Statistical Mechanics: An Introductory Graduate Course (Graduate Texts in Physics)

Preface
Contents
About the Authors
Part I Preliminaries
1 Introduction
	1.1 The Role of Statistical Mechanics
	1.2 Examples of Interacting Many-Body Systems
		1.2.1 Solid–Liquid–Gas
		1.2.2 Electron Liquid
		1.2.3 Classical Spins
		1.2.4 Superfluids
		1.2.5 Superconductors
		1.2.6 Quantum Spins
		1.2.7 Liquid Crystals, Polymers, Copolymers
		1.2.8 Quenched Randomness
		1.2.9 Cosmology and Astrophysics
	1.3 Challenges
	1.4 Some Key References
2 Phase Diagrams
	2.1 Examples of Phase Diagrams
		2.1.1 Solid–Liquid–Gas
		2.1.2 Ferromagnets
		2.1.3 Antiferromagnets
		2.1.4 3He–4He Mixtures
		2.1.5 Pure 3He
		2.1.6 Percolation
	2.2 Conclusions
	2.3 Exercises
	References
3 Thermodynamic Properties  and Relations
	3.1 Preamble
	3.2 Laws of Thermodynamics
	3.3 Thermodynamic Variables
	3.4 Thermodynamic Potential Functions
	3.5 Thermodynamic Relations
		3.5.1 Response Functions
		3.5.2 Mathematical Relations
		3.5.3 Applications
		3.5.4 Consequences of the Third Law
	3.6 Thermodynamic Stability
		3.6.1 Internal Energy as a Thermodynamic Potential
		3.6.2 Stability of a Homogeneous System
		3.6.3 Extremal Properties of the Free Energy
	3.7 Legendre Transformations
	3.8 N as a Thermodynamic Variable
		3.8.1 Two-Phase Coexistence and P/T Along the Melting Curve
		3.8.2 Physical Interpretation of the Chemical Potential
		3.8.3 General Structure of Phase Transitions
	3.9 Multicomponent Systems
		3.9.1 Gibbs–Duhem Relation
		3.9.2 Gibbs Phase Rule
	3.10 Exercises
	References
Part II Basic Formalism
4 Basic Principles
	4.1 Introduction
	4.2 Density Matrix for a System with Fixed Energy
		4.2.1 Macroscopic Argument
		4.2.2 Microscopic Argument
		4.2.3 Density of States of a Monatomic Ideal Gas
	4.3 System in Contact with an Energy Reservoir
		4.3.1 Two Subsystems in Thermal Contact
		4.3.2 System in Contact with a Thermal Reservoir
	4.4 Thermodynamic Functions for the Canonical Distribution
		4.4.1 The Entropy
		4.4.2 The Internal Energy and the Helmholtz Free Energy
	4.5 Classical Systems
		4.5.1 Classical Density Matrix
		4.5.2 Gibbs Entropy Paradox
		4.5.3 Irrelevance of Classical Kinetic Energy
	4.6 Summary
	4.7 Exercises
	4.8 Appendix Indistinguishability
	References
5 Examples
	5.1 Noninteracting Subsystems
	5.2 Equipartition Theorem
	5.3 Two-Level System
	5.4 Specific Heat—Finite-Level Scheme
	5.5 Harmonic Oscillator
		5.5.1 The Classical Oscillator
		5.5.2 The Quantum Oscillator
		5.5.3 Asymptotic Limits
	5.6 Free Rotator
		5.6.1 Classical Rotator
		5.6.2 Quantum Rotator
	5.7 Grüneisen Law
	5.8 Summary
	5.9 Exercises
6 Basic Principles (Continued)
	6.1 Grand Canonical Partition Function
	6.2 The Fixed Pressure Partition Function
	6.3 Grand and Fixed Pressure Partition Functions  for a Classical Ideal Gas
		6.3.1 Grand Partition Function of a Classical Ideal Gas
		6.3.2 Constant Pressure Partition Function of a Classical Ideal Gas
	6.4 Overview of Various Partition Functions
	6.5 Product Rule
	6.6 Variational Principles
		6.6.1 Entropy Functional
		6.6.2 Free Energy Functional
	6.7 Thermal Averages as Derivatives of the Free Energy
		6.7.1 Order Parameters
		6.7.2 Susceptibilities of Classical Systems
		6.7.3 Correlation Functions
	6.8 Summary
	6.9 Exercises
7 Noninteracting Gases
	7.1 Preliminaries
	7.2 The Noninteracting Fermi Gas
		7.2.1 High Temperature
		7.2.2 Low Temperature
		7.2.3 Spin Susceptibility at Low Temperature
		7.2.4 White Dwarfs
	7.3 The Noninteracting Bose Gas
		7.3.1 High Temperature
		7.3.2 Low Temperature
	7.4 Bose–Einstein Condensation, Superfluidity, and Liquid 4He
	7.5 Sound Waves (Phonons)
	7.6 Summary
	7.7 Exercises
	References
Part III Mean Field Theory, Landau Theory
8 Mean-Field Approximation for the Free Energy
	8.1 Introduction
	8.2 Ferromagnetic Ising Model
		8.2.1 Graphical Analysis of Self-consistency
		8.2.2 High Temperature
		8.2.3 Just Below the Ordering Temperature for H=0
		8.2.4 At the Critical Temperature
		8.2.5 Heat Capacity in Mean-Field Theory
	8.3 Scaling Analysis of Mean-Field Theory
	8.4 Further Applications of Mean-Field Theory
		8.4.1 Arbitrary Bilinear Hamiltonian
		8.4.2 Vector Spins
		8.4.3 Liquid Crystals
	8.5 Summary and Discussion
	8.6 Exercises
	References
9 Density Matrix Mean-Field Theory  and Landau Expansions
	9.1 The General Approach
	9.2 Order Parameters
	9.3 Example: The Ising Ferromagnet
		9.3.1 Landau Expansion for the Ising Model for h=0
	9.4 Classical Systems: Liquid Crystals
		9.4.1 Analysis of the Landau Expansion for Liquid Crystals
	9.5 General Expansions for a Single-Order Parameter
		9.5.1 Case 1: Disordered Phase at Small Field
		9.5.2 Case 2: Even-Order Terms with Positive Coefficients
		9.5.3 Case 3: f Has a Nonzero σ3 Term
		9.5.4 Case 4: Only Even Terms in f, But the Coefficient of σ4 is Negative
		9.5.5 Case 5: Only Even Terms in f, But the Coefficient of σ4 is Zero
	9.6 Phase Transitions and Mean-Field Theory
		9.6.1 Phenomenology of First-Order Transitions
		9.6.2 Limitations of Mean-Field Theory
	9.7 Summary
	9.8 Exercises
10 Landau Theory for Two or More Order Parameters
	10.1 Introduction
	10.2 Coupling of Two Variables at Quadratic Order
		10.2.1 General Remarks
		10.2.2 The Ising Antiferromagnet
		10.2.3 Landau Expansion for the Antiferromagnet
	10.3 Landau Theory and Lattice Fourier Transforms
		10.3.1 The First Brillouin Zone
	10.4 Wavevector Selection for Ferro-  and Antiferromagnetism
	10.5 Cubic Coupling
	10.6 Vector Order Parameters
	10.7 Potts Models
	10.8 The Lattice Gas: An Ising-Like System
		10.8.1 Disordered Phase of the Lattice Gas
		10.8.2 Lithium Intercalation Batteries
	10.9 Ordered Phase of the Lattice Gas
		10.9.1 Example: The Hexagonal Lattice with Repulsive Nearest Neighbor Interactions
		10.9.2 The Hexagonal Brillouin Zone
	10.10 Landau Theory of Multiferroics
		10.10.1 Incommensurate Order
		10.10.2 NVO—a multiferroic material
		10.10.3 Symmetry Analysis
	10.11 Summary
	10.12 Exercises
	References
11 Quantum Fluids
	11.1 Introduction
	11.2 The Interacting Fermi Gas
	11.3 Fermi Liquid Theory
	11.4 Spin-Zero Bose Gas with Short-Range Interactions
	11.5 The Bose Superfluid
		11.5.1 Diagonalizing the Effective Hamiltonian
		11.5.2 Minimizing the Free Energy
		11.5.3 Discussion
		11.5.4 Approximate Solution
		11.5.5 Comments on This Theory
	11.6 Superfluid Flow
	11.7 Summary
	11.8 Exercises
	11.9 Appendix—The Pseudopotential
	References
12 Superconductivity: Hartree–Fock  for Fermions with Attractive Interactions
	12.1 Introduction
	12.2 Fermion Pairing
	12.3 Nature of the Attractive Interaction
	12.4 Mean-Field Theory for Superconductivity
	12.5 Minimizing the Free Energy
	12.6 Solution to Self-consistent Equations
		12.6.1 The Energy Gap at T=0
		12.6.2 Solution for the Transition Temperature
	12.7 Free Energy of a BCS Superconductor
	12.8 Anderson Spin Model
	12.9 Suggestions for Further Reading
	12.10 Summary
	12.11 Exercises
	References
13 Qualitative Discussion of Fluctuations
	13.1 Spatial Correlations Within Mean-Field Theory
	13.2 Scaling
		13.2.1 Exponents and Scaling
		13.2.2 Relations Between Exponents
		13.2.3 Scaling in Temperature and Field
	13.3 Kadanoff Length Scaling
	13.4 The Ginzburg Criterion
	13.5 The Gaussian Model
	13.6 Hubbard–Stratonovich Transformation
	13.7 Summary
	13.8 Exercises
	13.9 Appendix—Scaling of Gaussian Variables
	References
14 The Cayley Tree
	14.1 Introduction
	14.2 Exact Solution for the Ising Model
	14.3 Exact Solution for the Percolation Model
		14.3.1 Susceptibility in the Disordered Phase
		14.3.2 Percolation Probability
	14.4 Exact Solution for the Spin Glass
	14.5 General Development of the Tree Approximation
		14.5.1 General Formulation
		14.5.2 Ising Model
		14.5.3 Hard-Core Dimers
		14.5.4 Discussion
	14.6 Questions
	14.7 Summary
	14.8 Exercises
	References
Part IV Beyond Mean-Field Theory
15 Exact Mappings
	15.1 q-State Potts Model
		15.1.1 Percolation and the q rightarrow1 Limit
		15.1.2 Critical Properties of Percolation
	15.2 Self-avoiding Walks and the n-Vector Model
		15.2.1 Phenomenology of SAWs
		15.2.2 Mapping
		15.2.3 Flory's Estimate
	15.3 Quenched Randomness
		15.3.1 Mapping onto the n-Replica Hamiltonian
		15.3.2 The Harris Criterion
	15.4 Summary
	15.5 Exercises
	References
16 Series Expansions
	16.1 Introduction
	16.2 Cumulant Expansion
		16.2.1 Summary
	16.3 Nonideal Gas
		16.3.1 Higher Order Terms
		16.3.2 Van der Waals Equation of State
	16.4 High-Temperature Expansions
		16.4.1 Inverse Susceptibility
	16.5 Enumeration of Diagrams
		16.5.1 Illustrative Construction of Series
	16.6 Analysis of Series
	16.7 Summary
	16.8 Exercises
	16.9 Appendix: Analysis of Nonideal Gas Series
	References
17 The Ising Model: Exact Solutions
	17.1 The One-Dimensional Ising Model
		17.1.1 Transfer Matrix Solution
		17.1.2 Correlation Functions
	17.2 Ising Model in a Transverse Field
		17.2.1 Mean-Field Theory (T=0)
		17.2.2 Duality
		17.2.3 Exact Diagonalization
		17.2.4 Finite Temperature and Order Parameters
		17.2.5 Correlation Functions
	17.3 The Two-Dimensional Ising Model
		17.3.1 Exact Solution via the Transfer Matrix
	17.4 Duality
		17.4.1 The Dual Lattice
		17.4.2 2D Ising Model
		17.4.3 2D q-State Potts Model
		17.4.4 Duality in Higher Dimensions
	17.5 Summary
	17.6 Exercises
	References
18 Monte Carlo
	18.1 Motivation
	18.2 The Method: How and Why It Works
	18.3 Example: The Ising Ferromagnet on a Square Lattice
	18.4 The Histogram Method
	18.5 Correlation Functions
	18.6 Finite-Size Scaling for the Correlation Function
	18.7 Finite-Size Scaling for the Magnetization
	18.8 Summary
	18.9 Exercises
	References
19 Real Space Renormalization Group
	19.1 One-Dimensional Ising Model
	19.2 Two-Dimensional Ising Model
	19.3 Formal Theory
	19.4 Finite Cluster RG
		19.4.1 Formalism
		19.4.2 Application to the 2D Square Lattice
	19.5 Summary
	19.6 Exercises
	References
20 The Epsilon Expansion
	20.1 Introduction
	20.2 Role of Spatial Dimension, d
	20.3 Qualitative Description of the RG ε-Expansion
		20.3.1 Gaussian Variables
		20.3.2 Qualitative Description of the RG
		20.3.3 Gaussian Model
		20.3.4 Scaling of Higher Order Couplings
	20.4 RG Calculations for the φ4 Model
		20.4.1 Preliminaries
		20.4.2 First Order in u
		20.4.3 Second Order in u
		20.4.4 Calculation of η
		20.4.5 Generalization to the n-Component Model and Higher Order in ε Terms
	20.5 Universality Classes
	20.6 Summary
	20.7 Exercises
	20.8 Appendix: Wick's Theorem
	References
21 Kosterlitz-Thouless Physics
	21.1 Introduction
	21.2 Phonons in Crystal Lattices
	21.3 2D Harmonic Crystals
	21.4 Disordering by Topological Defects
	21.5 Related Problems
	21.6 The 2D XY Model
		21.6.1 The Villain Approximation
		21.6.2 Duality Transformation
		21.6.3 Generalized Villain Model
		21.6.4 Spin Waves and Vortices
		21.6.5 The K-T Transition
		21.6.6 Disconituity of the Order Parameter at Tc
		21.6.7 Correlation Length
		21.6.8 Free Energy and Specific Heat
	21.7 Conclusion
	21.8 Exercises
	References
Index