STATISTICAL PHYSICS. Statics, Dynamics and Renormalization. Leo P Kadanoff. Departments of Physics & Mathematics University of Chicago

Transcription

1 STATISTICAL PHYSICS Statics, Dynamics and Renormalization Leo P Kadanoff Departments of Physics & Mathematics University of Chicago \o * World Scientific Singapore»New Jersey London»HongKong

2 Contents Introduction xi I Fundamentals of Statistical Physics 1 1 The Lectures A Survey The Journey: Many Different Approaches The Main Sights Is the Trip Worthwhile? 9 2 One Particle and Many Formulation The Ising Model iv Independent Particles Quantum Description Averages Prom Derivatives N Independent Particles in a Box Fluctuations Big and Small The Problems of Statistical Physics 21 3 Gaussian Distributions Introduction One Variable Many Gaussian Variables Lattice Green Function Gaussian Random Functions Central Limit Theorem Distribution of Energies Large Deviations On Almost Gaussian Integrals Three Versions of Gaussian Problems 42 v

3 VI Contents 4 Quantum Mechanics and Lattices All of Quantum Mechanics in One Brief Section Prom d = 1 Models to Quantum Mechanics An Example: The Linear Ising Chain One-Dimensional Gaussian Model Coherence Length Operator Averages Correlation Functions Ising Correlations Two-Dimensional Ising Model 64 II Random Dynamics 69 5 Diffusion and Hopping Random Walk on a Lattice Formulating This Problem The Diffusion of Probability and Particles Prom Conservation to Hydrodynamic Equations Distribution Functions Cascade Processes and Securities Prices Reprints on Dynamics Forest and Witten: Smoke Particle Aggregates Witten and Sander: Diffusion Limited Aggregation Kadanoff: Chaos and Complexity From Hops to Statistical Mechanics Random Walk in Momentum The Diffusion Equation Again Time Dependence of Probability Time Dependence in Deterministic Case Equilibrium Solutions Back to Collisions From Fokker-Planck to Equilibrium Properties of Fokker-Planck Equation Reprints on Organization Chao Tang et al.: Phase Organization Bak et al.: Self-Organized Criticality Carlson et al: Singular Diffusion Jaeger et al.: Experimental Studies 151

4 Contents vii 7 Correlations and Response Time Independent Response Hamiltonian Time-Dependence SumRules Non-Interacting Particles Plasma Behavior 164 III More Statistical Mechanics Statistical Thermodynamics The Chemical Potential Defined Barometer Formula Sharing Energy Ensemble Theory Temperatures and Energy Flow Fermi, Böse, and Other Quantum Formulation Statistical Mechanics of Non-Interacting Degenerate Particles The Non-Degenerate Limit Degenerate Fermions Degenerate Bosons I. Photons and Phonons Degenerate Bosons IL One-Dimensional Phonons Degenerate Bosons III. Böse Phase Transition Entropies 203 IV Phase Transitions Overview of Phase Transitions Thermodynamic Phases Phase Transitions Two Kinds of Transitions Back to the Ising Model Mean Field Theory of Magnets The Phases Low Temperature Result Free Energy Selection Argument Behaviors of Different Phases Mean Field Theory of Critical Behavior The Infinite Range Model Mean Field Theory Near the Critical Point 227

5 Vlll Contents 11.3 Critical Indices Scaling Function for Magnetization Spatial Correlations Analyticity Mean Field Theory for the Free Energy When Mean Field Theory Fails Continuous Phase Transitions Historical Background Widom Scaling Theory The Ising Model: Rescaled Fixed Points Phenomenology of Scaling Fields Theory of Scaling Fields Scaling Relations for Operators Transforming Operators Universality Operator Product Expansions Reprints on Critical Correlations Kadanoff: Correlations Along a Line Kadanoff-Wegner: Marginal Behavior Renormalization in One Dimension Introduction Decimation The Ising Example Phase Diagrams, Flow Diagrams, and the Coherence Length The Gaussian Model Analysis of Recursion Relation Fixed Point Analysis for the Gaussian Model Two-Dimensional Ising Model Real Space Renormalization Techniques Introduction Decimation: An Exact Calculation The Method of Neglect Potential Moving Further Work Reprints on Real Space RG Niemeijer and van Leeuwen: Triangulär Lattice R.G David Nelson's Early Summary Kadanoff: Bond-moving, and a Variational Method Kadanoff: Migdal's Simple and Versatile Method Migdal's Original Papers 348

6 Contents ix 15 Duality Doing Sums Two Dimensions Direct Coupling and Dual Coupling Two-Dimensional Calculation Ising Model XY is Connected to SOS Gaussian goes into Gaussian Dual Correlations Planar Model and Coulomb Systems Why Study a Planar Model? One-Dimensional Case Phases of the Planar Model The Gaussian Approximation Two-Dimensional Coulomb Systems Multipole Expansion Reprint on Spin Waves V. L. Berezinskii: An Overview of Problems with Continuous Symmetry XY Model, Renormalization, and Duality PlanofAction Villain Representation of the Basic Bonds Duality Transformation Two Limits Vortex Representation The Magnetically Charged System Correlation Calculation The Renormalization Calculation Spatial Averages The Actual Renormalization Reprints on Planar Model The Kosterlitz-Thouless Theory Kosterlitz: On Renormalization of the Planar Model Jorge V. Jose, Leo P. Kadanoff, Scott Kirkpatrick, David R. Nelson: Renormalization and Vortices 454 Index 479