algorithms exhibit a superior performance with regard to attribute reduction when compared with the non-incremental attribute reduction algorithm, which in turn is superior to other relative incremental algorithms.

Multivariate Shannon's entropy for adaptive IIR filtering via kernel density estimators

In supervised infinite impulse response adaptive filtering, approximate gradient-based approaches are the usual option among optimisation methods. When based on the mean squared error (MSE) criterion, however, these approaches may present biased solutions in noisy scenarios. In that sense, instead of the MSE, the authors propose the use of Shannon's error entropy, an information theoretic learning criterion, which is able to extract higher order statistics from the underlying signals. In particular, a multivariate entropy definition is considered, which is applied to derive a Recursive Prediction Error-based algorithm. The performance analyses are carried out in the context of the supervised channel equalisation problem, with results very favourable to the proposal, in high and low noise level environments.

Signatures of Quantum Mechanics in Chaotic Systems.

We examine the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system’s set of cupolets, which are highly-accurate stabilizations of its unstable periodic orbits. Our discussion is motivated by the bound or entangled states that we have recently detected between interacting chaotic systems, wherein pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external controls. This state is known as chaotic entanglement as it has been shown to exhibit several properties consistent with quantum entanglement. For instance, should the interaction be disturbed, the chaotic entanglement would then be broken. In this paper, we further describe chaotic entanglement and go on to address the capacity for chaotic systems to exhibit other characteristics that are conventionally associated with quantum mechanics, namely analogs to wave function collapse, various entropy definitions, the superposition of states, and the measurement problem. In doing so, we argue that these characteristics need not be regarded exclusively as quantum mechanical. We also discuss several characteristics of quantum systems that are not fully compatible with chaotic entanglement and that make quantum entanglement unique.

Turbine Passage Design Methodology to Minimize Entropy Production-A Two-Step Optimization Strategy.

Rapid aerodynamic design and optimization is essential for the development of future turbomachinery. The objective of this work is to demonstrate a methodology from 1D mean-line-design to a full 3D aerodynamic optimization of the turbine stage using a parameterization strategy that requires few parameters. The methodology is tested by designing a highly loaded and efficient turbine for the Purdue Experimental Turbine Aerothermal Laboratory. This manuscript describes the entire design process including the 2D/3D parameterization strategy in detail. The objective of the design is to maximize the entropy definition of efficiency while simultaneously maximizing the stage loading. Optimal design trends are highlighted for both the stator and rotor for several turbine characteristics in terms of pitch-to-chord ratio as well as the blades metal and stagger angles. Additionally, a correction term is proposed for the Horlock efficiency equation to maximize the accuracy based on the measured blade kinetic losses. Finally, the design and performance of optimal profiles along the Pareto front are summarized, featuring the highest aerodynamic performance and stage loading.

Unifying Entropies of Quantum Logic Based on Rényi Entropies

The aim of this study it to unify entropies of quantum logic. For this purpose, Renyi entropy of partition of quantum logic corresponding to a state is presented and formal properties of a proper definition of an entropy are detected. In addition, an appropriate definition of conditional Renyi entropy of quantum logic having the expected properties such as monotonicity is proposed. Finally, Renyi entropy of quantum dynamical system is mentioned and it is proved that it is invariant under isomorphism.

Reduction algorithm based on finding the maximum mutual information in incomplete information systems

Aiming at such drawbacks of the existing incomplete information system attribute reduction algorithms as high time complexity and insufficiency of the reduction result completeness, this paper proposes an attribute reduction algorithm based on finding the maximum mutual information via weakening the equivalence class in classical rough set theory into tolerance class and through combining the definition of information entropy with the tolerance class. Taking the mutual information of condition attribute and decision attribute as the iterative criterion, and the empty set as the initial reduction, the algorithm superposes the condition attribute set corresponding to the maximum value of the mutual information of condition attributes and decision attributes, thus equating the mutual information of condition attribute set and decision attribute with the information entropy of decision attribute, obtaining the relative reduction of the incomplete decision system and guaranteeing the completeness of the reduction result. In the solution of the attribute set tolerance class, an algorithm with lower time complexity is adopted, which can reduce the time complexity of the whole algorithm. The feasibility of the algorithm is illustrated by an example.

Exploring entropy for continuous vibrating systems using modal decomposition and truncation

This paper examines the concept of entropy for continuous systems undergoing simple harmonic oscillation. The paper begins with a discussion of entropy in the context of vibrating structural systems, including the constraints and assumptions of the Khinchin definition of entropy. Then, it is shown that the Khinchin entropy can be applied to vibrating continuous systems using modal analysis and under weak coupling condition. It is shown that under these conditions the results obtained for continuous systems are consistent with those obtained for discrete oscillators. Several numerical examples are presented to illustrate these findings. The governing equations for these systems are derived using the Lagrangian method, and solved using the Galerkin method.

Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

Given a partially hyperbolic diffeomorphism defined on a compact Riemannian manifold M, in this paper we define the concept of unstable topological entropy of f on a set not necessarily compact. Using recent results of Yang (2016 (arXiv:1601.05504)) and Hu et al (2017 Adv. Math. 321 31–68) we extend a theorem of Bowen (1973 Trans. Am. Math. Soc. 184 125–36) proving that, for an ergodic f -invariant measure , the unstable measure theoretical entropy of f is upper bounded by the unstable topological entropy of f on any set of positive -measure. We define a notion of unstable topological entropy of f using a Hausdorff dimension like characterization and we prove that this definition coincides with the definition of unstable topological entropy introduced in Hu et al (2017 Adv. Math. 321 31–68).

Spillover effects of Great Recession on Hong-Kong’s Real Estate Market: An analysis based on Causality Plane and Tsallis Curves of Complexity–Entropy

This paper investigates the impact of the sub-prime loan crisis on the Real Estate Market of Hong-Kong. Based on permutation entropy, complexity–entropy causality plane and Tsallis complexity–entropy curve, we characterize the complexity of the housing indices-both in terms of size and region-and distinguish the level of informational efficiency. By calculating the quantifiers we report that most indices exhibit a behavior equivalent to a persistent stochastic dynamics with Hurst exponents between 0.5 and 0.7. The outbreak of the crisis had changed the dynamical structure of the indices decreasing the level of randomness and increasing considerably their regularity and predictability. Only the index of the Kowloon area seems not impacted by the crisis, exhibiting higher levels of informational efficiency. The results are robust based on the utilization of two different entropy definitions: The Shannon and Tsallis-q entropy. Lastly, with the temporal evolution of the indices, we identify periods where the underlying dynamical structure of the market was impacted by certain events like the SARS epidemic and the imposition of Special Stamp Duty on housing.

Configurational entropy of glass-forming liquids.

The configurational entropy is one of the most important thermodynamic quantities characterizing supercooled liquids approaching the glass transition. Despite decades of experimental, theoretical, and computational investigation, a widely accepted definition of the configurational entropy is missing, its quantitative characterization remains fraught with difficulties, misconceptions, and paradoxes, and its physical relevance is vividly debated. Motivated by recent computational progress, we offer a pedagogical perspective on the configurational entropy in glass-forming liquids. We first explain why the configurational entropy has become a key quantity to describe glassy materials, from early empirical observations to modern theoretical treatments. We explain why practical measurements necessarily require approximations that make its physical interpretation delicate. We then demonstrate that computer simulations have become an invaluable tool to obtain precise, nonambiguous, and experimentally relevant measurements of the configurational entropy. We describe a panel of available computational tools, offering for each method a critical discussion. This perspective should be useful to both experimentalists and theoreticians interested in glassy materials and complex systems.

Sharma–Mittal quantum discord

We demonstrate a generalization of quantum discord using a generalized definition of von-Neumann entropy, which is Sharma-Mittal entropy; and the new definition of discord is called Sharma-Mittal quantum discord. Its analytic expressions are worked out for two qubit quantum states as well as Werner, isotropic, and pointer states as special cases. The R{\'e}nyi, Tsallis, and von-Neumann entropy based quantum discords can be expressed as limiting cases for of Sharma-Mittal quantum discord. We also numerically compare all these discords and entanglement negativity.

Origin of negative temperatures in systems interacting with external fields

The controversial existence of negative temperatures has stirred interesting debates that have reached the foundations of thermodynamics, including questions on the second law, the Carnot efficiency and the statistical definition of entropy. Here we show that for systems interacting with an external field, negative temperatures may arise when the interaction energy with the field is treated as a form of internal energy. We discuss how negative temperatures are avoided when using a thermodynamic formalism that accounts for the intensive and extensive variables associated to the external field. We use the paramagnetic system and a perfect gas in a gravitational field to illustrate these ideas. Considerations about the isothermal and adiabatic work done by the field or the system also shed light on the inconsistency of super-Carnot efficiencies.