Many people, including physicists, are confused about what the Second Law of thermodynamics really means, about how it relates to the arrow of time, and about whether it can be derived from classical mechanics. They also wonder what entropy really is: Is it all about information? But, if so, then, what is its relation to fluxes of heat?
One might ask similar questions about probabilities: Do they express subjective judgments by us, humans, or do they reflect facts about the world, i.e. frequencies. And what notion of probability is used in the natural sciences, in particular statistical mechanics?
This book addresses all of these questions in the clear and pedagogical style for which the author is known. Although valuable as accompaniment to an undergraduate course on statistical mechanics or thermodynamics, it is not a standard course book. Instead it addresses both the essentials and the many subtle questions that are usually brushed under the carpet in such courses. As one of the most lucid accounts of the above questions, it provides enlightening reading for all those seeking answers, including students, lecturers, researchers and philosophers of science.
Author(s): Jean Bricmont
Series: Undergraduate Lecture Notes in Physics
Edition: 1
Publisher: Springer Nature Switzerland AG
Year: 2022
Language: English
Pages: 368
City: Cham, Switzerland
Tags: Thermodynamics, Statistical Mechanics, Equilibrium, Entropy, Phase Transitions
Contents
1 Introduction
2 Probability
2.1 Introduction
2.2 ``Subjective'' Versus ``Objective'' Probabilities
2.2.1 The Indifference Principle
2.2.2 Cox' ``Axioms'' and Theorem
2.2.3 Bayesian Updating
2.2.4 Objections to the ``Subjective'' Approach
2.2.5 Bertrand's Paradox
2.3 The Law of Large Numbers
2.3.1 A Simple Example
2.3.2 A More General Result
2.3.3 Corrections to the Law of Large Numbers
2.4 The Law of Large Numbers and the Frequentist Interpretation
2.5 Explanations and Probabilistic Explanations
2.6 Final Remarks
2.7 Summary
2.8 Exercises
2.A Appendix A: Measure Theory
2.A.1 Definition of a Measure
2.A.2 Constructions of Measures
2.A.3 Integration
2.A.4 Approximation of Integrals
2.A.5 Invariant Measures
2.A.6 Probability Densities, Marginal and Conditional Probabilities
2.A.7 Cantor Sets and Measures
2.B Appendix B: Proofs of the Law of Large Numbers and of the Central Limit Theorem
3 Classical Mechanics
3.1 Introduction
3.2 Newton's Laws
3.3 Hamilton's Equations
3.3.1 The Hamiltonian Flow
3.3.2 Conservation of Energy
3.4 Liouville's Theorem and Measure
3.4.1 Liouville's Theorem
3.4.2 The Liouville Measure
3.5 Time Reversibility
3.6 Summary
3.7 Exercises
4 Dynamical Systems
4.1 Introduction
4.2 Poincaré's Recurrence Theorem, or The Eternal Return
4.2.1 Proof of Poincaré's Recurrence Theorem
4.3 Ergodic Theorems
4.3.1 Examples and Applications
4.3.2 Ergodicity and the Law of Large Numbers
4.4 Mixing
4.5 Sensitive Dependence on Initial Conditions
4.6 Statistical Theory of Dynamical Systems
4.6.1 Itineraries and Coding
4.6.2 Strange Attractors
4.7 Dynamical Entropies
4.8 Determinism and Predictability
4.9 Summary
4.10 Exercises
5 Thermodynamics
5.1 Introduction
5.2 The Zeroth Law
5.3 The First Law
5.3.1 Work
5.3.2 Heat
5.4 The Second Law
5.4.1 The Carnot Cycle
5.4.2 Proof of the Equivalence Between Lord Kelvin's Version and Clausius' Version of the Second Law
5.5 The Thermodynamic Entropy
5.5.1 Definition of the Entropy Function
5.5.2 The Second Law or the Increase of the Thermodynamic Entropy
5.6 Other Thermodynamic Functions
5.6.1 The Particle Number
5.6.2 Fundamental Relations
5.6.3 The Helmholtz Free Energy
5.6.4 The Grand Potential
5.6.5 The Principle of Maximum Entropy and Equilibrium Conditions
5.7 An Example: The Ideal Gas
5.7.1 Mathematical Identities and the Gibbs–Duhem Relations
5.7.2 Derivation of the Fundamental Relation for an Ideal Gas
5.7.3 The Fundamental Relation in Other Variables for an Ideal Gas
5.7.4 Adiabatic Transformations
5.7.5 Be Careful with Derivatives!
5.8 What Is All This Good For?
5.9 Summary
5.10 Exercises
5.A Differential Forms
5.B Legendre Transforms
5.B.1 Mathematical Definition
5.B.2 Physical Applications
5.C Physical Units and Boltzmann's Constant
6 Equilibrium Statistical Mechanics
6.1 Microstates and Macrostates
6.2 Dominance of the Equilibrium Macrostate
6.3 Typicality
6.4 Entropy in Equilibrium
6.5 Other Equilibrium Potentials
6.6 The Equilibrium Ensembles
6.6.1 Definition of the Ensembles
6.6.2 The Gibbs Entropy
6.6.3 Equivalence of Ensembles
6.6.4 The Meaning of Ensembles
6.7 Justification of the Entropy Formula (6.4.2)
6.8 What Justifies the Microcanonical Distribution?
6.9 Summary
6.10 Exercises
6.A Asymptotics
6.A.1 Laplace's Method
6.A.2 Stirling's Formula
6.B ``Derivation'' of Formula (6.5.4)
6.C An Intuitive Formula for the Entropy
7 Information-Theoretic and Predictive Statistical Mechanics
7.1 Introduction
7.2 The Shannon Entropy
7.3 The Maximum Entropy Principle
7.4 The Ensembles
7.5 Continuous Distributions and Relative Entropy
7.6 ``Derivation'' of the Second Law
7.7 Shannon's Entropy and Communication
7.7.1 Information Content of a Message
7.7.2 Encoding Messages
7.8 Evaluation of the Information Theoretic Approach
7.8.1 What Do Probabilities Mean Here?
7.8.2 Jaynes' Approach
7.9 Summary
7.10 Exercises
7.A Proof that (7.2.6) Implies A(N)= klogN
7.B Jensen's Inequality
8 Approach to Equilibrium
8.1 Boltzmann's Scheme
8.1.1 Microstates and Macrostates
8.1.2 Derivation of Macroscopic Laws from Microscopic Ones
8.1.3 Solution of the (Apparent) Reversibility Paradox
8.1.4 Irreversibility and Probabilistic Explanations
8.1.5 Time Dependent Boltzmann Entropy and the Second Law
8.2 Answers to the Classical Objections
8.2.1 Objections from Loschmidt and Zermelo
8.2.2 An Objection from Poincaré
8.2.3 Objections to Typicality Arguments
8.3 Ergodicity, Mixing, and Other Wrong Turns
8.3.1 Ergodicity
8.3.2 Mixing
8.3.3 The Brussels-Austin School
8.3.4 Real Systems Are Never Isolated
8.4 Maxwell's Demon
8.5 The Origins of the Low Entropy States
8.6 The Boltzmann Equation
8.6.1 The Relation Between the Boltzmann Equation and the Full Evolution of Measures
8.7 Simple Models
8.7.1 Ehrenfest's Urns
8.7.2 Kac Ring Model
8.7.3 Uncoupled Baker's Maps
8.7.4 Ideal Gas in a Box
8.7.5 The Abiabatic Piston
8.8 Boltzmann Versus Gibbs Entropies
8.8.1 Another Time Evolution of Measures
8.8.2 The Example of the Uncoupled Baker's Maps of Sect.8.7.3
8.9 Conclusion: Are Entropy and Irreversibility Subjective?
8.10 Dynamical Systems Vs Statistical Mechanics
8.11 Summary
8.12 Exercises
8.A Boltzmannian Quotes
8.B Quotes Critical of Boltzmann
9 Phase Transitions
9.1 Phenomenology
9.2 Lattice Models and Gibbs States
9.3 Mean Field Theory
9.4 One Dimension
9.5 High Temperature Expansions
9.6 Low Temperatures and the Peierls Argument
9.7 Other Models
9.7.1 Trivial Extensions
9.7.2 Long Range Interactions
9.7.3 Many Body Interactions
9.7.4 Continuous Spins
9.7.5 Models with a Continuous Symmetry
9.7.6 Non Translation Invariant Gibbs States
9.8 The Pirogov–Sinai Theory
9.8.1 A Simple Example
9.8.2 The General Pirogov–Sinai Theory
9.9 Critical Points
9.9.1 Phenomenology
9.9.2 Mean Field Theory
9.9.3 More Realistic Models
9.10 Summary
9.11 Exercises
10 Conclusion: Statistical Mechanics and Reductionism
11 Hints and Solutions for the Exercises
11.1 Exercises of Chap.2摥映數爠eflinkchap222
11.2 Exercises of Chap.3摥映數爠eflinkchap333
11.3 Exercises of Chap.4摥映數爠eflinkchap444
11.4 Exercises of Chap.5摥映數爠eflinkchap555
11.5 Exercises of Chap.6摥映數爠eflinkchap666
11.6 Exercises of Chap.7摥映數爠eflinkchap777
11.7 Exercises of Chap.8摥映數爠eflinkchap888
11.8 Exercises of Chap.9摥映數爠eflinkchap999
Appendix Glossary
Appendix Suggested Reading
Appendix References
Index
