Entropy

Princeton University Press
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The concept of entropy arose in the physical sciences during the nineteenth century, particularly in thermodynamics and statistical physics, as a measure of the equilibria and evolution of thermodynamic systems. Two main views developed: the macroscopic view formulated originally by Carnot, Clausius, Gibbs, Planck, and Caratheodory and the microscopic approach associated with Boltzmann and Maxwell. Since then both approaches have made possible deep insights into the nature and behavior of thermodynamic and other microscopically unpredictable processes. However, the mathematical tools used have later developed independently of their original physical background and have led to a plethora of methods and differing conventions.

The aim of this book is to identify the unifying threads by providing surveys of the uses and concepts of entropy in diverse areas of mathematics and the physical sciences. Two major threads, emphasized throughout the book, are variational principles and Ljapunov functionals. The book starts by providing basic concepts and terminology, illustrated by examples from both the macroscopic and microscopic lines of thought. In-depth surveys covering the macroscopic, microscopic and probabilistic approaches follow. Part I gives a basic introduction from the views of thermodynamics and probability theory. Part II collects surveys that look at the macroscopic approach of continuum mechanics and physics. Part III deals with the microscopic approach exposing the role of entropy as a concept in probability theory, namely in the analysis of the large time behavior of stochastic processes and in the study of qualitative properties of models in statistical physics. Finally in Part IV applications in dynamical systems, ergodic and information theory are presented.


The chapters were written to provide as cohesive an account as possible, making the book accessible to a wide range of graduate students and researchers. Any scientist dealing with systems that exhibit entropy will find the book an invaluable aid to their understanding.

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About the author

Andreas Greven and Gerhard Kellerare Professors of Mathematics at the University of Erlangen. Gerald Warnecke is Professor of Numerical Mathematics at the University of Magdeburg.
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Additional Information

Publisher
Princeton University Press
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Published on
Dec 31, 2003
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Pages
358
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ISBN
9780691113388
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Best For
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Language
English
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Genres
Mathematics / Applied
Science / Mechanics / Thermodynamics
Science / Physics / Mathematical & Computational
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Content Protection
This content is DRM protected.
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Eligible for Family Library

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Gerald Warnecke
Whatdoasupernovaexplosioninouterspace,?owaroundanairfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite di?erent. So are the motivations for studying them scienti?cally. The super- 8 nova is a thermo-nuclear explosion on a scale of 10 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In ?ows around airfoils of commercial airliners at the scale of 3 10 cm shock waves occur that in?uence the stability of the wings as well as fuel consumption in ?ight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical 1 process, and must be avoided since it damages motors. The scale is 10 cm and these processes must be optimized for e?ciency and environmental conside- tions. The common thread is that the underlying ?uid ?ows may at a certain scale of observation be described by basically the same type of hyperbolic s- tems of partial di?erential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scienti?c progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial di?erential equationsareamajor?eldofresearchinmathematics. Asubstantialportionof mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more spacedimensionsstillposeoneofthemainchallengestomodernmathematics.
Andreas Greven
The concept of entropy arose in the physical sciences during the nineteenth century, particularly in thermodynamics and statistical physics, as a measure of the equilibria and evolution of thermodynamic systems. Two main views developed: the macroscopic view formulated originally by Carnot, Clausius, Gibbs, Planck, and Caratheodory and the microscopic approach associated with Boltzmann and Maxwell. Since then both approaches have made possible deep insights into the nature and behavior of thermodynamic and other microscopically unpredictable processes. However, the mathematical tools used have later developed independently of their original physical background and have led to a plethora of methods and differing conventions.

The aim of this book is to identify the unifying threads by providing surveys of the uses and concepts of entropy in diverse areas of mathematics and the physical sciences. Two major threads, emphasized throughout the book, are variational principles and Ljapunov functionals. The book starts by providing basic concepts and terminology, illustrated by examples from both the macroscopic and microscopic lines of thought. In-depth surveys covering the macroscopic, microscopic and probabilistic approaches follow. Part I gives a basic introduction from the views of thermodynamics and probability theory. Part II collects surveys that look at the macroscopic approach of continuum mechanics and physics. Part III deals with the microscopic approach exposing the role of entropy as a concept in probability theory, namely in the analysis of the large time behavior of stochastic processes and in the study of qualitative properties of models in statistical physics. Finally in Part IV applications in dynamical systems, ergodic and information theory are presented.

The chapters were written to provide as cohesive an account as possible, making the book accessible to a wide range of graduate students and researchers. Any scientist dealing with systems that exhibit entropy will find the book an invaluable aid to their understanding.

Gerald Warnecke
Whatdoasupernovaexplosioninouterspace,?owaroundanairfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite di?erent. So are the motivations for studying them scienti?cally. The super- 8 nova is a thermo-nuclear explosion on a scale of 10 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In ?ows around airfoils of commercial airliners at the scale of 3 10 cm shock waves occur that in?uence the stability of the wings as well as fuel consumption in ?ight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical 1 process, and must be avoided since it damages motors. The scale is 10 cm and these processes must be optimized for e?ciency and environmental conside- tions. The common thread is that the underlying ?uid ?ows may at a certain scale of observation be described by basically the same type of hyperbolic s- tems of partial di?erential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scienti?c progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial di?erential equationsareamajor?eldofresearchinmathematics. Asubstantialportionof mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more spacedimensionsstillposeoneofthemainchallengestomodernmathematics.
Heinrich Freistühler
The Eighth International Conference on Hyperbolic Problems - Theory, Nu merics, Applications, was held in Magdeburg, Germany, from February 27 to March 3, 2000. It was attended by over 220 participants from many European countries as well as Brazil, Canada, China, Georgia, India, Israel, Japan, Taiwan, und the USA. There were 12 plenary lectures, 22 further invited talks, and around 150 con tributed talks in parallel sessions as well as posters. The speakers in the parallel sessions were invited to provide a poster in order to enhance the dissemination of information. Hyperbolic partial differential equations describe phenomena of material or wave transport in physics, biology and engineering, especially in the field of fluid mechanics. Despite considerable progress, the mathematical theory is still strug gling with fundamental open problems concerning systems of such equations in multiple space dimensions. For various applications the development of accurate and efficient numerical schemes for computation is of fundamental importance. Applications touched in these proceedings concern one-phase and multiphase fluid flow, phase transitions, shallow water dynamics, elasticity, extended ther modynamics, electromagnetism, classical and relativistic magnetohydrodynamics, cosmology. Contributions to the abstract theory of hyperbolic systems deal with viscous and relaxation approximations, front tracking and wellposedness, stability ofshock profiles and multi-shock patterns, traveling fronts for transport equations. Numerically oriented articles study finite difference, finite volume, and finite ele ment schemes, adaptive, multiresolution, and artificial dissipation methods.
Gerhard Keller
Die Gestaltung und Realisierung integrierter Infonnationssysteme in Untemehmen ist eine umfangreiche und komplexe Aufgabe. Viele Integrationsansatze sind techniklastig und berucksichtigen organisatorische Aspekte nur ungeniigend Dariiber hinaus dominieren bei Planung und Entwurf von Infonnationssystemen haufig die rein funktionalen Anforderungen der Fachbereiche - mit der Folge, daB funktionale Organisationsstrukturen verfestigt werden. Fortschritte auf dem Gebiet der Infonnationstechnologie, wie die Vemetzung von DV Systemen oder effiziente Datenbanksysteme, ennoglichen eine flexiblere Gestaltung der Informationssysteme. In Verbindung mit neuen Organisationskonzepten kann sich die Infonnationssystemgestaltung zunehmend von dem technisch-funktionalen Detenninismus lOsen. Ziel muB es sein, Planung und Realisierung integrierter Infonnationssysteme starker an prozeBorientierten und untemehmensspezifischen Anforderungen auszurichten. Infonnations und Organisationsstrukturen sind so umzugestalten, daB sie einer flexiblen und ganzheitlichen Aufgabenbearbeitung Rechnung tragen. Die vorliegende Arbeit beschreibt ein Konzept zur Planung und Realisierung objektorientierter Organisationsstrukturen in Industriebetrieben. Anhand der zentralen Objekte "Auf trag" und "Produkt" werden die logisch zusammengehOrenden Aufgaben innerhalb der planenden Bereiche von Industrieunternehmen analysien und Gestaltungsempfehlungen ffir Organisationsstrukturen mit kurzen Kommunikationswegen abgeleitet. Zur Darstellung der oben genannten Zielsetzung wird im ersten Schritt ein allgemein giiltiges Konzept zur Untemehmensmodellierung entwickelt und der prozeBorientierte Aspekt vertiefend diskutien. In einem zweiten Schritt wird aus den marktorientierten, den DV -technischen und den organisatorischen Entwicklungen auf dem Sektor der Industriebetriebe heraus der Handlungsbedarf zur Bildung von objektorientienen Organisationsstrukturen aufgezeigt und das anwendungsbezogene objektorientiene Fachkonzept der "Planungsinsel" entworfen. AnschlieBend werden mit Hilfe des entwickelten Methodenkonzepts die entworfenen Aspekte des Anwendungskonzepts in ProzeBmodelle umgesetzt und unterschiedliche objektorientiene Organisationskonzepte aufgezeigt.
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