Statistical Thermodynamics: An Information Theory Approach

An accessible and rigorous approach to thermodynamics and statistical mechanics In Statistical Thermodynamics: An Information Theory Approach, distinguished physicist Dr. Christopher Aubin delivers an accessible and comprehensive treatment of the subject from a statistical mechanics perspective. The author discusses the most challenging concept, entropy, using an information theory approach, allowing readers to build a solid foundation in an oft misunderstood and critically important physics concept. This text offers readers access to complimentary online materials, including animations, simple code, and more, that supplement the discussions of complex topics in the book. It provides calculations not usually provided in comparable textbooks that demonstrate how to perform the mathematics of thermodynamics in a systematic way. Readers will also find authoritative explorations of relevant theory accompanied by clear examples of applications and experiments, as well as: A brief introduction to information theory, as well as discussions of statistical systems, phase space, and the Microcanonical Ensemble Comprehensive explorations of the laws and mathematics of thermodynamics, as well as free expansion, Joule-Thomson expansion, heat??engines, and refrigerators Practical discussions of classical and quantum statistics, quantum ideal gases, and blackbody radiation Fulsome treatments of novel topics, including Bose-Einstein condensation, the Fermi gas, and black hole thermodynamics Perfect for upper-level undergraduate students studying statistical mechanics and thermodynamics, Statistical Thermodynamics: An Information Theory Approach provides an alternative and accessible approach to the subject.

Table of Contents:
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 Ē 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 Ƶ  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

About the Author :
Christopher Aubin, PhD, is an Associate Professor in the Department of Physics and Engineering Physics at Fordham University in the Bronx, New York, USA. He earned his doctorate in Physics at Washington University in 2004. His research is focused on the area of lattice QCD.