Statistical Physics - An Entropic Approach

Preface xiii 1. Disorder or Uncertainty? 1 2. Classical Thermodynamics 5 2.1 The Classical Laws of Thermodynamics 5 2.2 Macroscopic State Variables and Thermodynamic Processes 6 2.3 Properties of the Ideal Classical Gas 8 2.4 Thermodynamic Processing of the Ideal Gas 10 2.5 Entropy of the Ideal Gas 13 2.6 Entropy Change in Free Expansion of an Ideal Gas 15 2.7 Entropy Change due to Nonquasistatic Heat Transfer 17 2.8 Cyclic Thermodynamic Processes, the Clausius Inequality and Carnot s Theorem 19 2.9 Generality of the Clausius Expression for Entropy Change 21 2.10 Entropy Change due to Nonquasistatic Work 23 2.11 Fundamental Relation of Thermodynamics 25 2.12 Entropy Change due to Nonquasistatic Particle Transfer 28 2.13 Entropy Change due to Nonquasistatic Volume Exchange 30 2.14 General Thermodynamic Driving 31 2.15 Reversible and Irreversible Processes 32 2.16 Statements of the Second Law 33 2.17 Classical Thermodynamics: the Salient Points 35 Exercises 35 3. Applications of Classical Thermodynamics 37 3.1 Fluid Flow and Throttling Processes 37 3.2 Thermodynamic Potentials and Availability 39 3.2.1 Helmholtz Free Energy 40 3.2.2 Why Free Energy? 43 3.2.3 Contrast between Equilibria 43 3.2.4 Gibbs Free Energy 44 3.2.5 Grand Potential 46 3.3 Maxwell Relations 47 3.4 Nonideal Classical Gas 48 3.5 Relationship between Heat Capacities 49 3.6 General Expression for an Adiabat 50 3.7 Determination of Entropy from a Heat Capacity 50 3.8 Determination of Entropy from an Equation of State 51 3.9 Phase Transitions and Phase Diagrams 52 3.9.1 Conditions for Coexistence 53 3.9.2 Clausius Clapeyron Equation 55 3.9.3 The Maxwell Equal Areas Construction 57 3.9.4 Metastability and Nucleation 59 3.10 Work Processes without Volume Change 59 3.11 Consequences of the Third Law 60 3.12 Limitations of Classical Thermodynamics 61 Exercises 62 4. Core Ideas of Statistical Thermodynamics 65 4.1 The Nature of Probability 65 4.2 Dynamics of Complex Systems 68 4.2.1 The Principle of Equal a Priori Probabilities 68 4.2.2 Microstate Enumeration 71 4.3 Microstates and Macrostates 72 4.4 Boltzmann s Principle and the Second Law 75 4.5 Statistical Ensembles 77 4.6 Statistical Thermodynamics: the Salient Points 78 Exercises 79 5. Statistical Thermodynamics of a System of Harmonic Oscillators 81 5.1 Microstate Enumeration 81 5.2 Microcanonical Ensemble 83 5.3 Canonical Ensemble 84 5.4 The Thermodynamic Limit 88 5.5 Temperature and the Zeroth Law of Thermodynamics 91 5.6 Generalisation 91 Exercises 92 6. The Boltzmann Factor and the Canonical Partition Function 95 6.1 Simple Applications of the Boltzmann Factor 95 6.1.1 Maxwell Boltzmann Distribution 95 6.1.2 Single Classical Oscillator and the Equipartition Theorem 97 6.1.3 Isothermal Atmosphere Model 98 6.1.4 Escape Problems and Reaction Rates 99 6.2 Mathematical Properties of the Canonical Partition Function 99 6.3 Two-Level Paramagnet 101 6.4 Single Quantum Oscillator 103 6.5 Heat Capacity of a Diatomic Molecular Gas 104 6.6 Einstein Model of the Heat Capacity of Solids 105 6.7 Vacancies in Crystals 106 Exercises 108 7. The Grand Canonical Ensemble and Grand Partition Function 111 7.1 System of Harmonic Oscillators 111 7.2 Grand Canonical Ensemble for a General System 115 7.3 Vacancies in Crystals Revisited 116 Exercises 117 8. Statistical Models of Entropy 119 8.1 Boltzmann Entropy 119 8.1.1 The Second Law of Thermodynamics 120 8.1.2 The Maximum Entropy Macrostate of Oscillator Spikiness 122 8.1.3 The Maximum Entropy Macrostate of Oscillator Populations 122 8.1.4 The Third Law of Thermodynamics 126 8.2 Gibbs Entropy 127 8.2.1 Fundamental Relation of Thermodynamics and Thermodynamic Work 129 8.2.2 Relationship to Boltzmann Entropy 130 8.2.3 Third Law Revisited 131 8.3 Shannon Entropy 131 8.4 Fine and Coarse Grained Entropy 132 8.5 Entropy at the Nanoscale 133 8.6 Disorder and Uncertainty 134 Exercises 135 9. Statistical Thermodynamics of the Classical Ideal Gas 137 9.1 Quantum Mechanics of a Particle in a Box 137 9.2 Densities of States 138 9.3 Partition Function of a One-Particle Gas 140 9.4 Distinguishable and Indistinguishable Particles 141 9.5 Partition Function of an N-Particle Gas 145 9.6 Thermal Properties and Consistency with Classical Thermodynamics 146 9.7 Condition for Classical Behaviour 147 Exercises 149 10. Quantum Gases 151 10.1 Spin and Wavefunction Symmetry 151 10.2 Pauli Exclusion Principle 152 10.3 Phenomenology of Quantum Gases 153 Exercises 154 11. Boson Gas 155 11.1 Grand Partition Function for Bosons in a Single Particle State 155 11.2 Bose Einstein Statistics 156 11.3 Thermal Properties of a Boson Gas 158 11.4 Bose Einstein Condensation 161 11.5 Cooper Pairs and Superconductivity 166 Exercises 167 12. Fermion Gas 169 12.1 Grand Partition Function for Fermions in a Single Particle State 169 12.2 Fermi Dirac Statistics 170 12.3 Thermal Properties of a Fermion Gas 171 12.4 Maxwell Boltzmann Statistics 173 12.5 The Degenerate Fermion Gas 176 12.6 Electron Gas in Metals 177 12.7 White Dwarfs and the Chandrasekhar Limit 179 12.8 Neutron Stars 182 12.9 Entropy of a Black Hole 183 Exercises 184 13. Photon Gas 187 13.1 Electromagnetic Waves in a Box 187 13.2 Partition Function of the Electromagnetic Field 189 13.3 Thermal Properties of a Photon Gas 191 13.3.1 Planck Energy Spectrum of Black-Body Radiation 191 13.3.2 Photon Energy Density and Flux 193 13.3.3 Photon Pressure 193 13.3.4 Photon Entropy 194 13.4 The Global Radiation Budget and Climate Change 195 13.5 Cosmic Background Radiation 197 Exercises 198 14. Statistical Thermodynamics of Interacting Particles 201 14.1 Classical Phase Space 201 14.2 Virial Expansion 203 14.3 Harmonic Structures 206 14.3.1 Triatomic Molecule 207 14.3.2 Einstein Solid 208 14.3.3 Debye Solid 209 Exercises 211 15. Thermodynamics away from Equilibrium 213 15.1 Nonequilibrium Classical Thermodynamics 213 15.1.1 Energy and Particle Currents and their Conjugate Thermodynamic Driving Forces 213 15.1.2 Entropy Production in Constrained and Evolving Systems 218 15.2 Nonequilibrium Statistical Thermodynamics 220 15.2.1 Probability Flow and the Principle of Equal a Priori Probabilities 220 15.2.2 The Dynamical Basis of the Principle of Entropy Maximisation 222 Exercises 223 16. The Dynamics of Probability 225 16.1 The Discrete Random Walk 225 16.2 Master Equations 226 16.2.1 Solution to the Random Walk 228 16.2.2 Entropy Production during a Random Walk 229 16.3 The Continuous Random Walk and the Fokker Planck Equation 230 16.3.1 Wiener Process 232 16.3.2 Entropy Production in the Wiener Process 233 16.4 Brownian Motion 235 16.5 Transition Probability Density for a Harmonic Oscillator 236 Exercises 238 17. Fluctuation Relations 241 17.1 Forward and Backward Path Probabilities: a Criterion for Equilibrium 241 17.2 Time Asymmetry of Behaviour and a Definition of Entropy Production 243 17.3 The Relaxing Harmonic Oscillator 245 17.4 Entropy Production Arising from a Single Random Walk 247 17.5 Further Fluctuation Relations 249 17.6 The Fundamental Basis of the Second Law 253 Exercises 253 18. Final Remarks 255 Further Reading 261 Index 263