Entropy

by Greven, Andreas

• ISBN13: 9780691113388
• ISBN10: 0691113386
• Format: Hardcover
• Copyright: 2003-10-06
• Publisher: Princeton Univ Pr

Summary

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	(16)
1.1 Outline of the Book	4	(10)
1.2 Notations	14	(3)
PART I. FUNDAMENTAL CONCEPTS	17	(38)
Chapter 2. Entropy: a Subtle Concept in Thermodynamics
I. Müller	19	(18)
2.1 Origin of Entropy in Thermodynamics	19	(4)
2.2 Mechanical Interpretation of Entropy in the Kinetic Theory of Gases	23	(5)
2.2.1 Configurational Entropy	25	(3)
2.3 Entropy and Potential Energy of Gravitation	28	(2)
2.3.1 Planetary Atmospheres	28	(1)
2.3.2 Pfeffer Tube	29	(1)
2.4 Entropy and Intermolecular Energies	30	(2)
2.5 Entropy and Chemical Energies	32	(2)
2.6 Omissions	34	(1)
References	35	(2)
Chapter 3. Probabilistic Aspects of Entropy
H.-O. Georgii	37	(20)
3.1 Entropy as a Measure of Uncertainty	37	(2)
3.2 Entropy as a Measure of Information	39	(1)
3.3 Relative Entropy as a Measure of Discrimination	40	(3)
3.4 Entropy Maximization under Constraints	43	(2)
3.5 Asymptotics Governed by Entropy	45	(3)
3.6 Entropy Density of Stationary Processes and Fields	48	(4)
References	52	(3)
PART 2. ENTROPY IN THERMODYNAMICS	55	(142)
Chapter 4. Phenomenological Thermodynamics and Entropy Principles
K. Hutter and Y. Wang	57	(22)
4.1 Introduction	57	(1)
4.2 A Simple Classification of Theories of Continuum Thermodynamics	58	(5)
4.3 Comparison of Two Entropy Principles	63	(11)
4.3.1 Basic Equations	63	(3)
4.3.2 Generalized Coleman-Noll Evaluation of the Clausius-Duhem Inequality	66	(5)
4.3.3 Müller-Liu's Entropy Principle	71	(3)
4.4 Concluding Remarks	74	(1)
References	75	(4)
Chapter 5. Entropy in Nonequilibrium
I. Müller	79	(28)
5.1 Thermodynamics of Irreversible Processes and Rational Thermodynamics for Viscous, Heat-Conducting Fluids	79	(3)
5.2 Kinetic Theory of Gases, the Motivation for Extended Thermodynamics	82	(11)
5.2.1 A Remark on Temperature	82	(1)
5.2.2 Entropy Density and Entropy Flux	83	(1)
5.2.3 13-Moment Distribution. Maximization of Nonequilibrium Entropy	83	(1)
5.2.4 Balance Equations for Moments	84	(1)
5.2.5 Moment Equations for 13 Moments. Stationary Heat Conduction	85	(2)
5.2.6 Kinetic and Thermodynamic Temperatures	87	(2)
5.2.7 Moment Equations for 14 Moments. Minimum Entropy Production	89	(4)
5.3 Extended Thermodynamics	93	(10)
5.3.1 Paradoxes	93	(2)
5.3.2 Formal Structure	95	(3)
5.3.3 Pulse Speeds	98	(3)
5.3.4 Light Scattering	101	(2)
5.4 A Remark on Alternatives	103	(1)
References	104	(3)
Chapter 6. Entropy for Hyperbolic Conservation Laws
C.M. Dafermos	107	(14)
6.1 Introduction	107	(1)
6.2 Isothermal Thermoelasticity	108	(2)
6.3 Hyperbolic Systems of Conservation Laws	110	(3)
6.4 Entropy	113	(4)
6.5 Quenching of Oscillations	117	(2)
References	119	(2)
Chapter 7. Irreversibility and the Second Law of Thermodynamics
J. Uffink	121	(26)
7.1 Three Concepts of (Ir)reversibility	121	(3)
7.2 Early Formulations of the Second Law	124	(5)
7.3 Planck	129	(3)
7.4 Gibbs	132	(1)
7.5 Carathéodory	133	(7)
7.6 Lieb and Yngvason	140	(3)
7.7 Discussion	143	(2)
References	145	(2)
Chapter 8. The Entropy of Classical Thermodynamics
E.H. Lieb, J. Yngvason	147	(52)
8.1 A Guide to Entropy and the Second Law of Thermodynamics	148	(42)
8.2 Some Speculations and Open Problems	190	(2)
8.3 Some Remarks about Statistical Mechanics	192	(1)
References	193	(4)
PART 3. ENTROPY IN STOCHASTIC PROCESSES	197	(80)
Chapter 9. Large Deviations and Entropy
S.R.S. Varadhan	199	(16)
9.1 Where Does Entropy Come From?	199	(2)
9.2 Sanov's Theorem	201	(1)
9.3 What about Markov Chains?	202	(1)
9.4 Gibbs Measures and Large Deviations	203	(2)
9.5 Ventcel-Freidlin Theory	205	(1)
9.6 Entropy and Large Deviations	206	(3)
9.7 Entropy and Analysis	209	(2)
9.8 Hydrodynamic Scaling: an Example	211	(3)
References	214	(1)
Chapter 10. Relative Entropy for Random Motion in a Random Medium
F. den Hollander	215	(18)
10.1 Introduction	215	(9)
10.1.1 Motivation	215	(2)
10.1.2 A Branching Random Walk in a Random Environment	217	(1)
10.1.3 Particle Densities and Growth Rates	217	(2)
10.1.4 Interpretation of the Main Theorems	219	(1)
10.1.5 Solution of the Variational Problems	220	(3)
10.1.6 Phase Transitions	223	(1)
10.1.7 Outline	224	(1)
10.2 Two Extensions	224	(1)
10.3 Conclusion	225	(1)
10.4 Appendix: Sketch of the Derivation of the Main Theorems	226	(5)
10.4.1 Local Times of Random Walk	226	(2)
10.4.2 Large Deviations and Growth Rates	228	(2)
10.4.3 Relation between the Global and the Local Growth Rate	230	(1)
References	231	(2)
Chapter 11. Metastability and Entropy
E. Olivieri	233	(16)
11.1 Introduction	233	(2)
11.2 van der Waals Theory	235	(2)
11.3 Curie-Weiss Theory	237	(1)
11.4 Comparison between Mean-Field and Short-Range Models	237	(2)
11.5 The 'Restricted Ensemble'	239	(2)
11.6 The Pathwise Approach	241	(1)
11.7 Stochastic Ising Model. Metastability and Nucleation	241	(3)
11.8 First-Exit Problem for General Markov Chains	244	(2)
11.9 The First Descent Tube of Trajectories	246	(2)
11.10 Concluding Remarks	248	(1)
References	249	(2)
Chapter 12. Entropy Production in Driven Spatially Extended Systems
C. Maes	251	(18)
12.1 Introduction	251	(1)
12.2 Approach to Equilibrium	252	(2)
12.2.1 Boltzmann Entropy	253	(1)
12.2.2 Initial Conditions	254	(1)
12.3 Phenomenology of Steady-State Entropy Production	254	(1)
12.4 Multiplicity under Constraints	255	(3)
12.5 Gibbs Measures with an Involution	258	(3)
12.6 The Gibbs Hypothesis	261	(2)
12.6.1 Pathspace Measure Construction	262	(1)
12.6.2 Space-Time Equilibrium	262	(1)
12.7 Asymmetric Exclusion Processes	263	(3)
12.7.1 MEP for ASEP	263	(1)
12.7.2 LFT for ASEP	264	(2)
References	266	(3)
Chapter 13. Entropy: a Dialogue
J.L. Lebowitz, C. Maes	269	(10)
References	275	(2)
PART 4. ENTROPY AND INFORMATION	277	(60)
Chapter 14. Classical and Quantum Entropies: Dynamics and Information
F. Benatti	279	(20)
14.1 Introduction	279	(1)
14.2 Shannon and von Neumann Entropy	280	(3)
14.2.1 Coding for Classical Memoryless Sources	281	(1)
14.2.2 Coding for Quantum Memoryless Sources	282	(1)
14.3 Kolmogorov-Sinai Entropy	283	(4)
14.3.1 KS Entropy and Classical Chaos	285	(1)
14.3.2 KS Entropy and Classical Coding	285	(1)
14.3.3 KS Entropy and Algorithmic Complexity	286	(1)
14.4 Quantum Dynamical Entropies	287	(6)
14.4.1 Partitions of Unit and Decompositions of States	290	(1)
14.4.2 CNT Entropy: Decompositions of States	290	(2)
14.4.3 AF Entropy: Partitions of Unit	292	(1)
14.5 Quantum Dynamical Entropies: Perspectives	293	(3)
14.5.1 Quantum Dynamical Entropies and Quantum Chaos	295	(1)
14.5.2 Dynamical Entropies and Quantum Information	296	(1)
14.5.3 Dynamical Entropies and Quantum Randomness	296	(1)
References	296	(3)
Chapter 15. Complexity and Information in Data
J. Rissanen	299	(14)
15.1 Introduction	299	(2)
15.2 Basics of Coding	301	(2)
15.3 Kolmogorov Sufficient Statistics	303	(3)
15.4 Complexity	306	(2)
15.5 Information	308	(3)
15.6 Denoising with Wavelets	311	(1)
References	312	(1)
Chapter 16. Entropy in Dynamical Systems
L. -S. Young	313	(16)
16.1 Background	313	(3)
16.1.1 Dynamical Systems	313	(1)
16.1.2 Topological and Metric Entropies	314	(2)
16.2 Summary	316	(1)
16.3 Entropy, Lyapunov Exponents, and Dimension	317	(5)
16.3.1 Random Dynamical Systems	321	(1)
16.4 Other Interpretations of Entropy	322	(5)
16.4.1 Entropy and Volume Growth	322	(1)
16.4.2 Growth of Periodic Points and Horseshoes	323	(2)
16.4.3 Large Deviations and Rates of Escape	325	(2)
References	327	(2)
Chapter 17. Entropy in Ergodic Theory
M. Keane	329	(8)
References	335	(2)
Combined References	337	(14)
Index	351