Statistical Mechanics

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1 Statistical Mechanics Entropy, Order Parameters, and Complexity James P. Sethna Laboratory of Atomic and Solid State Physics Cornell University, Ithaca, NY OXFORD UNIVERSITY PRESS

2 Contents List of figures xv 1 What is statistical mechanics? 1 Exercises Quantum dice Probability distributions Waiting times Stirling's approximation Stirling and asymptotic series Random matrix theory Six degrees of separation Satisfactory map colorings 12 2 Random walks and emergent properties Random walk examples: universality and scale invariante The diffusion equation Currents and external forces Solving the diffusion equation Fourier Green 23 Exercises Random walks in grade space Photon diffusion in the Sun Molecular motors and random walks Perfume walk Generating random walks Fourier and Green Periodic diffusion Thermal diffusion Frying pan Polymers and random walks Stocks, volatility, and diversification Computational finance: pricing derivatives Building a percolation network 33 3 Temperature and equilibrium The microcanonical ensemble The microcanonical ideal gas Configuration space 39

3 x Contents Momentum space What is temperature? Pressure and chemical potential Advanced topic: pressure in mechanics and statistical mechanics Entropy, the ideal gas, and phase-space refinements 51 Exercises Temperature and energy Large and very large numbers Escape velocity Pressure computation Hard sphere gas Connecting two macroscopic systems Gas mixture Microcanonical energy fluctuations Gauss and Poisson Triple product relation Maxwell relations Solving differential equations: the pendulum 58 4 Phase-space dynamics and ergodicity Liouville's theorem Ergodicity 65 Exercises Equilibration Liouville vs. the damped pendulum Invariant measures Jupiter! and the KAM theorem 72 5 Entropy Entropy as irreversibility: engines and the heat death of the Universe Entropy as disorder Entropy of mixing: Maxwell's demon and osmotic pressure Residual entropy of glasses: the roads not taken Entropy as ignorante: information and memory Non-equilibrium entropy Information entropy 87 Exercises Life and the heat death of the Universe Burning information and Maxwellian demons Reversible computation Black hole thermodynamics Pressure-volume diagram Carnot refrigerator Does entropy increase? The Arnol'd cat map 95

4 Contents xi 5.9 Chaos, Lyapunov, and entropy increase Entropy increases: diffusion Entropy of glasses Rubber band How many shufiles? Information entropy Shannon entropy Fractal dimensions Deriving entropy Free energies The canonical ensemble Uncoupled systems and canonical ensembles Grand canonical ensemble What is thermodynamics? Mechanics: friction and fluctuations Chemical equilibrium and reaction rates Free energy density for the ideal gas 121 Exercises Exponential atmosphere Two-state system Negative temperature Molecular motors and free energies Laplace Lagrange Legendre Euler Gibbs-Duhem Clausius-Clapeyron Barrier crossing Michaelis-Menten and Hill Pollen and hard squares Statistical mechanics and statistics Quantum statistical mechanics Mixed states and density matrices Advanced topic: density matrices Quantum harmonic oscillator Bose and Fermi statistics Non-interacting bosons and fermions Maxwell-Boltzmann `quantum' statistics Black-body radiation and Bose condensation Free particles in a box Black-body radiation Bose condensation Metals and the Fermi gas 150 Exercises Ensembles and quantum statistics 151

5 xii Contents 7.2 Phonons and photons are bosons Phase-space units and the zero of entropy Does entropy increase in quantum systems? Photon density matrices Spin density matrix Light emission and absorption Einstein's A and B Bosons are gregarious: superfluids and Lasers Crystal defects Phonons an a string Semiconductors Bose condensation in a band Bose condensation: the experiment The photon-dominated Universe White dwarfs, neutron stars, and black holes Calculation and computation The Ising model Magnetism Binary alloys Liquids, gases, and the critical point How to solve the Ising model Markov chains What is a phase? Perturbation theory 171 Exercises The Ising model Ising fluctuations and susceptibilities Waiting for Godot, and Markov Red and green bacteria Detailed balance Metropolis Implementing Ising Wolff Implementing Wolff Stochastic cells The repressilator Entropy increases! Markov chains Hysteresis and avalanches Hysteresis algorithms NP-completeness and ksat Order parameters, broken syrnmetry, and topology Identify the broken symmetry Define the order parameter Examine the elementary excitations Classify the topological defects 198 Exercises Topological defects in nematic liquid crystals 203

6 Contents xiii 9.2 Topological defects in the XY model Defect energetics and total divergente terms Domain walls in magnets Landau theory for the Ising model Symmetries and wave equations Superfluid order and vortices Superfluids: density matrices and ODLRO Correlations, response, and dissipation Correlation functions: motivation Experimental probes of correlations Equal-time correlations in the ideal gas Onsager's regression hypothesis and time correlations Susceptibility and linear response Dissipation and the imaginary part Static susceptibility The fluctuation-dissipation theorem Causality and Kramers-Krönig 229 Exercises Microwave background radiation Pair distributions and molecular dynamics Damped oscillator Spin Telegraph noise in nanojunctions Fluctuation-dissipation: Ising Noise and Langevin equations Magnetic dynamics Quasiparticle poles and Goldstone's theorem Abrupt phase transitions Stable and metastable phases Maxwell construction Nucleation: critical droplet theory Morphology of abrupt transitions Coarsening Martensites Dendritic growth 250 Exercises Maxwell and van der Waals The van der Waals critical point Interfaces and van der Waals Nucleation in the Ising model Nucleation of dislocation pairs Coarsening in the Ising model Origami microstructure Minimizing sequences and microstructure Snowflakes and linear stability 259

7 v Contents 12 Continuous phase transitions Universality Scale invariante Examples of critical points Equilibrium criticality: energy versus entropy Quantum criticality: zero-point fluctuations versus energy Dynamical systems and the onset of chaos Glassy systems: random but frozen Perspectives 281 Exercises Ising self-similarity Scaling and corrections to scaling Scaling and coarsening Bifurcation theory Mean-field theory The onset of lasing Renormalization-group trajectories Superconductivity and the renormalization group Period doubling The renormalization group and the central limit theorem: short The renormalization group and the central limit theorem: long Percolation and universality 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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