1 The Methodology of Statistical Mechanics 1
1.1 Terminology and Methodology 1
1.1.1 Approaches to the subject 1
1.1.2 Description of states 3
1.1.3 Extensivity and the thermodynamic limit 3
1.2 The Fundamental Principles 4
1.2.1 The laws of thermodynamics 4
1.2.2 Probabilistic interpretation of the First Law 6
1.2.3 Microscopic basis for entropy 7
1.3 Interactions—The Conditions for Equilibrium 8
1.3.1 Thermal interaction—Temperature 8
1.3.2 Volume change—Pressure 10
1.3.3 Particle interchange—Chemical potential 12
1.3.4 Thermal interaction with the rest of the world—The Boltzmann factor 13
1.3.5 Particle and energy exchange with the rest of the world—The Gibbs factor 15
1.4 Thermodynamic Averages 17
1.4.1 The partition function 17
1.4.2 Generalised expression for entropy 18
1.4.3 Free energy 20
1.4.4 Thermodynamic variables 21
1.4.5 Fluctuations 21
1.4.6 The grand partition function 23
1.4.7 The grand potential 24
1.4.8 Thermodynamic variables 25
1.5 Quantum Distributions 25
1.5.1 Bosons and fermions 25
1.5.2 Grand potential for identical particles 28
1.5.3 The Fermi distribution 29
1.5.4 The Bose distribution 30
1.5.5 The classical limit—The Maxwell distribution 30
1.6 Classical Statistical Mechanics 31
1.6.1 Phase space and classical states 31
1.6.2 Boltzmann and Gibbs phase spaces 33
1.6.3 The Fundamental Postulate in the classical case 34
1.6.4 The classical partition function 35
1.6.5 The equipartition theorem 35
1.6.6 Consequences of equipartition 37
1.6.7 Liouville's theorem 38
1.6.8 Boltzmann's H theorem 40
1.7 The Third Law of Thermodynamics 42
1.7.1 History of the Third Law 42
1.7.2 Entropy 43
1.7.3 Quantum viewpoint 44
1.7.4 Unattainability of absolute zero 46
1.7.5 Heat capacity at low temperatures 46
1.7.6 Other consequences of the Third Law 48
1.7.7 Pessimist's statement of the laws of thermodynamics 50
2 Practical Calculations with Ideal Systems 54
2.1 The Density of States 54
2.1.1 Non-interacting systems 54
2.1.2 Converting sums to integrals 54
2.1.3 Enumeration of states 55
2.1.4 Counting states 56
2.1.5 General expression for the density of states 58
2.1.6 General relation between pressure and energy 59
2.2 Identical Particles 61
2.2.1 Indistinguishability 61
2.2.2 Classical approximation 62
2.3 Ideal Classical Gas 62
2.3.1 Quantum approach 62
2.3.2 Classical approach 64
2.3.3 Thermodynamic properties 64
2.3.4 The 1/N!term in the partition function 66
2.3.5 Entropy of mixing 67
2.4 Ideal Fermi Gas 69
2.4.0 Methodology for quantum gases 69
2.4.1 Fermi gas at zero temperature 70
2.4.2 Fermi gas at low temperatures—simple model 72
2.4.3 Fermi gas at low temperatures—series expansion 75
Chemical potential 78
Internal energy 80
Thermal capacity 81
2.4.4 More general treatment of low temperature heat capacity 81
2.4.5 High temperature behaviour—the classical limit 84
2.5 Ideal Bose Gas 87
2.5.1 General procedure for treating the Bose gas 87
2.5.2 Number of particles—chemical potential 88
2.5.3 Low temperature behaviour of Bose gas 89
2.5.4 Thermal capacity of Bose gas—below Tc 91
2.5.5 Comparison with superfluid 4He and other systems 93
2.5.6 Two-fluid model of superfluid 4He 95
2.5.7 Elementary excitations 96
2.6 Black Body Radiation—The Photon Gas 98
2.6.1 Photons as quantised electromagnetic waves 98
2.6.2 Photons in thermal equilibrium—black body radiation 99
2.6.3 Planck's formula 100
2.6.4 Internal energy and heat capacity 102
2.6.5 Black body radiation in one dimension 103
2.7 Ideal Paramagnet 105
2.7.1 Partition function and free energy 105
2.7.2 Thermodynamic properties 106
2.7.3 Negative temperatures 110
2.7.4 Thermodynamics of negative temperatures 112
3 Non-Ideal Gases 120
3.1 Statistical Mechanics 120
3.1.1 The partition function 120
3.1.2 Cluster expansion 121
3.1.3 Low density approximation 122
3.1.4 Equation of state 123
3.2 The Virial Expansion 124
3.2.1 Virial coefficients 124
3.2.2 Hard core potential 124
3.2.3 Square-well potential 126
3.2.4 Lennard-Jones potential 127
3.2.5 Second virial coefficient for Bose and Fermi gas 130
3.3 Thermodynamics 130
3.3.1 Throttling 130
3.3.2 Joule-Thomson coefficient 131
3.3.3 Connection with the second virial coefficient 132
3.3.4 Inversion temperature 134
3.4 Van der Waals Equation of State 134
3.4.1 Approximating the partition function 134
3.4.2 Van der Waals equation 135
3.4.3 Microscopic"derivation"of parameters 137
3.4.4 Virial expansion 138
3.5 Other Phenomenological Equations of State 139
3.5.1 The Dieterici equation 139
3.5.2 Virial expansion 139
3.5.3 The Berthelot equation 140
4 Phase Transitions 143
4.1 Phenomenology 143
4.1.1 Basic ideas 143
4.1.2 Phase diagrams 145
4.1.3 Symmetry 147
4.1.4 Order of phase transitions 148
4.1.5 The order parameter 149
4.1.6 Conserved and non-conserved order parameters 151
4.1.7 Critical exponents 152
4.1.8 Scaling theory 154
4.1.9 Scaling of the free energy 158
4.2 First-Order Transition—An Example 159
4.2.1 Coexistence 159
4.2.2 Van der Waals fluid 162
4.2.3 The Maxwell construction 163
4.2.4 The critical point 165
4.2.5 Corresponding states 166
4.2.6 Dieterici's equation 168
4.2.7 Quantum mechanical effects 169
4.3 Second-Order Transition—An Example 170
4.3.1 The ferromagnet 170
4.3.2 The Weiss model 172
4.3.3 Spontaneous magnetisation 173
4.3.4 Critical behaviour 176
4.3.5 Magnetic susceptibility 177
4.3.6 Goldstone modes 178
4.4 The Ising and Other Models 180
4.4.1 Ubiquity of the Ising model 180
4.4.2 Magnetic case of the Ising model 182
4.4.3 Ising model in one dimension 184
4.4.4 Ising model in two dimensions 185
4.4.5 Mean field critical exponents 188
4.4.6 The XY model 190
4.4.7 The spherical model 191
4.5 Landau Treatment of Phase Transitions 191
4.5.1 Landau free energy 191
4.5.2 Landau free energy for the ferromagnet 193
4.5.3 Landau theory—second-order transitions 196
4.5.4 Thermal capacity in the Landau model 198
4.5.5 Ferromagnet in a magnetic field 199
4.6 Ferroelectricity 201
4.6.1 Description of the phenomenon 201
4.6.2 Landau free energy 202
4.6.3 Second-order case 203
4.6.4 First-order case 204
4.6.5 Entropy and latent heat at the transition 208
4.6.6 Soft modes 209
4.7 Binary Mixtures 210
4.7.1 Basic ideas 210
4.7.2 Model calculation 211
4.7.3 System energy 212
4.7.4 Entropy 213
4.7.5 Free energy 214
4.7.6 Phase separation—the lever rule 215
4.7.7 Phase separation curve—the binodal 217
4.7.8 The spinodal curve 219
4.7.9 Entropy in the ordered phase 220
4.7.10 Thermal capacity in the ordered phase 222
4.7.11 Order of the transition and the critical point 223
4.7.12 The critical exponentβ 225
4.8 Quantum Phase Transitions 226
4.8.1 Introduction 226
4.8.2 The transverse Ising model 228
4.8.3 Revision of mean field Ising model 228
4.8.4 Application of a transverse field 230
4.8.5 Transition temperature 232
4.8.6 Quantum critical behaviour 233
4.8.7 Dimensionality and critical exponents 234
4.9 Retrospective 236
4.9.1 The existence of order 236
4.9.2 Validity of mean field theory 237
4.9.3 Features of different phase transition models 238
5 Fluctuations and Dynamics 243
5.1 Fluctuations 244
5.1.1 Probability distribution functions 244
5.1.2 Mean behaviour of fluctuations 246
5.1.3 The autocorrelation function 250
5.1.4 The correlation time 253
5.2 Brownian Motion 254
5.2.1 Kinematics of a Brownian particle 255
5.2.2 Short time limit 257
5.2.3 Long time limit 258
5.3 Langevin's Equation 260
5.3.1 Introduction 260
5.3.2 Separation of forces 261
5.3.3 The Langevin equation 263
5.3.4 Mean square velocity and equipartition 264
5.3.5 Velocity autocorrelation function 265
5.3.6 Electrical analogue of the Langevin equation 267
5.4 Linear Response—Phenomenology 268
5.4.1 Definitions 268
5.4.2 Response to a sinusoidal excitation 270
5.4.3 Fourier representation 271
5.4.4 Response to a step excitation 272
5.4.5 Response to a delta function excitation 273
5.4.6 Consequence of the reality of X(t) 274
5.4.7 Consequence of causality 275
5.4.8 Energy considerations 277
5.4.9 Static susceptibility 278
5.4.10 Relaxation time approximation 280
5.5 Linear Response—Microscopics 281
5.5.1 Onsager's hypothesis 281
5.5.2 Nyquist's theorem 283
5.5.3 Calculation of the step response function 285
5.5.4 Calculation of the autocorrelation function 286
Appendixes 291
Appendix 1 The Gibbs-Duhem Relation 291
A.1.1 Homogeneity of the fundamental relation 291
A.1.2 The Euler relation 291
A.1.3 A caveat 292
A.1.4 The Gibbs-Duhem relation 292
Appendix 2 Thermodynamic Potentials 293
A.2.1 Equilibrium states 293
A.2.2 Constant temperature(and volume):the Helmholtz potential 295
A.2.3 Constant pressure and energy:the Enthalpy function 296
A.2.4 Constant pressure and temperature:the Gibbs free energy 296
A.2.5 Differential expressions for the potentials 297
A.2.6 Natural variables and the Maxwell relations 298
Appendix 3 Mathematica Notebooks 299
A.3.1 Chemical potential of Fermi gas at low temperatures 299
A.3.2 Internal energy of the Fermi gas at low temperatures 301
A.3.3 Fugacity of the ideal gas at high temperatures—Fermi,Maxwell and Bose cases 303
A.3.4 Internal energy of the ideal gas at high temperatures—Fermi,Maxwell and Bose cases 307
Appendix 4 Evaluation of the Correlation Function Integral 310
A.4.1 Initial domain of integration 310
A.4.2 Transformation of variables 310
A.4.3 Jacobian of the transformation 311
Index 313