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 ex
Chapter
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
PART 2. ENTROPY IN THERMODYNAMICS
Chapter 4. Phenomenological Thermodynamics and Entropy Principles
4.2 A Simple Classification of Theories of Continuum Thermodynamics
4.3 Comparison of Two Entropy Principles
4.3.2 Generalized Coleman–Noll Evaluation of the Clausius–Duhem Inequality
4.3.3 Müller–Liu's Entropy Principle
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
5.4 A Remark on Alternatives
Chapter 6. Entropy for Hyperbolic Conservation Laws
6.2 Isothermal Thermoelasticity
6.3 Hyperbolic Systems of Conservation Laws
6.5 Quenching of Oscillations
Chapter 7. Irreversibility and the Second Law of Thermodynamics
7.1 Three Concepts of (Ir)reversibility
7.2 Early Formulations of the Second Law
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
PART 3. ENTROPY IN STOCHASTIC PROCESSES
Chapter 9. Large Deviations and Entropy
9.1 Where Does Entropy Come From?
9.3 What about Markov Chains?
9.4 Gibbs Measures and Large Deviations
9.5 Ventcel–Freidlin Theory
9.6 Entropy and Large Deviations
9.8 Hydrodynamic Scaling: an Example
Chapter 10. Relative Entropy for Random Motion in a Random Medium
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
10.4 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
Chapter 11. Metastability and Entropy
11.2 van der Waals 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
Chapter 12. Entropy Production in Driven Spatially Extended Systems
12.2 Approach to Equilibrium
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
Chapter 13. Entropy: a Dialogue
PART 4. ENTROPY AND INFORMATION
Chapter 14. Classical and Quantum Entropies: Dynamics and Information
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
Chapter 15. Complexity and Information in Data
15.3 Kolmogorov Sufficient Statistics
15.6 Denoising with Wavelets
Chapter 16. Entropy in Dynamical Systems
16.1.2 Topological and Metric Entropies
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
Chapter 17. Entropy in Ergodic Theory
Entropy
：Entropy
( Princeton Series in Applied Mathematics )
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