Making sense of statistical mechanics / / J. Bricmont

• Autore: Bricmont J (Jean)
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (375 pages)
• Disciplina: 530.13
• Soggetto topico: Statistical mechanics
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Making Sense of Statistical Mechanics
• ISBN: 9783030917944; 9783030917937
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910544865103321
• Lo trovi qui: Univ. Federico II
• Serie: Undergraduate lecture notes in physics.