Preface xiii
Acknowledgments xv
About the Companion Website xvii
1 Introduction 1
1.1 What is Thermodynamics? 2
1.2 What Is Statistical Mechanics? 5
1.3 Our Approach 6
2 Introduction to Probability Theory 9
2.1 Understanding Probability 9
2.2 Randomness, Fairness, and Probability 10
2.3 Mean Values 15
2.4 Continuous Probability Distributions 18
2.5 Common Probability Distributions 20
2.5.1 Binomial Distribution 20
2.5.2 Gaussian Distribution 21
2.6 Summary 22
Problems 23
References 28
3 Introduction to Information Theory 31
3.1 Missing Information 31
3.2 Missing Information for a General Probability Distribution 37
3.3 Summary 41
Problems 42
References 45
Further Reading 45
4 Statistical Systems and the Microcanonical Ensemble 47
4.1 From Probability and Information Theory to Physics 47
4.2 States in Statistical Systems 48
4.3 Ensembles in Statistical Systems 50
4.4 From States to Information 54
4.5 Microcanonical Ensemble: Counting States 59
4.5.1 Discrete Systems 59
4.5.2 Continuous Systems 62
4.5.3 From φ → Ω 64
4.5.4 Classical Ideal Gas 67
4.6 Interactions Between Systems 70
4.6.1 Thermal Interaction 70
4.6.2 Mechanical Interaction 71
4.7 Quasistatic Processes 73
4.7.1 Exact vs. Inexact Differentials 74
4.7.2 Physical Examples 77
4.8 Summary 79
Problems 79
References 85
5 Equilibrium and Temperature 87
5.1 Equilibrium and the Approach to it 87
5.1.1 Equilibrium 87
5.1.2 Irreversible and Reversible Processes 89
5.1.3 Two Systems in Equilibrium 90
5.1.4 Approaching Thermal Equilibrium 93
5.2 Temperature 95
5.3 Properties of Temperature 96
5.3.1 Negative Absolute Temperature 97
5.3.2 Temperature Scales 98
5.4 Summary 101
Problems 101
References 103
6 Thermodynamics: The Laws and the Mathematics 105
6.1 Interactions Between Systems 105
6.1.1 Quasistatic Thermal Interaction 105
6.1.2 The Heat Reservoir 106
6.1.3 General Interactions Between Systems 108
6.1.4 The Entropy in the Ground state 116
6.2 The First Derivatives 119
6.2.1 Heat Capacity 120
6.2.2 Coefficient of Thermal Expansion 125
6.2.3 Isothermal Compressibility 125
6.3 The Legendre Transform and Thermodynamic Potentials 125
6.3.1 Naturally Independent Variables 126
6.3.2 Legendre Transform 127
6.3.3 Thermodynamic Potentials 130
6.3.4 Fundamental Relations and the Equations of State 135
6.4 Derivative Crushing 136
6.5 More About the Classical Ideal Gas 142
6.6 First Derivatives Near Absolute Zero 145
6.7 Empirical Determination of the Entropy and Internal Energy 146
6.8 Summary 150
Problems 150
References 157
7 Applications of Thermodynamics 159
7.1 Adiabatic Expansion 159
7.2 Cooling Gases 162
7.2.1 Free Expansion 162
7.2.2 Throttling (Joule–Thomson) Process 165
7.3 Heat Engines 168
7.3.1 Carnot Cycle 171
7.4 Refrigerators 173
7.5 Summary 175
Problems 175
References 180
Further Reading 180
8 The Canonical Distribution 181
8.1 Restarting Our Study of Systems 181
8.1.1 A as an Isolated System 182
8.1.2 System in Contact with a Heat Reservoir 182
8.2 Connecting to the Microcanonical Ensemble 188
8.2.1 Mean Energy 189
8.2.2 Variance in ? 189
8.2.3 Mean Pressure 190
8.3 Thermodynamics and the Canonical Ensemble 191
8.4 Classical Ideal Gas (Yet Again) 193
8.5 Fudged Classical Statistics 196
8.6 Non-ideal Gases 198
8.7 Specified Mean Energy 203
8.8 Summary 204
Problems 205
9 Applications of the Canonical Distribution 211
9.1 Equipartition Theorem 211
9.2 Specific Heat of Solids 213
9.2.1 The Classical Case 214
9.2.2 The Einstein Model 216
9.2.3 A More Realistic Model 218
9.2.4 The Debye Model 220
9.3 Paramagnetism 221
9.4 Introduction to Kinetic Theory 226
9.4.1 Maxwell Velocity Distribution 226
9.4.2 Molecules Striking a Surface 231
9.4.3 Effusion 233
9.5 Summary 234
Problems 234
References 238
10 Phase Transitions and Chemical Equilibrium 241
10.1 Introduction to Phases 241
10.2 Equilibrium Conditions 243
10.2.1 Isolated System 243
10.2.2 A System in Contact with a Heat and Work Reservoir 245
10.3 Phase Equilibrium 247
10.3.1 Phase Diagram of Water 250
10.3.2 Vapor Pressure of an Ideal Gas 251
10.4 From the Equation of State to a Phase Transition 252
10.4.1 Stable Equilibrium Requirements 254
10.4.2 Back to Our Phase Transition 256
10.4.3 Density Fluctuations 262
10.5 Different Phases as Different Substances 263
10.5.1 Systems with Many Components 265
10.5.2 Gibbs–Duhem Relation 266
10.6 Chemical Equilibrium 268
10.7 Chemical Equilibrium Between Ideal Gases 270
10.8 Summary 275
Problems 275
References 281
11 Quantum Statistics 283
11.1 Grand Canonical Ensemble 283
11.1.1 A System in Contact with a Particle Reservoir 283
11.1.2 Connecting ? to Thermodynamics 286
11.2 Classical vs. Quantum Statistics 288
11.2.1 Symmetry Requirements 289
11.3 The Occupation Number 294
11.3.1 Maxwell–Boltzmann Distribution Function 295
11.3.2 Photon Distribution Function 297
11.3.3 Bose–Einstein Statistics 298
11.3.4 Fermi–Dirac Statistics 299
11.4 Classical Limit 301
11.4.1 From Quantum States to Classical Phase Space 304
11.5 Quantum Partition Function in the Classical Limit 307
11.6 Vapor Pressure of a Solid 308
11.6.1 General Expression for the Vapor Pressure 309
11.6.2 Vapor Pressure of a Solid in the Einstein Model 311
11.7 Partition Function of Ideal Polyatomic Molecules 312
11.7.1 Translational Motion of the Center of Mass 313
11.7.2 Electronic States 314
11.7.3 Rotation 314
11.7.4 Vibration 316
11.7.5 Molar Specific Heat of a Diatomic Molecule 317
11.8 Summary 317
Problems 318
Reference 320
12 Applications of Quantum Statistics 321
12.1 Blackbody Radiation 321
12.1.1 From E&M to Photons 321
12.1.2 Photon Gas 323
12.1.3 Radiation Pressure 326
12.1.4 Radiation from a Hot Object 327
12.2 Bose–Einstein Condensation 329
12.3 Fermi Gas 333
12.4 Summary 337
Problems 338
References 340
13 Black Hole Thermodynamics 341
13.1 Brief Introduction to General Relativity 341
13.1.1 Geometrized Units 341
13.1.2 Black Holes 343
13.1.3 Hawking Radiation 345
13.2 Black Hole Thermodynamics 345
13.2.1 Black Hole Heat Engine 346
13.2.2 The Math of Black Hole Thermodynamics 348
13.3 Heat Capacity of a Black Hole 351
13.4 Summary 352
Problems 352
References 353
Appendix A Important Constants and Units 355
References 357
Appendix B Periodic Table of Elements 359
Appendix C Gaussian Integrals 361
Appendix D Volumes in n-Dimensions 363
Appendix E Partial Derivatives in Thermodynamics 367
Reference 371
Index 373
