# Statistical Physics. An Entropic Approach

- ID: 2330678
- May 2013
- 280 Pages
- John Wiley and Sons Ltd

This undergraduate textbook provides a statistical mechanical foundation to the classical laws of thermodynamics via a comprehensive treatment of the basics of classical thermodynamics, equilibrium statistical mechanics, irreversible thermodynamics, and the statistical mechanics of non-equilibrium phenomena.

This timely book has a unique focus on the concept of entropy, which is studied starting from the well-known ideal gas law, employing various thermodynamic processes, example systems and interpretations to expose its role in the second law of thermodynamics. This modern treatment of statistical physics includes studies of neutron stars, superconductivity and the recently developed fluctuation theorems. It also presents figures and problems in a clear and concise way, aiding the student’s understanding.

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

“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)