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Table of Contents More Information Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information Contents Preface page xxi Acknowledgements xxiii Self-Study Guide xxiv List of Symbols xxvi Part I Atomic and Mesoscopic Viewpoint 1 1Introduction 3 1.1 Atomisms, Ancient and Modern 4 1.2 What was beyond Philosophers’ Grasp? 4 1.3 How Numerous are Atoms and Molecules? 5 1.4 Why are Molecules so Small? 5 1.5 Our World is Lawful to the Extent of Allowing the Evolution of Intelligence 6 1.6 Microscopic World is Unpredictable 7 1.7 Why our Macroscopic World is Lawful: the Law of Large Numbers 7 1.8 We Live in a Rather Gentle World 8 1.9 Thermodynamics, Statistical Mechanics, and Phase Transition 9 1.10 The Mesoscopic World 10 1.11 Large Deviation and Fluctuation 10 Problems 11 2 Atomic Picture of Gases 12 2.1 Aristotelian Physics and Galileo’s Struggle 12 2.2 Boyle: The True Pioneer of Kinetic Theory of Heat 13 2.3 Discovery of Atmospheric Pressure and Vacuum 13 2.4 Daniel Bernoulli’s Modern Dynamic Atomism and a 100 Year Hiatus 14 2.5 Between Bernoulli and Maxwell 14 2.6 Daniel Bernoulli’s Kinetic Theory 15 2.7 Equipartition of Kinetic Energy 17 Problems 18 3 Introduction to Probability 20 3.1 Probability is a Measure of Confidence Level 21 3.2 Events and Sets 21 vii © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information viii Contents 3.3 Probability 22 3.4 Additivity 22 3.5 Probability Measure 23 3.6 Relation to Combinatorics 24 3.7 Objectivity of Subjective Probability 24 3.8 Conditional Probability 25 3.9 Statistical Independence 25 3.10 Stochastic Variables 25 3.11 Expectation Value 26 3.12 Variance 27 3.13 Indicator 27 3.14 Independence of Stochastic Variables 28 3.15 Independence and Correlation 28 3.16 Stochastic Process 29 Problems 30 Appendix 3A Rudiments of Combinatorics 34 4 The Law of Large Numbers 38 4.1 How can we Measure Probability? 38 4.2 Precise Statement of the Law of Large Numbers 39 4.3 Why is the Law of Large Numbers Plausible? 40 4.4 Chebyshev’s Inequality and a Proof of LLN 40 4.5 Almost no Fluctuation of Internal Energy, an Example 41 4.6 Monte Carlo Integration 42 4.7 Central Limit Theorem 43 4.8 There is no Further Refinement of LLN 44 Problems 44 5 Maxwell’s Distribution 47 5.1 Density Distribution Function 47 5.2 Maxwell’s Derivation of the Distribution Function 48 5.3 Gaussian Integral 49 5.4 Gaussian Distribution Function and Maxwell Distribution 49 5.5 Moment Generating Function 50 5.6 Bernoulli Revisited 51 5.7 How to Derive the Boltzmann Factor 52 5.8 Elementary Derivation of Maxwell’s Distribution 53 Problems 54 6 Delta Function and Density Distribution Functions 58 6.1 What Average Gives the Density Distribution? 58 6.2 Introducing the δ-function 59 6.3 A Formal Expression of Density Distribution 60 6.4 Distribution Function: The 1D Case 60 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information ix Contents 6.5 Density Distribution Function: The 1D Case 61 6.6 The 1D δ-function 61 6.7 Key Formula 1 62 6.8 Key Formula 2 63 6.9 Some Practice 63 6.10 How to Compute Density Distribution Functions 63 6.11 Kinetic Energy Density Distribution of 2D Ideal Gas 64 Problems 64 Appendix 6A Extreme Rudiments of Distributions 66 7 MeanFreePathandDifusion 70 7.1 Mean Free Path 70 7.2 Effect of Molecular Collisions 71 7.3 Spatial Spread of Random Walks 73 7.4 Mixing Process of Two Distinct Gases: Each Molecule Walks Randomly 74 7.5 Characterizing the Equilibrium Macrostate 75 7.6 Equilibrium Macrostate Is Most Probable 76 7.7 The Equilibrium Macrostate Is Stable 77 7.8 Entropic Force 77 7.9 How we can Observe Entropic Force 79 Problems 79 8 Introduction to Transport Phenomena 81 8.1 What is a (Linear) Transport Phenomenon? 81 8.2 Density 82 8.3 Flux 82 8.4 Linear Transport Law 83 8.5 Gradient 84 8.6 Conservation Law and Divergence 84 8.7 Local Expression of Conservation Law 85 8.8 The Diffusion Equation 86 8.9 The Meaning of the Laplacian 86 8.10 Intuitive Computation of Transport Coefficient 87 8.11 The Diffusion Constant 88 8.12 Shear Viscosity 88 8.13 Maxwell Estimated the Molecular Size 90 8.14 Thermal Conductivity 91 8.15 Dimensional Analysis of Transport Coefficients 91 8.16 Significance of J ∝ grad xˆ 92 Problem 92 9 Brownian Motion 94 9.1 How Mesoscopic Particles Behave 94 9.2 Brown Discovered a Universal Motion (now Called Brownian Motion) 95 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information x Contents 9.3 General Properties of Brownian Motion 96 9.4 Langevin’s Explanation of Brownian Motion 97 9.5 Relation to Random Walk 98 9.6 Einstein’s Theory of Brownian Particle Flux 99 9.7 Einstein’s Formula 100 9.8 The Einstein–Stokes Relation: Dimensional Analysis 101 9.9 Displacement of Particles by Diffusion 101 9.10 Einstein’s Fundamental Idea: Summary 102 9.11 Summary of the Boltzmann Constant, kB 102 Problems 103 Appendix 9A How to Obtain Equation (9.23) 105 10 The Langevin Equation and Large Deviation 107 10.1 Overdamped Langevin Equation 108 10.2 Langevin Noise: Qualitative Features 108 10.3 Langevin Noise 109 10.4 Smoluchowski Equation 109 10.5 Relation Between the Langevin Noise and D 110 10.6 Smoluchowski Equation and Fluctuation–Dissipation Relation 111 10.7 Significance of the Fluctuation–Dissipation Relation 112 10.8 Mesoscopics and Large Deviation Principle 112 10.9 Three “Infinitesimal Times,” dt, δt, t 113 10.10 Time Averaging to Get Mesoscopic Results 114 10.11 Langevin Equation As a Result of Large Deviation Theory 115 10.12 Langevin Equation: Practical Summary 116 Problems 116 Part II Statistical Thermodynamics: Basics 121 11 Macrosystems in Equilibrium 123 11.1 How to Describe a Macroscopic System in Mechanics 123 11.2 Conservation of Mechanical Energy and the First Law of Thermodynamics 124 11.3 What Mechanics? 124 11.4 Additivity of Energy 125 11.5 Why We Are Interested in Additive Quantities 126 11.6 The Fourth Law of Thermodynamics 127 11.7 Time-Reversal Symmetry of Mechanics 127 11.8 Irreversibility from Mechanics? 128 11.9 The Boltzmann Equation 129 11.10 What is the Lesson? 129 11.11 Purely Mechanical Toy Model and Arrow of Time 130 11.12 The Law of Large Numbers Wins 131 Problem 131 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information xi Contents 12 Thermodynamic Description 133 12.1 What is Phenomenology? 133 12.2 Equilibrium States 135 12.3 External Disturbance: Mechanics vs. Thermodynamics 135 12.4 Thermodynamic Coordinates: a Privileged Set of Variables 136 12.5 Thermodynamic Space 137 12.6 Quasistatic Processes 138 12.7 State Quantities/Functions 139 12.8 Simple System 139 12.9 Compound System 139 12.10 Why Thermodynamics Can Be Useful 139 12.11 Partitioning–Rejoining Invariance 140 12.12 Thermal Contact 141 12.13 The Zeroth Law of Thermodynamics 141 13 Basic Thermodynamics 142 13.1 The First Law of Thermodynamics 142 13.2 Sign Convention 143 13.3 Volume Work 143 13.4 Magnetic Work 144 13.5 Convention for Electromagnetic Field 145 13.6 Prehistory of the Second Law 146 13.7 The Second Law of Thermodynamics 147 13.8 Planck’s Principle, Kelvin’s Principle, and Clausius’ Principle are Equivalent 148 13.9 Clausius Recognizes Entropy 148 13.10 Entropy Exists 148 13.11 Entropy Stratifies the Thermodynamic Space 149 13.12 Entropy and Heat 150 13.13 Entropy Principle 151 13.14 The Gibbs Relation 151 Problems 152 14 Thermodynamics: General Consequences 156 14.1 Summary of Basic Principles 156 14.2 Entropy Maximization Principle 157 14.3 Entropy is Concave 158 14.4 Internal Energy Minimization Principle 159 14.5 Internal Energy is Convex 160 14.6 Extension to Non-Adiabatic Systems 160 14.7 Clausius’ Inequality 161 14.8 Equilibrium Conditions between Two Systems 161 Problem 162 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-15401-8 — Perspectives on Statistical Thermodynamics Yoshitsugu Oono Table of Contents More Information xii Contents 15 Entropy Through Examples 165 15.1 Mayer’s Relation 165 15.2 Poisson’s Relation 166 15.3 Reversible Engine: Carnot’s Theorem 167 15.4 The Original Carnot’s Argument using the Carnot Cycle of an Ideal Gas 168 15.5 Fundamental Equation of Ideal Gas 170 15.6 Adiabatic Heat Exchange between Two Blocks at Different Temperatures 171 15.7 Sudden Doubling of Volume 173 15.8 Entropy and Information: Preview 173 15.9 Mixing Entropy 174 15.10 Entropy Changes due to Phase Transition 175 Problems 175 16 Free Energy and Convex Analysis 181 16.1 Isothermal System 181 16.2 The Free Energy Change, A, by an Irreversible Process 182 16.3 Clausius’ Inequality and Work Principle 182 16.4 Free Energy Minimum Principle 183 16.5 Gibbs Relation for A 183 16.6 Gibbs Free Energy 184 16.7 Enthalpy 184 16.8 Legendre Transformation 185 16.9 Convex Function 186 16.10 Geometrical Meaning of Legendre Transformation 186 16.11 The Function f ∗ is Also Convex and f (x) = f ∗∗(x) 187 16.12 Legendre Transformation Applied to E: Summary 188 Problems 188 17 Introduction to Statistical Mechanics
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