Statistical mechanics : entropy, order parameters, and complexity

Why Study Statistical Mechanics?......Page 8
Quantum Dice.......Page 12
Waiting times.......Page 13
Stirling\'s Approximation and Asymptotic Series.......Page 14
Random Matrix Theory.......Page 15
Random Walk Examples: Universality and Scale Invariance......Page 18
The Diffusion Equation......Page 22
Currents and External Forces.......Page 24
Fourier......Page 26
Green......Page 27
Exercises......Page 28
Ratchet and Molecular Motors.......Page 29
Frying Pan......Page 31
Polymers and Random Walks.......Page 32
The Microcanonical Ensemble......Page 34
The Microcanonical Ideal Gas......Page 36
Configuration Space......Page 37
Momentum Space......Page 38
What is Temperature?......Page 42
Pressure and Chemical Potential......Page 45
Entropy, the Ideal Gas, and Phase Space Refinements......Page 48
Exercises......Page 51
Connecting Two Macroscopic Systems.......Page 52
Gauss and Poisson.......Page 53
Microcanonical Thermodynamics......Page 54
Microcanonical Energy Fluctuations.......Page 55
Liouville\'s Theorem......Page 56
Ergodicity......Page 59
Jupiter! and the KAM Theorem......Page 63
Invariant Measures.......Page 65
Entropy as Irreversibility: Engines and Heat Death......Page 68
Mixing: Maxwell\'s Demon and Osmotic Pressure......Page 72
Residual Entropy of Glasses: The Roads Not Taken......Page 74
Entropy as Ignorance: Information and Memory......Page 76
Nonequilibrium Entropy......Page 77
Information Entropy......Page 78
Exercises......Page 81
P-V Diagram.......Page 82
Does Entropy Increase?......Page 83
Shannon entropy.......Page 85
Entropy of Glasses.......Page 86
Rubber Band.......Page 87
Deriving Entropy.......Page 88
Black Hole Thermodynamics.......Page 89
Fractal Dimensions.......Page 90
Free Energies......Page 92
The Canonical Ensemble......Page 93
Uncoupled Systems and Canonical Ensembles......Page 97
Grand Canonical Ensemble......Page 100
What is Thermodynamics?......Page 101
Mechanics: Friction and Fluctuations......Page 105
Chemical Equilibrium and Reaction Rates......Page 106
Free Energy Density for the Ideal Gas......Page 109
Exercises......Page 111
Barrier Crossing.......Page 112
Statistical Mechanics and Statistics.......Page 113
Euler, Gibbs-Duhem, and Clausius-Clapeyron.......Page 114
Laplace.......Page 115
Molecular Motors: Which Free Energy?......Page 116
Michaelis-Menten and Hill......Page 117
Pollen and Hard Squares.......Page 118
Mixed States and Density Matrices......Page 120
Bose and Fermi Statistics......Page 125
Non-Interacting Bosons and Fermions......Page 126
Maxwell-Boltzmann ``Quantum\'\' Statistics......Page 130
Free Particles in a Periodic Box......Page 132
Black Body Radiation......Page 133
Bose Condensation......Page 134
Metals and the Fermi Gas......Page 136
Phase Space Units and the Zero of Entropy.......Page 137
Does Entropy Increase in Quantum Systems?......Page 138
Density Matrices.......Page 139
Bosons are Gregarious: Superfluids and Lasers......Page 140
Einstein\'s A and B......Page 141
Phonons and Photons are Bosons.......Page 142
Bose Condensation in a Parabolic Potential.......Page 143
Light Emission and Absorption.......Page 144
Fermions in Semiconductors.......Page 145
White Dwarves, Neutron Stars, and Black Holes.......Page 146
What is a Phase? Perturbation theory.......Page 148
Magnetism......Page 151
Binary Alloys......Page 152
Lattice Gas and the Critical Point......Page 153
How to Solve the Ising Model.......Page 154
Markov Chains......Page 155
The Ising Model.......Page 159
Red and Green Bacteria......Page 160
Heat Bath, Metropolis, and Wolff.......Page 161
Stochastic Cells.......Page 162
The Repressilator.......Page 164
Entropy Increases! Markov chains.......Page 166
Solving ODE\'s: The Pendulum......Page 167
Small World Networks.......Page 170
Building a Percolation Network.......Page 172
Hysteresis Model: Computational Methods.......Page 174
Order Parameters, Broken Symmetry, and Topology......Page 176
Define the Order Parameter......Page 177
Examine the Elementary Excitations......Page 181
Classify the Topological Defects......Page 183
Topological Defects in the XY Model.......Page 188
Superfluid Order and Vortices.......Page 189
Landau Theory for the Ising model.......Page 191
Superfluids: Density Matrices and ODLRO.......Page 195
Correlation Functions: Motivation......Page 200
Experimental Probes of Correlations......Page 202
Equal--Time Correlations in the Ideal Gas......Page 203
Onsager\'s Regression Hypothesis and Time Correlations......Page 205
Susceptibility and the Fluctuation--Dissipation Theorem......Page 208
Dissipation and the imaginary part \'\'()......Page 209
Static susceptibility \"03650(k)......Page 210
(r,t) and Fluctuation--Dissipation......Page 212
Causality and Kramers Krönig......Page 215
Fluctuations in Damped Oscillators.......Page 217
Telegraph Noise and RNA Unfolding.......Page 218
Coarse-Grained Magnetic Dynamics.......Page 219
Fluctuations, Correlations, and Response: Ising......Page 221
Spin Correlation Functions and Susceptibilities.......Page 222
Abrupt Phase Transitions......Page 224
Maxwell Construction.......Page 225
Nucleation: Critical Droplet Theory.......Page 226
Coarsening.......Page 228
Dendritic Growth.......Page 232
van der Waals Water.......Page 233
Nucleation in the Ising Model.......Page 234
Coarsening and Criticality in the Ising Model.......Page 235
Nucleation of Dislocation Pairs.......Page 236
Oragami Microstructure.......Page 237
Minimizing Sequences and Microstructure.......Page 239
Continuous Transitions......Page 242
Universality.......Page 244
Scale Invariance......Page 251
Quantum Criticality: Zero-point fluctuations versus energy.......Page 258
Glassy Systems: Random but Frozen.......Page 259
Scaling: Critical Points and Coarsening.......Page 261
Bifurcation Theory and Phase Transitions.......Page 262
Onset of Lasing as a Critical Point.......Page 264
Superconductivity and the Renormalization Group.......Page 265
RG and the Central Limit Theorem: Long.......Page 267
Period Doubling.......Page 269
Percolation and Universality.......Page 272
Hysteresis Model: Scaling and Exponent Equalities.......Page 274
Appendix: Fourier Methods......Page 278
Fourier Conventions......Page 279
Derivatives, Convolutions, and Correlations......Page 281
Fourier Methods and Function Space......Page 282
Fourier and Translational Symmetry......Page 284
Fourier Series: Computation.......Page 286
Fourier Transforms and Gaussians: Computation.......Page 287
Fourier Series and Gibbs Phenomenon.......Page 289