Statistical Physics

Ian FordDepartment of Physics and Astronomy, University College London, UK

• Preface xiii1. Disorder or Uncertainty? 12. Classical Thermodynamics 52.1 The Classical Laws of Thermodynamics 52.2 Macroscopic State Variables and Thermodynamic Processes 62.3 Properties of the Ideal Classical Gas 82.4 Thermodynamic Processing of the Ideal Gas 102.5 Entropy of the Ideal Gas 132.6 Entropy Change in Free Expansion of an Ideal Gas 152.7 Entropy Change due to Nonquasistatic Heat Transfer 172.8 Cyclic Thermodynamic Processes, the Clausius Inequality and Carnot’s Theorem 192.9 Generality of the Clausius Expression for Entropy Change 212.10 Entropy Change due to Nonquasistatic Work 232.11 Fundamental Relation of Thermodynamics 252.12 Entropy Change due to Nonquasistatic Particle Transfer 282.13 Entropy Change due to Nonquasistatic Volume Exchange 302.14 General Thermodynamic Driving 312.15 Reversible and Irreversible Processes 322.16 Statements of the Second Law 332.17 Classical Thermodynamics: the Salient Points 35Exercises 353. Applications of Classical Thermodynamics 373.1 Fluid Flow and Throttling Processes 373.2 Thermodynamic Potentials and Availability 393.2.1 Helmholtz Free Energy 403.2.2 Why Free Energy? 433.2.3 Contrast between Equilibria 433.2.4 Gibbs Free Energy 443.2.5 Grand Potential 463.3 Maxwell Relations 473.4 Nonideal Classical Gas 483.5 Relationship between Heat Capacities 493.6 General Expression for an Adiabat 503.7 Determination of Entropy from a Heat Capacity 503.8 Determination of Entropy from an Equation of State 513.9 Phase Transitions and Phase Diagrams 523.9.1 Conditions for Coexistence 533.9.2 Clausius–Clapeyron Equation 553.9.3 The Maxwell Equal Areas Construction 573.9.4 Metastability and Nucleation 593.10 Work Processes without Volume Change 593.11 Consequences of the Third Law 603.12 Limitations of Classical Thermodynamics 61Exercises 624. Core Ideas of Statistical Thermodynamics 654.1 The Nature of Probability 654.2 Dynamics of Complex Systems 684.2.1 The Principle of Equal a Priori Probabilities 684.2.2 Microstate Enumeration 714.3 Microstates and Macrostates 724.4 Boltzmann’s Principle and the Second Law 754.5 Statistical Ensembles 774.6 Statistical Thermodynamics: the Salient Points 78Exercises 795. Statistical Thermodynamics of a System of Harmonic Oscillators 815.1 Microstate Enumeration 815.2 Microcanonical Ensemble 835.3 Canonical Ensemble 845.4 The Thermodynamic Limit 885.5 Temperature and the Zeroth Law of Thermodynamics 915.6 Generalisation 91Exercises 926. The Boltzmann Factor and the Canonical Partition Function 956.1 Simple Applications of the Boltzmann Factor 956.1.1 Maxwell–Boltzmann Distribution 956.1.2 Single Classical Oscillator and the Equipartition Theorem 976.1.3 Isothermal Atmosphere Model 986.1.4 Escape Problems and Reaction Rates 996.2 Mathematical Properties of the Canonical Partition Function 996.3 Two-Level Paramagnet 1016.4 Single Quantum Oscillator 1036.5 Heat Capacity of a Diatomic Molecular Gas 1046.6 Einstein Model of the Heat Capacity of Solids 1056.7 Vacancies in Crystals 106Exercises 1087. The Grand Canonical Ensemble and Grand Partition Function 1117.1 System of Harmonic Oscillators 1117.2 Grand Canonical Ensemble for a General System 1157.3 Vacancies in Crystals Revisited 116Exercises 1178. Statistical Models of Entropy 1198.1 Boltzmann Entropy 1198.1.1 The Second Law of Thermodynamics 1208.1.2 The Maximum Entropy Macrostate of Oscillator Spikiness 1228.1.3 The Maximum Entropy Macrostate of Oscillator Populations 1228.1.4 The Third Law of Thermodynamics 1268.2 Gibbs Entropy 1278.2.1 Fundamental Relation of Thermodynamics and Thermodynamic Work 1298.2.2 Relationship to Boltzmann Entropy 1308.2.3 Third Law Revisited 1318.3 Shannon Entropy 1318.4 Fine and Coarse Grained Entropy 1328.5 Entropy at the Nanoscale 1338.6 Disorder and Uncertainty 134Exercises 1359. Statistical Thermodynamics of the Classical Ideal Gas 1379.1 Quantum Mechanics of a Particle in a Box 1379.2 Densities of States 1389.3 Partition Function of a One-Particle Gas 1409.4 Distinguishable and Indistinguishable Particles 1419.5 Partition Function of an N -Particle Gas 1459.6 Thermal Properties and Consistency with Classical Thermodynamics 1469.7 Condition for Classical Behaviour 147Exercises 14910. Quantum Gases 15110.1 Spin and Wavefunction Symmetry 15110.2 Pauli Exclusion Principle 15210.3 Phenomenology of Quantum Gases 153Exercises 15411. Boson Gas 15511.1 Grand Partition Function for Bosons in a Single Particle State 15511.2 Bose–Einstein Statistics 15611.3 Thermal Properties of a Boson Gas 15811.4 Bose–Einstein Condensation 16111.5 Cooper Pairs and Superconductivity 166Exercises 16712. Fermion Gas 16912.1 Grand Partition Function for Fermions in a Single Particle State 16912.2 Fermi–Dirac Statistics 17012.3 Thermal Properties of a Fermion Gas 17112.4 Maxwell–Boltzmann Statistics 17312.5 The Degenerate Fermion Gas 17612.6 Electron Gas in Metals 17712.7 White Dwarfs and the Chandrasekhar Limit 17912.8 Neutron Stars 18212.9 Entropy of a Black Hole 183Exercises 18413. Photon Gas 18713.1 Electromagnetic Waves in a Box 18713.2 Partition Function of the Electromagnetic Field 18913.3 Thermal Properties of a Photon Gas 19113.3.1 Planck Energy Spectrum of Black-Body Radiation 19113.3.2 Photon Energy Density and Flux 19313.3.3 Photon Pressure 19313.3.4 Photon Entropy 19413.4 The Global Radiation Budget and Climate Change 19513.5 Cosmic Background Radiation 197Exercises 19814. Statistical Thermodynamics of Interacting Particles 20114.1 Classical Phase Space 20114.2 Virial Expansion 20314.3 Harmonic Structures 20614.3.1 Triatomic Molecule 20714.3.2 Einstein Solid 20814.3.3 Debye Solid 209Exercises 21115. Thermodynamics away from Equilibrium 21315.1 Nonequilibrium Classical Thermodynamics 21315.1.1 Energy and Particle Currents and their Conjugate Thermodynamic Driving Forces 21315.1.2 Entropy Production in Constrained and Evolving Systems 21815.2 Nonequilibrium Statistical Thermodynamics 22015.2.1 Probability Flow and the Principle of Equal a Priori Probabilities 22015.2.2 The Dynamical Basis of the Principle of Entropy Maximisation 222Exercises 22316. The Dynamics of Probability 22516.1 The Discrete Random Walk 22516.2 Master Equations 22616.2.1 Solution to the Random Walk 22816.2.2 Entropy Production during a Random Walk 22916.3 The Continuous Random Walk and the Fokker–Planck Equation 23016.3.1 Wiener Process 23216.3.2 Entropy Production in the Wiener Process 23316.4 Brownian Motion 23516.5 Transition Probability Density for a Harmonic Oscillator 236Exercises 23817. Fluctuation Relations 24117.1 Forward and Backward Path Probabilities: a Criterion for Equilibrium 24117.2 Time Asymmetry of Behaviour and a Definition of Entropy Production 24317.3 The Relaxing Harmonic Oscillator 24517.4 Entropy Production Arising from a Single Random Walk 24717.5 Further Fluctuation Relations 24917.6 The Fundamental Basis of the Second Law 253Exercises 25318. Final Remarks 255Further Reading 261Index 263

“Summing Up: Recommended.  Upper-division undergraduates.”  (Choice, 1 March 2014)“The best choice is finally that the entropy is uncertainty commodified". The reviewer believes that the aim of the book is evident and it is worthwhile to make a detailed study of it from time to time.”  (Zentralblatt MATH, 1 October 2013)