Entropy.
Başlık:
Entropy.
Yazar:
Greven, Andreas.
ISBN:
9781400865222
Yazar Ek Girişi:
Fiziksel Tanımlama:
1 online resource (376 pages)
Seri:
Princeton Series in Applied Mathematics
İçerik:
Cover -- Title -- Copyright -- Contents -- Preface -- List of Contributors -- Chapter 1. Introduction -- 1.1 Outline of the Book -- 1.2 Notations -- PART 1. FUNDAMENTAL CONCEPTS -- Chapter 2. Entropy: a Subtle Concept in Thermodynamics -- 2.1 Origin of Entropy in Thermodynamics -- 2.2 Mechanical Interpretation of Entropy in the Kinetic Theory of Gases -- 2.2.1 Configurational Entropy -- 2.3 Entropy and Potential Energy of Gravitation -- 2.3.1 Planetary Atmospheres -- 2.3.2 Pfeffer Tube -- 2.4 Entropy and Intermolecular Energies -- 2.5 Entropy and Chemical Energies -- 2.6 Omissions -- References -- Chapter 3. Probabilistic Aspects of Entropy -- 3.1 Entropy as a Measure of Uncertainty -- 3.2 Entropy as a Measure of Information -- 3.3 Relative Entropy as a Measure of Discrimination -- 3.4 Entropy Maximization under Constraints -- 3.5 Asymptotics Governed by Entropy -- 3.6 Entropy Density of Stationary Processes and Fields -- References -- PART 2. ENTROPY IN THERMODYNAMICS -- Chapter 4. Phenomenological Thermodynamics and Entropy Principles -- 4.1 Introduction -- 4.2 A Simple Classification of Theories of Continuum Thermodynamics -- 4.3 Comparison of Two Entropy Principles -- 4.3.1 Basic Equations -- 4.3.2 Generalized Coleman-Noll Evaluation of the Clausius-Duhem Inequality -- 4.3.3 Müller-Liu's Entropy Principle -- 4.4 Concluding Remarks -- References -- Chapter 5. Entropy in Nonequilibrium -- 5.1 Thermodynamics of Irreversible Processes and Rational Thermodynamics for Viscous, Heat-Conducting Fluids -- 5.2 Kinetic Theory of Gases, the Motivation for Extended Thermodynamics -- 5.2.1 A Remark on Temperature -- 5.2.2 Entropy Density and Entropy Flux -- 5.2.3 13-Moment Distribution. Maximization of Nonequilibrium Entropy -- 5.2.4 Balance Equations for Moments -- 5.2.5 Moment Equations for 13 Moments. Stationary Heat Conduction.
5.2.6 Kinetic and Thermodynamic Temperatures -- 5.2.7 Moment Equations for 14 Moments. Minimum Entropy Production -- 5.3 Extended Thermodynamics -- 5.3.1 Paradoxes -- 5.3.2 Formal Structure -- 5.3.3 Pulse Speeds -- 5.3.4 Light Scattering -- 5.4 A Remark on Alternatives -- References -- Chapter 6. Entropy for Hyperbolic Conservation Laws -- 6.1 Introduction -- 6.2 Isothermal Thermoelasticity -- 6.3 Hyperbolic Systems of Conservation Laws -- 6.4 Entropy -- 6.5 Quenching of Oscillations -- References -- Chapter 7. Irreversibility and the Second Law of Thermodynamics -- 7.1 Three Concepts of (Ir)reversibility -- 7.2 Early Formulations of the Second Law -- 7.3 Planck -- 7.4 Gibbs -- 7.5 Carathéodory -- 7.6 Lieb and Yngvason -- 7.7 Discussion -- References -- Chapter 8. The Entropy of Classical Thermodynamics -- 8.1 A Guide to Entropy and the Second Law of Thermodynamics -- 8.2 Some Speculations and Open Problems -- 8.3 Some Remarks about Statistical Mechanics -- References -- PART 3. ENTROPY IN STOCHASTIC PROCESSES -- Chapter 9. Large Deviations and Entropy -- 9.1 Where Does Entropy Come From? -- 9.2 Sanov's Theorem -- 9.3 What about Markov Chains? -- 9.4 Gibbs Measures and Large Deviations -- 9.5 Ventcel-Freidlin Theory -- 9.6 Entropy and Large Deviations -- 9.7 Entropy and Analysis -- 9.8 Hydrodynamic Scaling: an Example -- References -- Chapter 10. Relative Entropy for Random Motion in a Random Medium -- 10.1 Introduction -- 10.1.1 Motivation -- 10.1.2 A Branching Random Walk in a Random Environment -- 10.1.3 Particle Densities and Growth Rates -- 10.1.4 Interpretation of the Main Theorems -- 10.1.5 Solution of the Variational Problems -- 10.1.6 Phase Transitions -- 10.1.7 Outline -- 10.2 Two Extensions -- 10.3 Conclusion -- 10.4 Appendix: Sketch of the Derivation of the Main Theorems -- 10.4.1 Local Times of Random Walk.
10.4.2 Large Deviations and Growth Rates -- 10.4.3 Relation between the Global and the Local Growth Rate -- References -- Chapter 11. Metastability and Entropy -- 11.1 Introduction -- 11.2 van der Waals Theory -- 11.3 Curie-Weiss Theory -- 11.4 Comparison between Mean-Field and Short-Range Models -- 11.5 The 'Restricted Ensemble' -- 11.6 The Pathwise Approach -- 11.7 Stochastic Ising Model. Metastability and Nucleation -- 11.8 First-Exit Problem for General Markov Chains -- 11.9 The First Descent Tube of Trajectories -- 11.10 Concluding Remarks -- References -- Chapter 12. Entropy Production in Driven Spatially Extended Systems -- 12.1 Introduction -- 12.2 Approach to Equilibrium -- 12.2.1 Boltzmann Entropy -- 12.2.2 Initial Conditions -- 12.3 Phenomenology of Steady-State Entropy Production -- 12.4 Multiplicity under Constraints -- 12.5 Gibbs Measures with an Involution -- 12.6 The Gibbs Hypothesis -- 12.6.1 Pathspace Measure Construction -- 12.6.2 Space-Time Equilibrium -- 12.7 Asymmetric Exclusion Processes -- 12.7.1 MEP for ASEP -- 12.7.2 LFT for ASEP -- References -- Chapter 13. Entropy: a Dialogue -- References -- PART 4. ENTROPY AND INFORMATION -- Chapter 14. Classical and Quantum Entropies: Dynamics and Information -- 14.1 Introduction -- 14.2 Shannon and von Neumann Entropy -- 14.2.1 Coding for Classical Memory1ess Sources -- 14.2.2 Coding for Quantum Memoryless Sources -- 14.3 Kolmogorov-Sinai Entropy -- 14.3.1 KS Entropy and Classical Chaos -- 14.3.2 KS Entropy and Classical Coding -- 14.3.3 KS Entropy and Algorithmic Complexity -- 14.4 Quantum Dynamical Entropies -- 14.4.1 Partitions of Unit and Decompositions of States -- 14.4.2 CNT Entropy: Decompositions of States -- 14.4.3 AF Entropy: Partitions of Unit -- 14.5 Quantum Dynamical Entropies: Perspectives -- 14.5.1 Quantum Dynamical Entropies and Quantum Chaos.
14.5.2 Dynamical Entropies and Quantum Information -- 14.5.3 Dynamical Entropies and Quantum Randomness -- References -- Chapter 15. Complexity and Information in Data -- 15.1 Introduction -- 15.2 Basics of Coding -- 15.3 Kolmogorov Sufficient Statistics -- 15.4 Complexity -- 15.5 Information -- 15.6 Denoising with Wavelets -- References -- Chapter 16. Entropy in Dynamical Systems -- 16.1 Background -- 16.1.1 Dynamical Systems -- 16.1.2 Topological and Metric Entropies -- 16.2 Summary -- 16.3 Entropy, Lyapunov Exponents, and Dimension -- 16.3.1 Random Dynamical Systems -- 16.4 Other Interpretations of Entropy -- 16.4.1 Entropy and Volume Growth -- 16.4.2 Growth of Periodic Points and Horseshoes -- 16.4.3 Large Deviations and Rates of Escape -- References -- Chapter 17. Entropy in Ergodic Theory -- References -- Combined References -- Index.
Özet:
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.
Notlar:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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