Entropy
1st Edition
Published on 8. September 2014
384 pages
978-1-4008-6522-2 (ISBN)
Description
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.
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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Andreas Greven | Gerhard Keller | Gerald Warnecke
Entropy
Book
10/2003
Princeton University Press
€109.60
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Persons
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.
Content
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. Müller 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 Müller-Liu's Entropy Principle 71
4.4 Concluding Remarks 74
References 75
Chapter 5. Entropy in Nonequilibrium
I. Müller 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 Carathéodory 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 Transitions 223
10.1.7 Outline 224
10.2 Two Extensions 224
10.3 Conclusion 225
10.4 Appendix: Sketch of the Derivation of the Main Theorems 226
10.4.1 Local Times of Random Walk 226
10.4.2 Large Deviations and Growth Rates 228
10.4.3 Relation between the Global and the Local Growth Rate 230
References 231
Chapter 11. Metastability and Entropy
E. Olivieri 233
11.1 Introduction 233
11.2 van der Waals Theory 235
11.3 Curie-Weiss Theory 237
11.4 Comparison between Mean-Field and Short-Range Models 237
11.5 The 'Restricted Ensemble' 239
11.6 The Pathwise Approach 241
11.7 Stochastic Ising Model. Metastability and Nucleation 241
11.8 First-Exit Problem for General Markov Chains 244
11.9 The First Descent Tube of Trajectories 246
11.10 Concluding Remarks 248
References 249
Chapter 12. Entropy Production in Driven Spatially Extended Systems
C. Maes 251
12.1 Introduction 251
12.2 Approach to Equilibrium 252
12.2.1 Boltzmann Entropy 253
12.2.2 Initial Conditions 254
12.3 Phenomenology of Steady-State Entropy Production 254
12.4 Multiplicity under Constraints 255
12.5 Gibbs Measures with an Involution 258
12.6 The Gibbs Hypothesis 261
12.6.1 Pathspace Measure Construction 262
12.6.2 Space-Time Equilibrium 262
12.7 Asymmetric Exclusion Processes 263
12.7.1 MEP for ASEP 263
12.7.2 LFT for ASEP 264
References 266
Chapter 13. Entropy: a Dialogue
J. L. Lebowitz, C. Maes 269
References 275
PART 4.ENTROPY AND INFORMATION 277
Chapter 14. Classical and Quantum Entropies:Dynamics and Information
F. Benatti 279
14.1 Introduction 279
14.2 Shannon and von Neumann Entropy 280
14.2.1 Coding for Classical Memoryless Sources 281
14.2.2 Coding for Quantum Memoryless Sources 282
14.3 Kolmogorov-Sinai Entropy 283
14.3.1 KS Entropy and Classical Chaos 285
14.3.2 KS Entropy and Classical Coding 285
14.3.3 KS Entropy and Algorithmic Complexity 286
14.4 Quantum Dynamical Entropies 287
14.4.1 Partitions of Unit and Decompositions of States 290
14.4.2 CNT Entropy: Decompositions of States 290
14.4.3 AF Entropy: Partitions of Unit 292
14.5 Quantum Dynamical Entropies: Perspectives 293
14.5.1 Quantum Dynamical Entropies and Quantum Chaos 295
14.5.2 Dynamical Entropies and Quantum Information 296
14.5.3 Dynamical Entropies and Quantum Randomness 296
References 296
Chapter 15. Complexity and Information in Data
J. Rissanen 299
15.1 Introduction 299
15.2 Basics of Coding 301
15.3 Kolmogorov Sufficient Statistics 303
15.4 Complexity 306
15.5 Information 308
15.6 Denoising with Wavelets 311
References 312
Chapter 16. Entropy in Dynamical Systems
L. -S. Young 313
16.1 Background 313
16.1.1 Dynamical Systems 313
16.1.2 Topological and Metric Entropies 314
16.2 Summary 316
16.3 Entropy, Lyapunov Exponents, and Dimension 317
16.3.1 Random Dynamical Systems 321
16.4 Other Interpretations of Entropy 322
16.4.1 Entropy and Volume Growth 322
16.4.2 Growth of Periodic Points and Horseshoes 323
16.4.3 Large Deviations and Rates of Escape 325
References 327
Chapter 17. Entropy in Ergodic Theory
M. Keane 329
References 335
Combined References 337
Index 351
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. Müller 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 Müller-Liu's Entropy Principle 71
4.4 Concluding Remarks 74
References 75
Chapter 5. Entropy in Nonequilibrium
I. Müller 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 Carathéodory 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 Transitions 223
10.1.7 Outline 224
10.2 Two Extensions 224
10.3 Conclusion 225
10.4 Appendix: Sketch of the Derivation of the Main Theorems 226
10.4.1 Local Times of Random Walk 226
10.4.2 Large Deviations and Growth Rates 228
10.4.3 Relation between the Global and the Local Growth Rate 230
References 231
Chapter 11. Metastability and Entropy
E. Olivieri 233
11.1 Introduction 233
11.2 van der Waals Theory 235
11.3 Curie-Weiss Theory 237
11.4 Comparison between Mean-Field and Short-Range Models 237
11.5 The 'Restricted Ensemble' 239
11.6 The Pathwise Approach 241
11.7 Stochastic Ising Model. Metastability and Nucleation 241
11.8 First-Exit Problem for General Markov Chains 244
11.9 The First Descent Tube of Trajectories 246
11.10 Concluding Remarks 248
References 249
Chapter 12. Entropy Production in Driven Spatially Extended Systems
C. Maes 251
12.1 Introduction 251
12.2 Approach to Equilibrium 252
12.2.1 Boltzmann Entropy 253
12.2.2 Initial Conditions 254
12.3 Phenomenology of Steady-State Entropy Production 254
12.4 Multiplicity under Constraints 255
12.5 Gibbs Measures with an Involution 258
12.6 The Gibbs Hypothesis 261
12.6.1 Pathspace Measure Construction 262
12.6.2 Space-Time Equilibrium 262
12.7 Asymmetric Exclusion Processes 263
12.7.1 MEP for ASEP 263
12.7.2 LFT for ASEP 264
References 266
Chapter 13. Entropy: a Dialogue
J. L. Lebowitz, C. Maes 269
References 275
PART 4.ENTROPY AND INFORMATION 277
Chapter 14. Classical and Quantum Entropies:Dynamics and Information
F. Benatti 279
14.1 Introduction 279
14.2 Shannon and von Neumann Entropy 280
14.2.1 Coding for Classical Memoryless Sources 281
14.2.2 Coding for Quantum Memoryless Sources 282
14.3 Kolmogorov-Sinai Entropy 283
14.3.1 KS Entropy and Classical Chaos 285
14.3.2 KS Entropy and Classical Coding 285
14.3.3 KS Entropy and Algorithmic Complexity 286
14.4 Quantum Dynamical Entropies 287
14.4.1 Partitions of Unit and Decompositions of States 290
14.4.2 CNT Entropy: Decompositions of States 290
14.4.3 AF Entropy: Partitions of Unit 292
14.5 Quantum Dynamical Entropies: Perspectives 293
14.5.1 Quantum Dynamical Entropies and Quantum Chaos 295
14.5.2 Dynamical Entropies and Quantum Information 296
14.5.3 Dynamical Entropies and Quantum Randomness 296
References 296
Chapter 15. Complexity and Information in Data
J. Rissanen 299
15.1 Introduction 299
15.2 Basics of Coding 301
15.3 Kolmogorov Sufficient Statistics 303
15.4 Complexity 306
15.5 Information 308
15.6 Denoising with Wavelets 311
References 312
Chapter 16. Entropy in Dynamical Systems
L. -S. Young 313
16.1 Background 313
16.1.1 Dynamical Systems 313
16.1.2 Topological and Metric Entropies 314
16.2 Summary 316
16.3 Entropy, Lyapunov Exponents, and Dimension 317
16.3.1 Random Dynamical Systems 321
16.4 Other Interpretations of Entropy 322
16.4.1 Entropy and Volume Growth 322
16.4.2 Growth of Periodic Points and Horseshoes 323
16.4.3 Large Deviations and Rates of Escape 325
References 327
Chapter 17. Entropy in Ergodic Theory
M. Keane 329
References 335
Combined References 337
Index 351
