Definition Of Entropy Research Articles

Matrix Liberation Process II: Relation to Orbital Free Entropy

AbstractWe investigate the concept of orbital free entropy from the viewpoint of the matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is essential to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.

Entropy: (1) The former trouble with particle-tracking simulation, and (2) A measure of computational information penalty

Traditional random-walk particle-tracking (PT) models of advection and dispersion do not track entropy, because particle masses remain constant. However, newer mass-transfer particle tracking (MTPT) models have the ability to do so because masses of all compounds may change along trajectories. Additionally, the probability mass functions (PMF) of these MTPT models may be compared to continuous solutions with probability density functions, when a consistent definition of entropy (or similarly, the dilution index) is constructed. This definition reveals that every discretized numerical model incurs a computational entropy. Similar to Akaike’s (1974, 1992) entropic penalty for larger numbers of adjustable parameters, the computational complexity of a model (e.g., number of nodes or particles) adds to the entropy and, as such, must be penalized. Application of a new computational information criterion reveals that increased accuracy is not always justified relative to increased computational complexity. The MTPT method can use a particle-collision based kernel or an adaptive kernel derived from smoothed-particle hydrodynamics (SPH). The latter is more representative of a locally well-mixed system (i.e., one in which the dispersion tensor equally represents mixing and solute spreading), while the former better represents the separate processes of mixing versus spreading. We use computational means to demonstrate the fitness of each of these methods for simulating 1-D advective-dispersive transport with uniform coefficients.

On properties of a new decomposable entropy of Dempster-Shafer belief functions

We define entropy of belief functions in the Dempster-Shafer (D-S) theory that satisfies a compound distributions property that is analogous to the property that characterizes Shannon's definitions of entropy and conditional entropy for probability mass functions. None of the existing definitions of entropy for belief functions in the D-S theory satisfy this property. We describe some important properties of our definition, and discuss its semantics as a measure of dissonance and not uncertainty. Finally, we compare our definition of entropy with some other definitions that are similar to ours in the sense that these definitions measure dissonance and not uncertainty.

ОСОБЕННОСТИ ЭНТРОПИЙНОЙ ОРГАНИЗОВАННОСТИ ПЕРЕВОЗОЧНОГО ПРОЦЕССА НА ТИПИЧНЫХ МАРШРУТАХ ГОРОДСКОГО ОБЩЕСТВЕННОГО ТРАНСПОРТА

The paper presents the possibility, necessity, ideology and essence of the entropy method of estimating the orderliness of the transportation process in relation to urban public transport. Examples of numerical definition of the relative entropy of the transportation process on typical public transport routes are given. The paper shows that the level of orderliness of the transportation process, identified by relative entropy, is determined by the specifics of the spatial location of public transport routes in terms of the urban area and mainly depends on the quality of users of public transport services.

Decision-Making Method Based on Grey Relation Analysis and Trapezoidal Fuzzy Neutrosophic Numbers Under Double Incomplete Information and its Application in Typhoon Disaster Assessment

Multi-attribute decision-making problems under the trapezoidal fuzzy neutrosophic numbers environment are complex, particularly when the attribute value data are incomplete, and the attribute weight is completely unknown. As a solution, this study proposes a decision-making method based on information entropy and grey theory. First, regarding the problem of attribute value incompleteness in the decision matrix, a method of defining the missing information using trapezoidal fuzzy neutrosophic numbers is proposed, which is simpler and more reasonable than the traditional method of complements. Regarding the problem of completely unknown attribute weights, the definition of new trapezoidal fuzzy neutrosophic entropy is proposed and used to determine the attribute weight. Thereafter, the grey relation analysis method of grey theory is used to rank alternatives and select the best one. Finally, an illustrative example about typhoon disaster assessment is presented to show the feasibility and effectiveness of the proposed method. Finally, the advantages of the proposed method are illustrated by comparison with other methods from multiple aspects.

Entropy on regular trees

We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is $ \mathbb Z $. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approximate the hard square constant on $ \mathbb Z^2 $, we show that the entropy of the hard square tree shift on the regular $ k $-tree increases with $ k $, in contrast to the case of $ \mathbb Z^k $. We prove that the strip entropy approximations increase strictly to the entropy of the golden mean tree shift for $ k = 2,\dots,8 $ and propose that this holds for all $ k \geq 2 $. We study the dynamics of the map of the simplex that advances the vector of ratios of symbol counts as the width of the approximating strip is increased, providing a fairly complete description for the golden mean subshift on the $ k $-tree for all $ k $. This map provides an efficient numerical method for approximating the entropies of tree shifts defined by nearest neighbor restrictions. Finally, we show that counting configurations over certain other patterns besides the natural finite subtrees yields the same value of entropy for tree SFT's.

Thermal processes in natural gas pipeline transport

A commonly used premise in pipeline hydraulics where the work of friction forces performed at the movement of real gas in the gas pipeline completely turns into thermal energy is verified in the article. By means of the integral definition of Clausius entropy, it is shown that the premise of the conversion of friction forces into thermal energy of gas flow is justified with an acceptable accuracy for engineering applications in relation to the one-dimensional formulation of the task regarding the determination of the longitudinal temperature field of gas.

On entropy for general quantum systems

In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.

An Extended VIKOR Method for Multiple Attribute Decision Making with Linguistic D Numbers Based on Fuzzy Entropy

The linguistic [Formula: see text] numbers (LDNs) can express the fuzzy evaluation information more easily and precisely by combining the advantages of linguistic terms (LTs) and [Formula: see text] numbers (DNs). Existing researches on the fuzzy entropy of LDNs are rare, and most of the definitions of fuzzy entropy for LDNs are unreasonable. In view of this research gap, this paper improves the definition of fuzzy entropy of LDNs, which simultaneously considers the effects of confidence degrees and LTs on the value of fuzzy entropy in LDNs. Then, the weights of attributes can be calculated by the improved fuzzy entropy. Further, a new combination rule for LDNs is also presented in this paper. Based on these studies, we extend the traditional Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method to the LDNs and develop the LD-VIKOR method. The proposed LD-VIKOR method is convenient to handle MADM problems with unknown attributes weights under the environment of LDNs. Finally, we verify the validity and reliability of the proposed method by an illustrative example, and analyze the advantages of the proposed method by comparing it with other existing MADM methods.

Some Notes on the Concepts of Entropy and Information

Abstract—The definition of entropy as the intensity of a probabilistic process is substantiated. The concept of an abstract process is introduced, on the basis of which information is defined as the internal relation of feedback systems. A new approach to determining the amount of information for deterministic systems is proposed.

The bifurcation measure has maximal entropy

Let A be a complex manifold and let $${({f_\lambda})_{\lambda \in {\rm{\Lambda}}}}$$ be a holomorphic family of rational maps of degree d ≥ 2of ℙ1. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion of a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdin’s bound of the volume of the image of a dynamical ball. Applying our results to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of ℙk.

Topological and metric entropy for group and semigroup actions

AbstractIt is well-known that the classical definition of topological entropy for group and semigroup actions is frequently zero in some rather interesting situations, e.g. smooth actions of ℤk+ (k >1) on manifolds. Different definitions have been considered by several authors. In the present article we describe the one proposed in 1995 by K.H.Hofmann and the author which produces topological entropy not trivially zero for such smooth actions. We discuss this particular approach, and also some of the main properties of the topological entropy defined in this way, its advantages and disadvantages compared with the classical definition. We also discuss some recent results, obtained jointly with Andrzej Biś, Dikran Dikranjan and Anna Giordano Bruno, of a similar definition of metric entropy, i.e. entropy with respect to an invariant measure for a group or a semigroup action, and some of its properties.

Visual Perception Entropy Measure

This paper is about a theoretical framework on perceptual reproduction quality assessment. Visual perception is the main theme for two approaches on image metrics: visual performance and image quality. Visual performance looks at the perception of standard elementary images, such as gratings, by different observers or by a single observer in different visual conditions; image quality looks at the perception of variations of the same complex scene by a standard observer. In the audiovisual field, all the configurations approved by the creative team must be reproduced in the final viewing. This cannot be measured by visual performance or image quality indexes. A third approach is needed, which may be called perceptual reproduction quality assessment. Visual perception assessment is not measuring defects, per se; it is built to evaluate the perceptual effectiveness of a given scene coding through a given reproduction technology. It is only dependent on the number of configurations being perceived through a given theater system relative to the various configurations reproduced and approved in the review room. As such, it is a statistical measure. It is, in fact, similar to the definition of relative entropy. One important fact is that this evaluation must include a model of visual perception. This paper illustrates how relative entropy and entropy loss can be estimated, while taking into account the coding transform, the projector performances, maximum light, and the characteristics of the human visual system. This scheme works easily for monochromatic content. Color, trichromacy, is quite challenging. The size of configurations goes from 2 12 to 2 36 , around 6 billion configurations. Calculation and memory requirements are huge. Nonlinearities are adding complexity, leading to calculation of an upper limit only of entropy loss. The presentation of results is also a challenge. One way to solve this is to separately process lightness and chromatic information. In summary, the main arguments in this paper are: 1) besides image quality and visual performance, there is a need for perceptual reproduction quality assessment; 2) entropy is the proper statistical estimator for evaluating the perceptual reproduction quality; and 3) color reproduction quality estimation requires processing lightness and chromatic attributes separately.

A new perspective of how to understand entropy in thermodynamics

Using the analogy between thermodynamics and electricity, two new concepts of thermal charge and quantity of thermal charge are introduced. A simple yet explicit definition of entropy is then derived—entropy is the quantity of thermal charge. As a result, quantity of thermal charge (entropy) and quantity of heat (energy) are now clearly distinguishable from each other. The former is an energy carrier while the latter is energy itself. This largely eliminates the difficulties in learning and applying thermodynamics, and therefore will be of considerable assistance to those who work, teach and study in the fields associated with thermodynamics. This also leads us to re-examine the validity of caloric theory and make a brief comment on it.

On the statistical mechanics of structural vibration

The analysis of the structural dynamics of a complex engineering structure has much in common with the subject of statistical mechanics. Both are concerned with the analysis of large systems in the presence of various sources of randomness, and both are concerned with the possibility of emergent laws that might be used to provide a simplified approach to the analysis of the system. The aim of the present work is to apply a number of the concepts of statistical mechanics to structural dynamic systems in order to provide new insights into the system behaviour under various conditions. The work is foundational, in that it is based on employing the fundamental equations of motion of the system in conjunction with various definitions of entropy, and no recourse is made to emergent laws that are accepted in thermodynamics. The analysis covers closed (undamped and unforced) and open (forced and damped) systems, linear and nonlinear systems, and both single systems and coupled systems. The fact that the system itself can be random leads to a number of results that differ from those found in classical statistical mechanics, where the initial conditions might be considered to be random but the Hamiltonian is taken to be well defined. For example, the occurrence of a stationary state in a closed system normally requires nonlinearity and coarse-graining of the statistical distribution, but neither condition is required for a random system. For coupled systems it is shown that under certain conditions both Statistical Energy Analysis (SEA) and Transient Statistical Energy Analysis (TSEA) are emergent laws, and insights are gained as to the validity of these laws. The analysis is supported by a number of numerical examples to illustrate key points.

Phase entropy: a new complexity measure for heart rate variability

Objective: Information entropy is generally employed for analysing the complexity of physiological signals. However, most definitions of entropy estimate the degree of compressibility and thus quantify the randomness. Physiological signals are very complex because of nonlinear relationships and interactions between various systems and subsystems of the body. Therefore, analysis of randomness may not be sufficient to describe this complexity. To analyse the complexity of heart rate variability (HRV), a new entropy method, phase entropy (PhEn), has been proposed as a quantification of two-dimensional phase space. Approach: The second-order difference plot (SODP), a two-dimensional phase space, provides a visual summary of the rate of variability. The distribution of scatter points in a SODP provides information about the dynamics of the underlying system. PhEn estimates the Shannon entropy of the weighted distribution in a coarse-grained SODP. Main results: The performance of PhEn has been evaluated using simulated signals, synthetic HRV signals and real HRV signals. PhEn shows a better discriminating power and stability than other entropy measures. It is computationally efficient. Moreover, it has the ability to assess temporal asymmetry of physiological signals. Significance: PhEn quantifies the multiplicity and rate of variability associated with physiological signals. It is sensitive to time irreversibility. Therefore, it appears to be a promising tool for analysing physiological signals such as HRV.

A Dynamical Systems-Based Hierarchy for Shannon, Metric and Topological Entropy

A rigorous dynamical systems-based hierarchy is established for the definitions of entropy of Shannon (information), Kolmogorov–Sinai (metric) and Adler, Konheim & McAndrew (topological). In particular, metric entropy, with the imposition of some additional properties, is proven to be a special case of topological entropy and Shannon entropy is shown to be a particular form of metric entropy. This is the first of two papers aimed at establishing a dynamically grounded hierarchy comprising Clausius, Boltzmann, Gibbs, Shannon, metric and topological entropy in which each element is ideally a special case of its successor or some kind of limit thereof.

Teaching Entropy from Phase Space Perspective: Connecting the Statistical and Thermodynamic Views Using a Simple One-Dimensional Model

Connecting the thermodynamic definition of entropy, dS = dQ/T (Clausius’s equation), with the statistical definition, S = kB ln Ω (Boltzmann’s equation), has been a persistent challenge in chemical education at the undergraduate level. Not meeting this challenge results in students taking away the meaning of entropy in a vague and subjective way as a measure of “disorder” or increase in number of configurations without any meaningful way of connecting it to heat. To address this challenge, we present a simple model that connects these two definitions. This approach relies centrally on emphasizing that the number of configurations, Ω, includes configurations in both real space and momentum space, collectively known as the phase space. Without including momentum configurations (i.e., how fast the particles move), connecting heat to entropy change is not possible. We construct the phase space for an ensemble of simple one-dimensional systems at equilibrium and show that delivery of heat dQ to the system results in an increase in the number of momentum configurations and consequently an expansion of the phase space area by dΩ. Relating dQ to dΩ is the linchpin between the two views and, when integrated, leads to Boltzmann’s equation. We further show that understanding entropy in terms of volume of phase space removes common ambiguities in teaching the subject. Among other examples, we show that understanding adiabatic compression, if treated using the usual approach, results in contradictions that are resolved if a phase space view is adopted. We propose this approach at the undergraduate physical chemistry and physics level.

Microcanonical analysis of Boltzmann and Gibbs entropies in trapped cold atomic gases

We analyze a gas of noninteracting fermions confined to a one-dimensional harmonic oscillator potential, with the aim of distinguishing between two proposed definitions of the thermodynamic entropy in the microcanonical ensemble, namely the standard Boltzmann entropy and the Gibbs (or volume) entropy. The distinction between these two definitions is crucial for systems with an upper bound on allowed energy levels, where the Boltzmann definition can lead to the notion of negative absolute temperature. Although negative temperatures do not exist for the system of fermions studied here, we still find a significant difference between the Boltzmann and Gibbs entropies, and between the corresponding temperatures with the Gibbs temperature being closer (for small particle number) to the temperature based on a grand canonical picture.

On the Conditional Smooth Rényi Entropy and its Applications in Guessing and Source Coding

A novel definition of the conditional smooth Renyi entropy, which is different from that of Renner and Wolf, is introduced. It is shown that our definition of the conditional smooth Renyi entropy is appropriate for providing lower and upper bounds on the optimal guessing moment in a guessing problem where the guesser is allowed to stop guessing and declare an error. Further a general formula for the optimal guessing exponent is presented. In particular, a single-letterized formula for a mixture of i.i.d. sources is obtained. It is also shown that our definition is appropriate to characterize the optimal exponential moment of the codeword length in the problem of source coding with common side-information available at the encoder and decoder under a constraint on the probability of a decoding error.