Entropy

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.

Table of Contents:
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. Muller 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 Muller-Liu's Entropy Principle 71 4.4 Concluding Remarks 74 References 75 Chapter 5. Entropy in Nonequilibrium I. Muller 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 Caratheodory 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