Buch, Englisch, 448 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 728 g
Buch, Englisch, 448 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 728 g
Reihe: Princeton Series in Applied Mathematics
ISBN: 978-0-691-11338-8
Verlag: Princeton University Press
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.
Autoren/Hrsg.
Fachgebiete
- Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
- Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Technische Thermodynamik
- Naturwissenschaften Physik Thermodynamik
- Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
- Naturwissenschaften Physik Angewandte Physik Statistische Physik, Dynamische Systeme
Weitere Infos & Material
Preface xi
List of Contributors xiii
Chapter 1. Introduction
A.Greven, G.Keller, G.Warnecke 1
1.1 Outline of the Book 4
1.2 Notations 14
PART 1. FUNDAMENTAL CONCEPTS 17
Chapter 2. Entropy: a Subtle Concept in Thermodynamics
I. M?ller 19
2.1 Origin of Entropy in Thermodynamics 19
2.2 Mechanical Interpretation of Entropy in the Kinetic Theory of Gases 23
2.2.1 Configurational Entropy 25
2.3 Entropy and Potential Energy of Gravitation 28
2.3.1 Planetary Atmospheres 28
2.3.2 Pfeffer Tube 29
2.4 Entropy and Intermolecular Energies 30
2.5 Entropy and Chemical Energies 32
2.6 Omissions 34
References 35
Chapter 3. Probabilistic Aspects of Entropy
H. -O.Georgii 37
3.1 Entropy as a Measure of Uncertainty 37
3.2 Entropy as a Measure of Information 39
3.3 Relative Entropy as a Measure of Discrimination 40
3.4 Entropy Maximization under Constraints 43
3.5 Asymptotics Governed by Entropy 45
3.6 Entropy Density of Stationary Processes and Fields 48
References 52
PART 2.ENTROPY IN THERMODYNAMICS 55
Chapter 4. Phenomenological Thermodynamics and Entropy Principles
K.Hutter and Y.Wang 57
4.1 Introduction 57
4.2 A Simple Classification of Theories of Continuum Thermodynamics 58
4.3 Comparison of Two Entropy Principles 63
4.3.1 Basic Equations 63
4.3.2 Generalized Coleman-Noll Evaluation of the Clausius-Duhem Inequality 66
4.3.3 M?ller-Liu's Entropy Principle 71
4.4 Concluding Remarks 74
References 75
Chapter 5. Entropy in Nonequilibrium
I. M?ller 79
5.1 Thermodynamics of Irreversible Processes and Rational Thermodynamics for Viscous, Heat-Conducting Fluids 79
5.2 Kinetic Theory of Gases, the Motivation for Extended Thermodynamics 82
5.2.1 A Remark on Temperature 82
5.2.2 Entropy Density and Entropy Flux 83
5.2.3 13-Moment Distribution. Maximization of Nonequilibrium Entropy 83
5.2.4 Balance Equations for Moments 84
5.2.5 Moment Equations for 13 Moments. Stationary Heat Conduction 85
5.2.6 Kinetic and Thermodynamic Temperatures 87
5.2.7 Moment Equations for 14 Moments. Minimum Entropy Production 89
5.3 Extended Thermodynamics 93
5.3.1 Paradoxes 93
5.3.2 Formal Structure 95
5.3.3 Pulse Speeds 98
5.3.4 Light Scattering 101
5.4 A Remark on Alternatives 103
References 104
Chapter 6. Entropy for Hyperbolic Conservation Laws
C.M.Dafermos 107
6.1 Introduction 107
6.2 Isothermal Thermoelasticity 108
6.3 Hyperbolic Systems of Conservation Laws 110
6.4 Entropy 113
6.5 Quenching of Oscillations 117
References 119
Chapter 7. Irreversibility and the Second Law of Thermodynamics
J.Uffink 121
7.1 Three Concepts of (Ir)reversibility 121
7.2 Early Formulations of the Second Law 124
7.3 Planck 129
7.4 Gibbs 132
7.5 Carath?odory 133
7.6 Lieb and Yngvason 140
7.7 Discussion 143
References 145
Chapter 8. The Entropy of Classical Thermodynamics
E. H. Lieb, J. Yngvason 147
8.1 A Guide to Entropy and the Second Law of Thermodynamics 148
8.2 Some Speculations and Open Problems 190
8.3 Some Remarks about Statistical Mechanics 192
References 193
PART 3.ENTROPY IN STOCHASTIC PROCESSES 197
Chapter 9. Large Deviations and Entropy
S. R. S. Varadhan 199
9.1 Where Does Entropy Come From? 199
9.2 Sanov's Theorem 201
9.3 What about Markov Chains? 202
9.4 Gibbs Measures and Large Deviations 203
9.5 Ventcel-Freidlin Theory 205
9.6 Entropy and Large Deviations 206
9.7 Entropy and Analysis 209
9.8 Hydrodynamic Scaling: an Example 211
References 214
Chapter 10. Relative Entropy for Random Motion in a Random Medium
F. den Hollander 215
10.1 Introduction 215
10.1.1 Motivation 215
10.1.2 A Branching Random Walk in a Random Environment 217
10.1.3 Particle Densities and Growth Rates 217
10.1.4 Interpretation of the Main Theorems 219
10.1.5 Solution of the Variational Problems 220
10.1.6 Phase
