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Entropy, Volume 12, Issue 3 (March 2010) – 20 articles , Pages 289-630

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1221 KiB  
Article
The Maximum Entropy Production Principle: Its Theoretical Foundations and Applications to the Earth System
by James Dyke and Axel Kleidon
Entropy 2010, 12(3), 613-630; https://doi.org/10.3390/e12030613 - 22 Mar 2010
Cited by 51 | Viewed by 16391
Abstract
The Maximum Entropy Production (MEP) principle has been remarkably successful in producing accurate predictions for non-equilibrium states. We argue that this is because the MEP principle is an effective inference procedure that produces the best predictions from the available information. Since all Earth [...] Read more.
The Maximum Entropy Production (MEP) principle has been remarkably successful in producing accurate predictions for non-equilibrium states. We argue that this is because the MEP principle is an effective inference procedure that produces the best predictions from the available information. Since all Earth system processes are subject to the conservation of energy, mass and momentum, we argue that in practical terms the MEP principle should be applied to Earth system processes in terms of the already established framework of non-equilibrium thermodynamics, with the assumption of local thermodynamic equilibrium at the appropriate scales. Full article
(This article belongs to the Special Issue What Is Maximum Entropy Production and How Should We Apply It?)
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310 KiB  
Article
Application of Thermoeconomics to Industrial Ecology
by Antonio Valero, Sergio Usón, César Torres and Alicia Valero
Entropy 2010, 12(3), 591-612; https://doi.org/10.3390/e12030591 - 22 Mar 2010
Cited by 59 | Viewed by 12226
Abstract
Industrial Ecology involves the transformation of industrial processes from linear to closed loop systems: matter and energy flows which were initially considered as wastes become now resources for existing or new processes. In this paper, Thermoeconomics, commonly used for the optimization and diagnosis [...] Read more.
Industrial Ecology involves the transformation of industrial processes from linear to closed loop systems: matter and energy flows which were initially considered as wastes become now resources for existing or new processes. In this paper, Thermoeconomics, commonly used for the optimization and diagnosis of energy systems, is proposed as a tool for the characterization of Industrial Ecology. Thermoeconomics is based on the exergy analysis (Thermodynamics) but goes further by introducing the concepts of purpose and cost (Economics). It is presented in this study as a systematic and general approach for the analysis of waste flow integration. The formulation is based on extending the thermoeconomic process of the cost formation of wastes in order to consider their use as input for other processes. Consequently, it can be applied to important Industrial Ecology issues such as identification of integration possibilities and efficiency improvement, quantification of benefits obtained by integration, or determination of fair prices based on physical roots. The capability of the methodology is demonstrated by means of a case study based on the integration of a power plant, a cement kiln and a gas-fired boiler. Full article
(This article belongs to the Special Issue Exergy: Analysis and Applications)
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132 KiB  
Article
Thermodynamic and Differential Entropy under a Change of Variables
by Vladimir Hnizdo and Michael K. Gilson
Entropy 2010, 12(3), 578-590; https://doi.org/10.3390/e12030578 - 16 Mar 2010
Cited by 35 | Viewed by 10938
Abstract
The differential Shannon entropy of information theory can change under a change of variables (coordinates), but the thermodynamic entropy of a physical system must be invariant under such a change. This difference is puzzling, because the Shannon and Gibbs entropies have the same [...] Read more.
The differential Shannon entropy of information theory can change under a change of variables (coordinates), but the thermodynamic entropy of a physical system must be invariant under such a change. This difference is puzzling, because the Shannon and Gibbs entropies have the same functional form. We show that a canonical change of variables can, indeed, alter the spatial component of the thermodynamic entropy just as it alters the differential Shannon entropy. However, there is also a momentum part of the entropy, which turns out to undergo an equal and opposite change when the coordinates are transformed, so that the total thermodynamic entropy remains invariant. We furthermore show how one may correctly write the change in total entropy for an isothermal physical process in any set of spatial coordinates. Full article
(This article belongs to the Special Issue Configurational Entropy)
455 KiB  
Article
Entropy Variation in the Two-dimensional Phase Transition of Anthracene Adsorbed at the Hg Electrode/Ethylene Glycol Solution Interface
by Claudio Fontanesi
Entropy 2010, 12(3), 570-577; https://doi.org/10.3390/e12030570 - 16 Mar 2010
Cited by 4 | Viewed by 6693
Abstract
The adsorption of anthracene (C14H10), at the mercury electrode/ethylene glycol (EG) solution interface, is characterized by a low and almost constant capacity (about 8 μF cm−2) region (capacitive “pit” or “plateau”) in capacity vs. potential curves, upon [...] Read more.
The adsorption of anthracene (C14H10), at the mercury electrode/ethylene glycol (EG) solution interface, is characterized by a low and almost constant capacity (about 8 μF cm−2) region (capacitive “pit” or “plateau”) in capacity vs. potential curves, upon selection of suitable values of temperature, bulk concentration and applied potential values. This result is rationalized assuming the occurrence of a 2D phase transition between two distinct adsorbed phases: (i) a “disordered” phase, characterized by a flat “parallel” disposition of the aromatic moiety on the electrode surface (ii) an “ordered” phase, characterized by a “perpendicular” disposition of the aromatic moiety on the electrode surface. The experimental evidence is rationalized by considering the chemical potential as an explicit function of the “electric field/adsorbed molecule” interaction. Such a modelistic approach enables the determination of the relevant standard entropy variation. Full article
(This article belongs to the Special Issue Entropy in Model Reduction)
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328 KiB  
Article
Turing Systems, Entropy, and Kinetic Models for Self-Healing Surfaces
by Eugene Kagan
Entropy 2010, 12(3), 554-569; https://doi.org/10.3390/e12030554 - 15 Mar 2010
Cited by 15 | Viewed by 7638
Abstract
The paper addresses the methods of description of friction-induced self-healing at the interface between two solid bodies. A macroscopic description of self-healing is based on a Turing system for the transfer of matter that leads to self-organization at the interface in the case [...] Read more.
The paper addresses the methods of description of friction-induced self-healing at the interface between two solid bodies. A macroscopic description of self-healing is based on a Turing system for the transfer of matter that leads to self-organization at the interface in the case of an unstable state. A microscopic description deals with a kinetic model of the process and entropy production during self-organization. The paper provides a brief overview of the Turing system approach and statistical kinetic models. The relation between these methods and the description of the self-healing surfaces is discussed, as well as results of their application. The analytical considerations are illustrated by numerical simulations. Full article
(This article belongs to the Special Issue Entropy and Friction Volume 2)
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469 KiB  
Article
Information and Signs: The Language of Images
by Inna Semetsky
Entropy 2010, 12(3), 528-553; https://doi.org/10.3390/e12030528 - 12 Mar 2010
Cited by 6 | Viewed by 12777
Abstract
Since time immemorial, philosophers and scientists were searching for a “machine code” of the so-called Mentalese language capable of processing information at the pre-verbal, pre-expressive level. In this paper I suggest that human languages are only secondary to the system of primitive extra-linguistic [...] Read more.
Since time immemorial, philosophers and scientists were searching for a “machine code” of the so-called Mentalese language capable of processing information at the pre-verbal, pre-expressive level. In this paper I suggest that human languages are only secondary to the system of primitive extra-linguistic signs which are hardwired in humans and serve as tools for understanding selves and others; and creating meanings for the multiplicity of experiences. The combinatorial semantics of the Mentalese may find its unorthodox expression in the semiotic system of Tarot images, the latter serving as the ”keys” to the encoded proto-mental information. The paper uses some works in systems theory by Erich Jantsch and Erwin Laszlo and relates Tarot images to the archetypes of the field of collective unconscious posited by Carl Jung. Our subconscious beliefs, hopes, fears and desires, of which we may be unaware at the subjective level, do have an objective compositional structure that may be laid down in front of our eyes in the format of pictorial semiotics representing the universe of affects, thoughts, and actions. Constructing imaginative narratives based on the expressive “language” of Tarot images enables us to anticipate possible consequences and consider a range of future options. The thesis advanced in this paper is also supported by the concept of informational universe of contemporary cosmology. Full article
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194 KiB  
Article
Recovering Matrices of Economic Flows from Incomplete Data and a Composite Prior
by Esteban Fernández-Vázquez
Entropy 2010, 12(3), 516-527; https://doi.org/10.3390/e12030516 - 12 Mar 2010
Cited by 3 | Viewed by 8241
Abstract
In several socioeconomic applications, matrices containing information on flows-trade, income or migration flows, for example–are usually not constructed from direct observation but are rather estimated, since the compilation of the information required is often extremely expensive and time-consuming. The estimation process takes as [...] Read more.
In several socioeconomic applications, matrices containing information on flows-trade, income or migration flows, for example–are usually not constructed from direct observation but are rather estimated, since the compilation of the information required is often extremely expensive and time-consuming. The estimation process takes as point of departure another matrix which is adjusted until it optimizes some divergence criterion and simultaneously is consistent with some partial information-row and column margins–of the target matrix. Among all the possible criteria to be considered, one of the most popular is the Kullback-Leibler divergence [1], leading to the well-known Cross-Entropy technique. This paper proposes the use of a composite Cross-Entropy approach that allows for introducing a mixture of two types of a priori information–two possible matrices to be included as point of departure in the estimation process. By means of a Monte Carlo simulation experiment, we will show that under some circumstances this approach outperforms other competing estimators. Besides, a real-world case with a matrix of interregional trade is included to show the applicability of the suggested technique. Full article
(This article belongs to the Special Issue Information and Entropy)
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228 KiB  
Article
Different Senses of Entropy—Implications for Education
by Jesper Haglund, Fredrik Jeppsson and Helge Strömdahl
Entropy 2010, 12(3), 490-515; https://doi.org/10.3390/e12030490 - 12 Mar 2010
Cited by 20 | Viewed by 13628
Abstract
A challenge in the teaching of entropy is that the word has several different senses, which may provide an obstacle for communication. This study identifies five distinct senses of the word ‘entropy’, using the Principled Polysemy approach from the field of linguistics. A [...] Read more.
A challenge in the teaching of entropy is that the word has several different senses, which may provide an obstacle for communication. This study identifies five distinct senses of the word ‘entropy’, using the Principled Polysemy approach from the field of linguistics. A semantic network is developed of how the senses are related, using text excerpts from dictionaries, text books and text corpora. Educational challenges such as the existence of several formal senses of entropy and the intermediary position of entropy as disorder along the formal/non-formal scale are presented using a two-Dimensional Semiotic/semantic Analysing Schema (2-D SAS). Full article
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870 KiB  
Article
Negentropy Generation and Fractality in the Dry Friction of Polished Surfaces
by Pablo Fleurquin, Hugo Fort, Mordechai Kornbluth, Roman Sandler, Mordecai Segall and Fredy Zypman
Entropy 2010, 12(3), 480-489; https://doi.org/10.3390/e12030480 - 11 Mar 2010
Cited by 17 | Viewed by 8746
Abstract
We consider the Robin Hood model of dry friction to study entropy transfer during sliding. For the polished surface (steady state) we study the probability distribution of slips and find an exponential behavior for all the physically relevant asperity interaction-distance thresholds. In addition, [...] Read more.
We consider the Robin Hood model of dry friction to study entropy transfer during sliding. For the polished surface (steady state) we study the probability distribution of slips and find an exponential behavior for all the physically relevant asperity interaction-distance thresholds. In addition, we characterize the time evolution of the sample by its spatial fractal dimension and by its entropy content. Starting from an unpolished surface, the entropy decreases during the Robin Hood process, until it reaches a plateau; thereafter the system fluctuates above the critical height. This validates the notion that friction increases information in the neighborhood of the contacting surface at the expense of losing information in remote regions. We explain the practical relevance of these results for engineering surface processing such as honing. Full article
(This article belongs to the Special Issue Entropy and Friction Volume 2)
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93 KiB  
Article
Relaxation Processes and the Maximum Entropy Production Principle
by Paško Županović, Srećko Botrić, Davor Juretić and Domagoj Kuić
Entropy 2010, 12(3), 473-479; https://doi.org/10.3390/e12030473 - 11 Mar 2010
Cited by 8 | Viewed by 6786
Abstract
Spontaneous transitions of an isolated system from one macroscopic state to another (relaxation processes) are accompanied by a change of entropy. Following Jaynes’ MaxEnt formalism, it is shown that practically all the possible microscopic developments of a system, within a fixed time interval, [...] Read more.
Spontaneous transitions of an isolated system from one macroscopic state to another (relaxation processes) are accompanied by a change of entropy. Following Jaynes’ MaxEnt formalism, it is shown that practically all the possible microscopic developments of a system, within a fixed time interval, are accompanied by the maximum possible entropy change. In other words relaxation processes are accompanied by maximum entropy production. Full article
(This article belongs to the Special Issue What Is Maximum Entropy Production and How Should We Apply It?)
447 KiB  
Article
Combined Effects of Pipe Diameter, Reynolds Number and Wall Heat Flux and on Flow, Heat Transfer and Second-Law Characteristics of Laminar-Transitional Micro-Pipe Flows
by A. Alper Ozalp
Entropy 2010, 12(3), 445-472; https://doi.org/10.3390/e12030445 - 9 Mar 2010
Cited by 7 | Viewed by 11209
Abstract
Fluid flow, heat transfer and entropy generation characteristics of micro-pipes are investigated computationally by considering the simultaneous effects of pipe diameter, wall heat flux and Reynolds number in detail. Variable fluid property continuity, Navier-Stokes and energy equations are numerically handled for wide ranges [...] Read more.
Fluid flow, heat transfer and entropy generation characteristics of micro-pipes are investigated computationally by considering the simultaneous effects of pipe diameter, wall heat flux and Reynolds number in detail. Variable fluid property continuity, Navier-Stokes and energy equations are numerically handled for wide ranges of pipe diameter (d = 0.50–1.00 mm), wall heat flux (q''= 1000–2000 W/m2) and Reynolds number (Re = 1 – 2000), where the relative roughness is kept constant at e/d = 0.001 in the complete set of the scenarios considered. Computations indicated slight shifts in velocity profiles from the laminar character at Re = 500 with the corresponding shape factor (H) and intermittency values (γ) of H = 3.293→3.275 and γ = 0.041→0.051 (d = 1.00→0.50 mm). Moreover, the onset of transition was determined to move down to Retra = 1,656, 1,607, 1,491, 1,341 and 1,272 at d = 1.00, 0.90, 0.75, 0.60 and 0.50 mm, respectively. The impacts of pipe diameter on friction mechanism and heat transfer rates are evaluated to become more significant at high Reynolds numbers, resulting in the rise of energy loss data at the identical conditions as well. In cases with low pipe diameter and high Reynolds number, wall heat flux is determined to promote the magnitude of local thermal entropy generation rates. Local Bejan numbers are inspected to rise with wall heat flux at high Reynolds numbers, indicating that the elevating role of wall heat flux on local thermal entropy generation is dominant to the suppressing function of Reynolds number on local thermal entropy generation. Cross-sectional total entropy generation is computed to be most influenced by pipe diameter at high wall heat flux and low Reynolds numbers. Full article
(This article belongs to the Special Issue Exergy: Analysis and Applications)
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136 KiB  
Article
Entropy Transport Equation in Large Eddy Simulation for Exergy Analysis of Turbulent Combustion Systems
by Mehdi Safari, M. Reza H. Sheikhi, Mohammad Janbozorgi and Hameed Metghalchi
Entropy 2010, 12(3), 434-444; https://doi.org/10.3390/e12030434 - 8 Mar 2010
Cited by 30 | Viewed by 9995
Abstract
The transport equation of entropy is introduced in large eddy simulation to perform exergy analysis of turbulent combustion systems. The sources of exergy destruction can be evaluated by analyzing entropy generation terms, which appear in unclosed forms in this equation. The closure is [...] Read more.
The transport equation of entropy is introduced in large eddy simulation to perform exergy analysis of turbulent combustion systems. The sources of exergy destruction can be evaluated by analyzing entropy generation terms, which appear in unclosed forms in this equation. The closure is based on the filtered density function (FDF) methodology. The primary advantage of FDF is that chemical reaction and its entropy generation effects appear in closed forms. This methodology involves a stochastic model, which is being developed to account for the subgrid scale transport of entropy. Full article
(This article belongs to the Special Issue Exergy: Analysis and Applications)
206 KiB  
Article
Representing Entropy with Dispersion Sets
by Bernhard Kolarczyk
Entropy 2010, 12(3), 420-433; https://doi.org/10.3390/e12030420 - 8 Mar 2010
Cited by 1 | Viewed by 6805
Abstract
A novel representation of entropy is introduced, based on the heuristic concept of heat dispersion. To this end dispersion sets are defined, which describe both the heat transferred to a system and the associated entropy change. Some applications are discussed. Full article
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2031 KiB  
Article
From Talking Heads to Communicating Bodies: Cybersemiotics and Total Communication
by Ole Nedergaard Thomsen
Entropy 2010, 12(3), 390-419; https://doi.org/10.3390/e12030390 - 5 Mar 2010
Cited by 9 | Viewed by 13164
Abstract
Current linguistics is biased towards considering as object of scientific study only verbal language, i.e., ordinary language whose basic entities are words, sentences, and texts. By having this focus, the crucial non-verbal semiotic contributions from acts of bodily communication are left out [...] Read more.
Current linguistics is biased towards considering as object of scientific study only verbal language, i.e., ordinary language whose basic entities are words, sentences, and texts. By having this focus, the crucial non-verbal semiotic contributions from acts of bodily communication are left out of consideration. On the face of it, this is a strange situation, because, phenomenologically, when observing a communicating dyad, what appears to the senses is a multimodal semiotic display–the interactants produce acts of total communication, the linguistic part of which has in fact to be disentangled from the integral semiotic behavior. That a human being should in the first place be conceptualized as a ‘talking head’, rather than a ‘communicating body’, stems from at least four historically interrelated fountains: ancient Greek philosophy with its emphasis on logos as meaning both rational mind and verbal language/speech as well as with its rejection of rhetoric (including body language); Cartesian dualistic rationalism where the body was the animal, mechanistic part of a human being, unworthy for the Geisteswissenschaften; Saussure’s formal structuralism with its defocusing of the individual’s performance, parole, and its high focus on societal langue; and Chomskyan linguistics with its neglect of actual, also bodily, performance, and its total focus on an ideal mental grammatical computational competence. With the recent philosophy (‘in the flesh’) of the ‘embodied mind’, time has now come for integrating the (linguistic) head with the (other part of the communicating) body and seeing communication as total communication of the whole body. This means that the communicating mind is no longer restricted to its ‘rational’ aspects but has to be conceived full-scale as integrating also all kinds of ‘irrational’ factors, like emotions and motivations. Another, no less important, implication of the above is that an individual’s ‘language faculty’ is to be understood rather as a faculty of total communication–verbal and non-verbal semiotic behavior is an integrated, multi-modal whole of total communication performed by whole human organisms. Cybersemiotics offers itself here as the meta-theoretical, transdisciplinary framework within which this new paradigm of total communication can be developed. Full article
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250 KiB  
Article
Second-Law Analysis to Improve the Energy Efficiency of Screw Liquid Chillers
by Tzong-Shing Lee
Entropy 2010, 12(3), 375-389; https://doi.org/10.3390/e12030375 - 4 Mar 2010
Cited by 20 | Viewed by 10651
Abstract
This work applies the second-law analysis of thermodynamics to quantify the exergy destruction of the components of screw liquid chiller, and to identify the potential for each component to contribute to improve the overall energy efficiency of the system. Three screw liquid chiller [...] Read more.
This work applies the second-law analysis of thermodynamics to quantify the exergy destruction of the components of screw liquid chiller, and to identify the potential for each component to contribute to improve the overall energy efficiency of the system. Three screw liquid chiller units were built to demonstrate the feasibility of the model presented herein. Unit A was a 100 RT water-cooled screw liquid chiller. Unit B was modified from Unit A by switching the old condenser for a new one with a greater heat transfer, and Unit C was modified from Unit B by exchanging the compressor for a more efficient one. The results indicate that the compressor has the largest potential to improve energy efficiency, followed in order by the condenser, and then the evaporator. The second law analysis may help engineers to focus on the components with higher exergy destruction and quantify the extent to which modifying such components can influence, favorably or unfavorably, the performance of other components of the screw liquid chiller. Full article
(This article belongs to the Special Issue Exergy: Analysis and Applications)
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282 KiB  
Article
Estimation of an Entropy-based Functional
by Brian N. Maurizi
Entropy 2010, 12(3), 338-374; https://doi.org/10.3390/e12030338 - 3 Mar 2010
Cited by 2 | Viewed by 6767
Abstract
Given a function f from [0, 1] to the real line, we consider the (nonlinear) functional h obtained by evaluating the continuous entropy of the “density function” of f. Motivated by an application in signal processing, we wish to estimate h(f [...] Read more.
Given a function f from [0, 1] to the real line, we consider the (nonlinear) functional h obtained by evaluating the continuous entropy of the “density function” of f. Motivated by an application in signal processing, we wish to estimate h(f). Our main tool is a decomposition of h into two terms, which each have favorable scaling properties. We show that, if functions f and g satisfy a regularity condition, then the smallness of ∥fg and ∥f′g′, along with some basic control on derivatives of f and g, is sufficient to imply that h(f) and h(g) are close. Full article
234 KiB  
Article
Comparative Analysis of Networks of Phonologically Similar Words in English and Spanish
by Samuel Arbesman, Steven H. Strogatz and Michael S. Vitevitch
Entropy 2010, 12(3), 327-337; https://doi.org/10.3390/e12030327 - 2 Mar 2010
Cited by 43 | Viewed by 14344
Abstract
Previous network analyses of several languages revealed a unique set of structural characteristics. One of these characteristics—the presence of many smaller components (referred to as islands)—was further examined with a comparative analysis of the island constituents. The results showed that Spanish words [...] Read more.
Previous network analyses of several languages revealed a unique set of structural characteristics. One of these characteristics—the presence of many smaller components (referred to as islands)—was further examined with a comparative analysis of the island constituents. The results showed that Spanish words in the islands tended to be phonologically and semantically similar to each other, but English words in the islands tended only to be phonologically similar to each other. The results of this analysis yielded hypotheses about language processing that can be tested with psycholinguistic experiments, and offer insight into cross-language differences in processing that have been previously observed. Full article
(This article belongs to the Special Issue Complexity of Human Language and Cognition)
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15 KiB  
Correction
Fuhrman, G. Rehabilitating Information. Entropy, 2010, 12, 164-196
by Gary Fuhrman
Entropy 2010, 12(3), 326; https://doi.org/10.3390/e12030326 - 2 Mar 2010
Viewed by 6078
Abstract
The author would like to change the phrase “the financial ‘meltdown’ of 1908” into “the financial ‘meltdown’ of 2008” on page 190 of the article. [...] Full article
191 KiB  
Article
From ƒ-Divergence to Quantum Quasi-Entropies and Their Use
by Dénes Petz
Entropy 2010, 12(3), 304-325; https://doi.org/10.3390/e12030304 - 1 Mar 2010
Cited by 23 | Viewed by 7723
Abstract
Csiszár’s ƒ-divergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting, positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasi-entropy, which is related to some [...] Read more.
Csiszár’s ƒ-divergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting, positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasi-entropy, which is related to some other important concepts as covariance, quadratic costs, Fisher information, Cram´er-Rao inequality and uncertainty relation. It is remarkable that in the quantum case theoretically there are several Fisher information and variances. Fisher information are obtained as the Hessian of a quasi-entropy. A conjecture about the scalar curvature of a Fisher information geometry is explained. The described subjects are overviewed in details in the matrix setting. The von Neumann algebra approach is also discussed for uncertainty relation. Full article
125 KiB  
Article
Measurement Invariance, Entropy, and Probability
by Steven A. Frank and D. Eric Smith
Entropy 2010, 12(3), 289-303; https://doi.org/10.3390/e12030289 - 26 Feb 2010
Cited by 32 | Viewed by 8266
Abstract
We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint. [...] Read more.
We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint. We argue that a very common measurement scale is linear at small magnitudes grading into logarithmic at large magnitudes, leading to observations that often follow Student’s probability distribution which has a Gaussian shape for small fluctuations from the mean and a power law shape for large fluctuations from the mean. An inverse scaling often arises in which measures naturally grade from logarithmic to linear as one moves from small to large magnitudes, leading to observations that often follow a gamma probability distribution. A gamma distribution has a power law shape for small magnitudes and an exponential shape for large magnitudes. The two measurement scales are natural inverses connected by the Laplace integral transform. This inversion connects the two major scaling patterns commonly found in nature. We also show that superstatistics is a special case of an integral transform, and thus can be understood as a particular way in which to change the scale of measurement. Incorporating information about measurement scale into maximum entropy provides a general approach to the relations between measurement, information and probability. Full article
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