Statistical Physics

Preface xi
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
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
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
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
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
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
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
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
10. Quantum Gases 151
10.1 Spin and Wavefunction Symmetry 151
10.2 Pauli Exclusion Principle 152
10.3 Phenomenology of Quantum Gases 153
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
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
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
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
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
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
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
18. Final Remarks 255
Further Reading 259
Index 261