Entropy.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.