Statistical Mechanics

The following sections are included:

• Preface

• Contents

The following sections are included:

• A Recollection of Basic Notions in Classical Thermodynamics
  • The Fundamental Equation of Thermodynamics

• Thermodynamic Potentials, Maxwell's Relations,Variational Principles and Stability Conditions

• A Mathematical Digression: Integrating Factors and the Geometry of Thermodynamics

• Appendix 1A An Exercise in the Use of the Gibbs-Duhem Relation: the Ideal Monoatomic Gas

• Appendix 1B Thermodynamics of Paramagnetic Bodies

• Appendix 1C Some Relations on Partial Derivatives & Jacobians

• Appendix 1D A Digression on: Integrability Conditions in the Language of Differential Geometry

The following sections are included:

• Foundations of Classical Statistical Mechanics: The Ergodic Hypothesis & All That
  • A rèsumè of Hamiltonian Dynamics
  • Canonical Transformations
  • The Heisenberg and Schrödinger Pictures in Classical Dynamics
  • Integrable Dynamical Systems and Perturbations
  • The Ergodic Hypothesis and the Foundations of Classical Statistical Mechanics

• Statistical Ensembles in CSM: Micro-canonical Ensemble
  • The Entropy Function

• Statistical Ensembles in CSM: Canonical and Grand-Canonical Ensembles
  • The Canonical Ensemble
  • The Grand-Canonical Ensemble
  • Some Applications
  • General Remarks

• Response, Correlations and Fluctuations: I Classical
  • Symmetry Properties of Correlation Functions
  • Fourier Transforms of Correlation Functions
  • Generating Functionals and Static Generalized Susceptibilities
  • Linear Response Theory
  • The Classical Fluctuation-Dissipation Theorem

• Appendix 2A Harmonic Oscillators & Ergodicity

• Appendix 2B The Volume Phase Space for a Perfect Gas

• Appendix 2C Density-Density Correlation Function of a Perfect Gas

The following sections are included:

• Spin Hamiltonians

• Gaussian Identities for Spin Hamiltonians

• Mean Field Theory and Phase Transitions
  • MFA for Ising Model
  • MFA for Heisenberg Model

• Linearized Spin Dynamics: Spin Waves, Response and Correlations

• Spontaneous Symmetry Breaking, the Goldstone and Classical Mermin-Wagner Theorems
  • The Goldstone Theorem
  • The Mermin-Wagner Theorem

• Appendix 3A Poisson Description of Spin Dynamics

• Appendix 3B Perturbation expansions and the Classical Analogue of Wick's Theorem

• Appendix 3C “Conventional” Mean Field Theory

• Appendix 3D Some Group-Theoretical Aspects Related to SSB

The following sections are included:

• Résumé of Quantum Mechanics
  • Basic Axioms of Quantum Mechanics
  • Mixed States and Density Matrices
  • Time Evolution

• Foundations of Quantum Statistical Mechanics: Ensembles, KMS Conditions and Ergodicity
  • Ensembles in Equilibrium QSM
  • The KMS conditions and the thermodynamic limit
  • Ergodicity in Quantum Mechanics

• Response, Correlations and Fluctuations II: Quantum
  • Static Susceptibilities and Correlation Functions
  • Linear Response Theory and the Fluctuation-Dissipation Theorem

• Appendix 4A Two-level Systems

The following sections are included:

• Statistics and Identical Particles in QSM
  • Inequivalent Quantizations in Multiply-Connected Spaces

• Fock Spaces & Second Quantization
  • Observables in Fock Space and Gauge Invariance

• Quantum Gases and Beyond
  • The Free Fermi Gas
  • The Free Bose Gas and the Bose-Einstein Condensation
  • Interacting Identical Particles. The Thermal Hartree-Fock Approximation

The following sections are included:

• The Heisenberg Model Hamiltonian
  • Free Spins

• Partition Function and Path Integrals
  • Correlation Functions, KMS Conditions and the Boundary Conditions on the Auxiliary Fields
  • The Static Approximation and the Classical Limit
  • The adiabatic expansion

• Mean-Field Approximations and SSB: Ferro and Anti-ferro Magnetism
  • Linearized Spin Dynamics. Spin Waves

The following sections are included:

• Introduction to Phase Transitions
  • Phase Coexistence and Phase Diagrams: Classification of Phase Transitions
  • Some thermodynamics across the coexistence line(s)
  • Phase Transitions in Magnetic Systems
  • Fluctuations and Correlations Near a Critical Point
  • Ornstein-Zernike Type Theories

• Some Exact Results on Phase Transitions
  • The Theorems of Lee and Yang
  • Critical Points, SSB, Order Parameters and the Goldstone Theorem
  • The Mermin-Wagner Theorem
  • Quantum Description of Phase Coexistence. Decomposition of KMS States

The following sections are included:

• Phase Transitions in Model Systems
  • The Gaussian Model
  • The Spherical Model
  • The 2D Ising Model
  • The XY Model and the Kosterlitz-Thouless Transition

• Critical Exponents, Exponent Inequalities and Scaling Assumptions
  • Critical Exponents and Universality
  • Exponent Inequalities
  • Scaling Assumptions and Scaling Laws

• The Landau Theory of Phase Transitions
  • Second-Order Transitions in the Landau Theory
  • Dimensional Analysis, Rôle of Fluctuations and the Ginzburg Criterion
  • Cubic Terms and First-Order Transitions. Tricritical points

The following sections are included:

• Superconductivity and the Ginzburg-Landau Theory
  • Introduction
  • Two-Fluid Theories
    • The Critical Magnetic Field H_c
  • The Ginzburg-Landau Theory of Superconductivity
  • The Ginzburg-Landau Parameter
  • Magnetic Properties of Type-II Superconductors
  • Field-theoretic aspects of the Ginzburg-Landau theory. Time-dependent Ginzburg-Landau theory

• Superfluids and Superfluidity
  • Some phenomenology of ⁴He. Two-fluid models
  • A microscopic approach to superfluidity
  • Vortices in He^II

The following sections are included:

• Introduction: a Recollection of Basic Facts

• Kadanoff's Block Scaling

• Introduction to the RG. Sketch of the Method

• Real Space Renormalization Group (RSRG). Results for Ising Spins
  • Introduction
  • Fixed Points and the Linearized RSRG
  • Results for Ising Systems

• Momentum Space Renormalization Group. Applications to Landau-Types Models
  • Fixed Points and Exponents
  • The Gaussian Model
  • The m⁴ Model

The following sections are included:

• Appendix A Mathematical Digression I: Differentiable Manifolds and Exterior Calculus

• Appendix B Mathematical Digression II: Some Mathematics of Hilbert Spaces

• Appendix C Linear Stability Theory

• Appendix D Eigenvalue and Eigenvector Problems for Non-Symmetric Matrices

• Bibliography

• Index