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Entropy : Princeton Series in Applied Mathematics - Andreas Greven


Princeton Series in Applied Mathematics

By: Andreas Greven (Editor), Gerhard Keller (Editor), Gerald Warnecke (Editor)


Published: 6th October 2003
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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.

"This book is that rare thing: an edited volume that will be a lasting contribution to the literature."--Ray Streater, King's College

Prefacep. xi
List of Contributorsp. xiii
Introductionp. 1
Outline of the Bookp. 4
Notationsp. 14
Fundamental Conceptsp. 17
Entropy: a Subtle Concept in Thermodynamicsp. 19
Origin of Entropy in Thermodynamicsp. 19
Mechanical Interpretation of Entropy in the Kinetic Theory of Gasesp. 23
Configurational Entropyp. 25
Entropy and Potential Energy of Gravitationp. 28
Planetary Atmospheresp. 28
Pfeffer Tubep. 29
Entropy and Intermolecular Energiesp. 30
Entropy and Chemical Energiesp. 32
Omissionsp. 34
Referencesp. 35
Probabilistic Aspects of Entropyp. 37
Entropy as a Measure of Uncertaintyp. 37
Entropy as a Measure of Informationp. 39
Relative Entropy as a Measure of Discriminationp. 40
Entropy Maximization under Constraintsp. 43
Asymptotics Governed by Entropyp. 45
Entropy Density of Stationary Processes and Fieldsp. 48
Referencesp. 52
Entropy in Thermodynamicsp. 55
Phenomenological Thermodynamics and Entropy Principlesp. 57
Introductionp. 57
A Simple Classification of Theories of Continuum Thermodynamicsp. 58
Comparison of Two Entropy Principlesp. 63
Basic Equationsp. 63
Generalized Coleman-Noll Evaluation of the Clausius-Duhem Inequalityp. 66
Muller-Liu's Entropy Principlep. 71
Concluding Remarksp. 74
Referencesp. 75
Entropy in Nonequilibriump. 79
Thermodynamics of Irreversible Processes and Rational Thermodynamics for Viscous, Heat-Conducting Fluidsp. 79
Kinetic Theory of Gases, the Motivation for Extended Thermodynamicsp. 82
A Remark on Temperaturep. 82
Entropy Density and Entropy Fluxp. 83
13-Moment Distribution. Maximization of Nonequilibrium Entropyp. 83
Balance Equations for Momentsp. 84
Moment Equations for 13 Moments. Stationary Heat Conductionp. 85
Kinetic and Thermodynamic Temperaturesp. 87
Moment Equations for 14 Moments. Minimum Entropy Productionp. 89
Extended Thermodynamicsp. 93
Paradoxesp. 93
Formal Structurep. 95
Pulse Speedsp. 98
Light Scatteringp. 101
A Remark on Alternativesp. 103
Referencesp. 104
Entropy for Hyperbolic Conservation Lawsp. 107
Introductionp. 107
Isothermal Thermoelasticityp. 108
Hyperbolic Systems of Conservation Lawsp. 110
Entropyp. 113
Quenching of Oscillationsp. 117
Referencesp. 119
Irreversibility and the Second Law of Thermodynamicsp. 121
Three Concepts of (Ir)reversibilityp. 121
Early Formulations of the Second Lawp. 124
Planckp. 129
Gibbsp. 132
Caratheodoryp. 133
Lieb and Yngvasonp. 140
Discussionp. 143
Referencesp. 145
The Entropy of Classical Thermodynamicsp. 147
A Guide to Entropy and the Second Law of Thermodynamicsp. 148
Some Speculations and Open Problemsp. 190
Some Remarks about Statistical Mechanicsp. 192
Referencesp. 193
Entropy in Stochastic Processesp. 197
Large Deviations and Entropyp. 199
Where Does Entropy Come From?p. 199
Sanov's Theoremp. 201
What about Markov Chains?p. 202
Gibbs Measures and Large Deviationsp. 203
Ventcel-Freidlin Theoryp. 205
Entropy and Large Deviationsp. 206
Entropy and Analysisp. 209
Hydrodynamic Scaling: an Examplep. 211
Referencesp. 214
Relative Entropy for Random Motion in a Random Mediump. 215
Introductionp. 215
Motivationp. 215
A Branching Random Walk in a Random Environmentp. 217
Particle Densities and Growth Ratesp. 217
Interpretation of the Main Theoremsp. 219
Solution of the Variational Problem
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691113388
ISBN-10: 0691113386
Series: Princeton Series in Applied Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 384
Published: 6th October 2003
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 3.18
Weight (kg): 0.68