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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 |
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