Ⅰ Fundamentals of Statistical Physics

1 The Lectures—A Survey

1.1 The Journey:Many Different Approaches

1.2 The Main Sights

1.3 Is the Trip Worthwhile?

2 One Particle and Many

2.1 Formulation

2.2 The Ising Model

2.3 N Independent Particles—Quantum Description

2.4 Averages From Derivatives

2.5 N Independent Particles in a Box

2.6 Fluctuations Big and Small

2.7 The Problems of Statistical Physics

3 Gaussian Distributions

3.1 Introduction

3.2 One Variable

3.3 Many Gaussian Variables

3.4 Lattice Green Function

3.5 Gaussian Random Functions

3.6 Central Limit Theorem

3.7 Distribution of Energies

3.8 Large Deviations

3.9 On Almost Gaussian Integrals

3.10 Three Versions of Gaussian Problems

4 Quantum Mechanics and Lattices

4.1 All of Quantum Mechanics in One Brief Section

4.2 From d=1 Models to Quantum Mechanics

4.3 An Example:The Linear Ising Chain

4.4 One-Dimensional Gaussian Model

4.5 Coherence Length

4.6 Operator Averages

4.7 Correlation Functions

4.8 Ising Correlations

4.9 Two-Dimensional Ising Model

Ⅱ Random Dynamics

5 Diffusion and Hopping

5.1 Random Walk on a Lattice

5.2 Formulating This Problem

5.3 The Diffusion of Probability and Particles

5.4 From Conservation to Hydrodynamic Equations

5.5 Distribution Functions

5.6 Cascade Processes and Securities Prices

5.7 Reprints on Dynamics

5.7.1 Forest and Witten:Smoke Particle Aggregates

5.7.2 Witten and Sander:Diffusion Limited Aggregation

5.7.3 Kadanoff:Chaos and Complexity

6 From Hops to Statistical Mechanics

6.1 Random Walk in Momentum

6.2 The Diffusion Equation Again

6.3 Time Dependence of Probability

6.4 Time Dependence in Deterministic Case

6.5 Equilibrium Solutions

6.6 Back to Collisions

6.7 From Fokker-Planck to Equilibrium

6.8 Properties of Fokker-Planck Equation

6.9 Reprints on Organization

6.9.1 Chao Tang et al.:Phase Organization

6.9.2 Baket al.:Self-Organized Criticality

6.9.3 Carlson et al.:Singular Diffusion

6.9.4 Jaeger et al.:Experimental Studies

7 Correlations and Response

7.1 Time Independent Response

7.2 Hamiltonian Time-Dependence

7.3 Sum Rules

7.4 Non-Interacting Particles

7.5 Plasma Behavior

Ⅲ More Statistical Mechanics

8 Statistical Thermodynamics

8.1 The Chemical Potential Defined

8.2 Barometer Formula

8.3 Sharing Energy

8.4 Ensemble Theory

8.5 Temperatures and Energy Flow

9 Fermi,Bose,and Other

9.1 Quantum Formulation

9.2 Statistical Mechanics of Non-Interacting Degenerate Particles

9.3 The Non-Degenerate Limit

9.4 Degenerate Fermions

9.5 Degenerate Bosons Ⅰ.Photons and Phonons

9.6 Degenerate Bosons Ⅱ.One-Dimensional Phonons

9.7 Degenerate Bosons Ⅲ.Bose Phase Transition

9.8 Entropies

Ⅳ Phase Transitions

10 Overview of Phase Transitions

10.1 Thermodynamic Phases

10.2 Phase Transitions

10.3 Two Kinds of Transitions

10.4 Back to the Ising Model

10.5 Mean Field Theory of Magnets

10.6 The Phases

10.7 Low Temperature Result

10.8 Free Energy Selection Argument

10.9 Behaviors of Different Phases

11 Mean Field Theory of Critical Behavior

11.1 The Infinite Range Model

11.2 Mean Field Theory Near the Critical Point

11.3 Critical Indices

11.4 Scaling Function for Magnetization

11.5 Spatial Correlations

11.6 Analyticity

11.7 Mean Field Theory for the Free Energy

11.8 When Mean Field Theory Fails

12 Continuous Phase Transitions

12.1 Historical Background

12.2 Widom Scaling Theory

12.3 The Ising Model:Rescaled

12.4 Fixed Points

12.5 Phenomenology of Scaling Fields

12.6 Theory of Scaling Fields

12.7 Scaling Relations for Operators

12.8 Transforming Operators

12.9 Universality

12.10 Operator Product Expansions

12.11 Reprints on Critical Correlations

12.11.1 Kadanoff:Correlations Along a Line

12.11.2 Kadanoff-Wegner:Marginal Behavior

13 Renormalization in One Dimension

13.1 Introduction

13.2 Decimation

13.3 The Ising Example

13.4 Phase Diagrams,Flow Diagrams,and the Coherence Length

13.5 The Gaussian Model

13.6 Analysis of Recursion Relation

13.7 Fixed Point Analysis for the Gaussian Model

13.8 Two-Dimensional Ising Model

14 Real Space Renormalization Techniques

14.1 Introduction

14.2 Decimation:An Exact Calculation

14.3 The Method of Neglect

14.4 Potential Moving

14.5 Further Work

14.6 Reprints on Real Space RG

14.6.1 Niemeijer and van Leeuwen:Triangular Lattice R.G

14.6.2 David Nelson's Early Summary

14.6.3 Kadanoff:Bond-moving,and a Variational Method

14.6.4 Kadanoff:Migdal's Simple and Versatile Method

14.6.5 Migdal's Original Papers

15 Duality

15.1 Doing Sums

15.2 Two Dimensions

15.3 Direct Coupling and Dual Coupling

15.4 Two-Dimensional Calculation

15.5 Ising Model

15.6 XY is Connected to SOS

15.7 Gaussian goes into Gaussian

15.8 Dual Correlations

16 Planar Model and Coulomb Systems

16.1 Why Study a Planar Model?

16.2 One-Dimensional Case

16.3 Phases of the Planar Model

16.4 The Gaussian Approximation

16.5 Two-Dimensional Coulomb Systems

16.6 Multipole Expansion

16.7 Reprint on Spin Waves

16.7.1 V.L.Berezinskii:An Overview of Problems with Continuous Symmetry

17 XY Model,Renormalization,and Duality

17.1 Plan of Action

17.2 Villain Representation of the Basic Bonds

17.3 Duality Transformation

17.4 Two Limits

17.5 Vortex Representation

17.6 The Magnetically Charged System

17.7 Correlation Calculation

17.8 The Renormalization Calculation

17.9 Spatial Averages

17.10 The Actual Renormalization

17.11 Reprints on Planar Model

17.11.1 The Kosterlitz-Thouless Theory

17.11.2 Kosterlitz:On Renormalization of the Planar Model

17.11.3 Jorge V.José,Leo P.Kadanoff,Scott Kirkpatrick,David R.Nelson:Renormalization and Vortices

Index