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Table of Contents
List of figures xv
1 What is statistical mechanics? 1
Exercises 4
1.1 Quantum dice 4
1.2 Probability distributions 5
1.3 Waiting times 6
1.4 Stirling\'s approximation 7
1.5 Stirling and asymptotic series 7
1.6 Random matrix theory 8
1.7 Six degrees of separation 9
1.8 Satisfactory map colorings 12
2 Random walks and emergent properties 15
2.1 Random walk examples: universality and scale invariance 15
2.2 The diffusion equation 19
2.3 Currents and external forces 20
2.4 Solving the diffusion equation 22
2.4.1 Fourier 23
2.4.2 Green 23
Exercises 25
2.1 Random walks in grade space 25
2.2 Photon diffusion in the Sun 26
2.3 Molecular motors and random walks 26
2.4 Perfume walk 27
2.5 Generating random walks 28
2.6 Fourier and Green 28
2.7 Periodic diffusion 29
2.8 Thermal diffusion 30
2.9 Frying pan 30
2.10 Polymers and random walks 30
2.11 Stocks, volatility, and diversification 31
2.12 Computational finance: pricing derivatives 32
2.13 Building a percolation network 33
3 Temperature and equilibrium 37
3.1 The microcanonical ensemble 37
3.2 The microcanonical ideal gas 39
3.2.1 Configuration space 39
3.2.2 Momentum space 41
3.3 What is temperature? 44
3.4 Pressure and chemical potential 47
3.4.1 Advanced topic: pressure in mechanics and statistical mechanics 48
3.5 Entropy, the ideal gas, and phase-space refinements 51
Exercises 53
3.1 Temperature and energy 54
3.2 Large and very large numbers 54
3.3 Escape velocity 54
3.4 Pressure computation 54
3.5 Hard sphere gas 55
3.6 Connecting two macroscopic systems 55
3.7 Gas mixture 56
3.8 Microcanonical energy fluctuations 56
3.9 Gauss and Poisson 57
3.10 Triple product relation 58
3.11 Maxwell relations 58
3.12 Solving differential equations: the pendulum 58
4 Phase-space dynamics and ergodicity 63
4.1 Liouville\'s theorem 63
4.2 Ergodicity 65
Exercises 69
4.1 Equilibration 69
4.2 Liouville vs. the damped pendulum 70
4.3 Invariant measures 70
4.4 Jupiter! and the KAM theorem 72
5 Entropy 77
5.1 Entropy as irreversibility: engines and the heat death of the Universe 77
5.2 Entropy as disorder 81
5.2.1 Entropy of mixing: Maxwell\'s demon and osmotic pressure 82
5.2.2 Residual entropy of glasses: the roads not taken 83
5.3 Entropy as ignorance: information and memory 85
5.3.1 Non-equilibrium entropy 86
5.3.2 Information entropy 87
Exercises 90
5.1 Life and the heat death of the Universe 91
5.2 Burning information and Maxwellian demons 91
5.3 Reversible computation 93
5.4 Black hole thermodynamics 93
5.5 Pressure-volume diagram 94
5.6 Carnot refrigerator 95
5.7 Does entropy increase? 95
5.8 The Arnol\'d cat map 95
5.9 Chaos, Lyapunov, and entropy increase 96
5.10 Entropy increases: diffusion 97
5.11 Entropy of glasses 97
5.12 Rubber band 98
5.13 How many shuffles? 99
5.14 Information entropy 100
5.15 Shannon entropy 100
5.16 Fractal dimensions 101
5.17 Deriving entropy 102
6 Free energies 105
6.1 The canonical ensemble 106
6.2 Uncoupled systems and canonical ensembles 109
6.3 Grand canonical ensemble 112
6.4 What is thermodynamics? 113
6.5 Mechanics: friction and fluctuations 117
6.6 Chemical equilibrium and reaction rates 118
6.7 Free energy density for the ideal gas 121
Exercises 123
6.1 Exponential atmosphere 124
6.2 Two-state system 125
6.3 Negative temperature 125
6.4 Molecular motors and free energies 126
6.5 Laplace 127
6.6 Lagrange 128
6.7 Legendre 128
6.8 Euler 128
6.9 Gibbs-Duhem 129
6.10 Clausius-Clapeyron 129
6.11 Barrier crossing 129
6.12 Michaelis-Menten and Hill 131
6.13 Pollen and hard squares 132
6.14 Statistical mechanics and statistics 133
7 Quantum statistical mechanics 135
7.1 Mixed states and density matrices 135
7.1.1 Advanced topic: density matrices 136
7.2 Quantum harmonic oscillator 139
7.3 Bose and Fermi statistics 140
7.4 Non-interacting bosons and fermions 141
7.5 Maxwell-Boltzmann \'quantum\' statistics 144
7.6 Black-body radiation and Bose condensation 146
7.6.1 Free particles in a box 146
7.6.2 Black-body radiation 147
7.6.3 Bose condensation 148
7.7 Metals and the Fermi gas 150
Exercises 151
7.1 Ensembles and quantum statistics 151
7.2 Phonons and photons are bosons 152
7.3 Phase-space units and the zero of entropy 153
7.4 Does entropy increase in quantum systems? 153
7.5 Photon density matrices 154
7.6 Spin density matrix 154
7.7 Light emission and absorption 154
7.8 Einstein\'s A and B 155
7.9 Bosons are gregarious: superfluids and lasers 156
7.10 Crystal defects 157
7.11 Phonons on a string 157
7.12 Semiconductors 157
7.13 Bose condensation in a band 158
7.14 Bose condensation: the experiment 158
7.15 The photon-dominated Universe 159
7.16 White dwarfs, neutron stars, and black holes 161
8 Calculation and computation 163
8.1 The Ising model 163
8.1.1 Magnetism 164
8.1.2 Binary alloys 165
8.1.3 Liquids, gases, and the critical point 166
8.1.4 How to solve the Ising model 166
8.2 Markov chains 167
8.3 What is a phase? Perturbation theory 171
Exercises 174
8.1 The Ising model 174
8.2 Ising fluctuations and susceptibilities 174
8.3 Waiting for Godot, and Markov 175
8.4 Red and green bacteria 175
8.5 Detailed balance 176
8.6 Metropolis 176
8.7 Implementing Ising 176
8.8 Wolff 177
8.9 Implementing Wolff 177
8.10 Stochastic cells 178
8.11 The repressilator 179
8.12 Entropy increases! Markov chains 182
8.13 Hysteresis and avalanches 182
8.14 Hysteresis algorithms 185
8.15 NP-completeness and kSAT 186
9 Order parameters, broken symmetry, and topology 191
9.1 Identify the broken symmetry 192
9.2 Define the order parameter 192
9.3 Examine the elementary excitations 196
9.4 Classify the topological defects 198
Exercises 203
9.1 Topological defects in nematic liquid crystals 203
9.2 Topological defects in the XY model 204
9.3 Defect energetics and total divergence terms 205
9.4 Domain walls in magnets 206
9.5 Landau theory for the Ising model 206
9.6 Symmetries and wave equations 209
9.7 Superfluid order and vortices 210
9.8 Superfluids: density matrices and ODLRO 211
10 Correlations, response, and dissipation 215
10.1 Correlation functions: motivation 215
10.2 Experimental probes of correlations 217
10.3 Equal-time correlations in the ideal gas 218
10.4 Onsager\'s regression hypothesis and time correlations 220
10.5 Susceptibility and linear response 222
10.6 Dissipation and the imaginary part 223
10.7 Static susceptibility 224
10.8 The fluctuation-dissipation theorem 227
10.9 Causality and Kramers-Kronig 229
Exercises 231
10.1 Microwave background radiation 231
10.2 Pair distributions and molecular dynamics 233
10.3 Damped oscillator 235
10.4 Spin 236
10.5 Telegraph noise in nanojunctions 236
10.6 Fluctuation-dissipation: Ising 237
10.7 Noise and Langevin equations 238
10.8 Magnetic dynamics 238
10.9 Quasiparticle poles and Goldstone\'s theorem 239
11 Abrupt phase transitions 241
11.1 Stable and metastable phases 241
11.2 Maxwell construction 243
11.3 Nucleation: critical droplet theory 244
11.4 Morphology of abrupt transitions 246
11.4.1 Coarsening 246
11.4.2 Martensites 250
11.4.3 Dendritic growth 250
Exercises 251
11.1 Maxwell and van der Waals 251
11.2 The van der Waals critical point 252
11.3 Interfaces and van der Waals 252
11.4 Nucleation in the Ising model 253
11.5 Nucleation of dislocation pairs 254
11.6 Coarsening in the Ising model 255
11.7 Origami microstructure 255
11.8 Minimizing sequences and microstructure 258
11.9 Snowflakes and linear stability 259
12 Continuous phase transitions 263
12.1 Universality 265
12.2 Scale invariance 272
12.3 Examples of critical points 277
12.3.1 Equilibrium criticality: energy versus entropy 278
12.3.2 Quantum criticality: zero-point fluctuations versus energy 278
12.3.3 Dynamical systems and the onset of chaos 279
12.3.4 Glassy systems: random but frozen 280
12.3.5 Perspectives 281
Exercises 282
12.1 Ising self-similarity 282
12.2 Scaling and corrections to scaling 282
12.3 Scaling and coarsening 282
12.4 Bifurcation theory 283
12.5 Mean-field theory 284
12.6 The onset of lasing 284
12.7 Renormalization-group trajectories 285
12.8 Superconductivity and the renormalization group 286
12.9 Period doubling 288
12.10 The renormalization group and the central limit theorem: short 291
12.11 The renormalization group and the central limit theorem: long 291
12.12 Percolation and universality 293
12.13 Hysteresis and avalanches: scaling 296
A Appendix: Fourier methods 299
A.1 Fourier conventions 299
A.2 Derivatives, convolutions, and correlations 302
A.3 Fourier methods and function space 303
A.4 Fourier and translational symmetry 305
Exercises 307
A.1 Sound wave 307
A.2 Fourier cosines 307
A.3 Double sinusoid 307
A.4 Fourier Gaussians 308
A.5 Uncertainty 309
A.6 Fourier relationships 309
A.7 Aliasing and windowing 310
A.8 White noise 311
A.9 Fourier matching 311
A.10 Gibbs phenomenon 311
References 313
Index 323
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