﻿ QHE

# Statistical Physics

### Text:

L.E.Reichl, "A Modern Course in Statistical Physics", 2nd ed., Wiley, 1998.

# Revised Mathematical Supplement

Files updated after Oct.17, 2019 have gone through the 1st stage proof-reading.

## Part I.   Thermodynamics

### Chapter 2: Introduction to Thermodynamics

Chap 2. Introduction to Thermodynamics       ( Oct 19, 2019 )
•    2.B. State Variables & Exact Differentials       ( Oct 17, 2019 )
•    2.C. Some Mechanical Equations of State       ( Oct 17, 2019 )
•    2.D. The Laws of Thermodynamics       ( Oct 17, 2019 )
•    2.E. Fundamental Equation of Thermodynamics       ( Oct 18, 2019 )
•     2.F. Thermodynamic Potentials
•    2.F.0. Overview       ( Oct 18, 2019 )
•    2.F.1. Internal Energy       ( Oct 18, 2019 )
•    2.F.2. Enthalpy       ( Oct 18, 2019 )
•    2.F.3. Helmholtz Free Energy       ( Oct 19, 2019 )
•    2.F.4. Gibbs Free Energy       ( Oct 19, 2019 )
•    2.F.5. Grand Potential       ( Oct 19, 2019 )
•    2.F.6. Mnemonics       ( Apr 08, 2019 )
•    2.G. Response Functions       ( Oct 19, 2019 )
•     2.H. Stability of the Equilibrium State
•    2.H.1. Conditions for Local Equilibrium in a PVT System       ( Oct 19, 2019 )
•    2.H.2. Conditions for Local Stability in a PVT System       ( Oct 19, 2019 )
•    2.H.3. Implications of the Stability Requirements for the Free Energies       ( Oct 19, 2019 )
•    Zip file for Chap. 2       ( Oct 19, 2019 )

### Special Topics

Chap 2. Special Topics       ( Oct 20, 2019 )
•    S2.A. Cooling & Liquefaction of Gases       ( Oct 19, 2019 )
•    S2.B. Entropy of Mixing and the Gibbs Paradox       ( Oct 20, 2019 )
•    S2.C. Osmotic Pressure in Dilute Solutions       ( Oct 20, 2019 )
•     S2.D. The Thermodynamics of Chemical Reactions
•    S2.D.1. The Affinity       ( Oct 20, 2019 )
•    S2.D.2. Stability       ( Oct 20, 2019 )
•    S2.E. The Thermodynamics of Electrolytes       ( Oct 20, 2019 )
•    Zip file for Special Topics       ( Oct 20, 2019 )

### Chapter 3: Thermodynamics of Phase Transitions

•    3.B. Coexistence Of Phases: Gibbs Phase Rule       ( Oct 21, 2019 )
•    3.C. Classification of Phase Transitions       ( Oct 21, 2019 )
•     3.D. Pure PVT Systems
•    3.D.1. Phase Diagrams       ( Oct 21, 2019 )
•    3.D.2. Coexistence Curves: Clausius-Clapyron Equation       ( Oct 21, 2019 )
•    3.D.3. Liquid-Vapor Coexistence Region       ( Oct 21, 2019 )
•    3.D.4. The van der Waals Equation       ( Oct 21, 2019 )
•    Supplement: Properties of PVT Systems       ( Jun 03, 2019 )
•    3.E._Superconductors       ( Jun 03, 2019 )
•    3.F. The Helium Liquids       ( Jun 06, 2019 )
•    3.G. Ginzburg-Landau Theory       ( Jun 17, 2019 )
•    3.H. Critical Exponents       ( Jun 21, 2019 )
•    Zip file for Chap. 3       ( Jun 24, 2019 )

### Special Topics

•    S3.A. Surface Tension       ( Jun 27, 2019 )
•    S3.B. Thermomechanical Effect       ( Jun 30, 2019 )
•    S3.C. The Critical Exponents for the Curie Point       ( Jun 30, 2019 )
•    S3.D. Tricritical Points       ( Jul 01, 2019 )
•    S3.E. Binary Mixtures       ( Jul 11, 2019 )
•    S3.F. The Ginzburg-Landau Theory of Superconductors       ( Jul 13, 2019 )
•    Zip file for Special Topics       ( Jul 13, 2019 )

## Part II.   Concepts from Probability Theory

### Chapter 4: Elementary Probability Theory and Limit Theorems

•    4.B. Permutations & Combinations       ( Jul 14, 2019 )
•    4.C. Definition of Probability       ( Jul 15, 2019 )
•     4.D. Stochastic Variables & Probability
•    4.D.0. Introduction       ( Jul 18, 2019 )
•    4.D.1. Distribution Functions       ( Jul 23, 2019 )
•    4.D.2. Moments       ( Jul 18, 2019 )
•    4.D.3. Characteristic Functions       ( Jul 18, 2019 )
•    4.D.4. Jointly Distributed Stochastic Variables       ( Jul 23, 2019 )
•     4.E. Binomial Distributions
•    4.E.1. The Binomial Distribution       ( Jul 24, 2019 )
•    4.E.2. The Gaussian ( Or Normal ) Distribution       ( Jul 23, 2019 )
•    4.E.3. The Poisson Distribution       ( Jul 24, 2019 )
•    4.E.4. Binomial Random Walk       ( Jul 25, 2019 )
•    4.F. The Central Limit Theorem and the Law of Large Numbers       ( Jul 28, 2019 )
•    Zip file for Chap. 4       ( Jul 29, 2019 )

### Special Topics

S4.A. Lattice Random Walk
•    S4.A.1. One Dimensional Lattice       ( Aug 01, 2019 )
•    S4.A.2. Random Walk in Higher Dimensions       ( Aug 04, 2019 )
•    S4.B. Infinitely Divisible Distributions       ( Aug 15, 2019 )
•    S4.C. The Central Limit Theorem       ( Aug 07, 2019 )
•    S4.C. The Central Limit Theorem       ( Aug 07, 2019 )
•     S4.D. Weierstrass Random Walk
•    S4.D.1. Discrete 1-D Random Walk       ( Aug 11, 2019 )
•    S4.D.2. Continuum Limit of 1-D Discrete Random Walk       ( Aug 11, 2019 )
•    S4.D.3. 2-D Discrete Random Walk (Levy Flight)       ( Aug 12, 2019 )
•    S4.E. General form of Infinitely Divisible Distributions       ( Aug 19, 2019 )
•    Zip file for Special Topics       ( Aug 19, 2019 )

### Chapter 5: Stochastic Dynamics and Brownian Motion

•    5.A. Introduction       ( Oct 25, 2017 )
•    5.B. General Theory       ( Aug 12, 2018 )
•    5.C. Markov Chains       ( May 26, 2018 )
•    5.C.1. Spectral Properties       ( May 26, 2018 )
•    5.C.2. Random Walk       ( May 22, 2018 )
•     5.D. The Master Equation
•    5.D.1. Derivation of the Master Equation       ( May 23, 2018 )
•    5.D.2. Detailed Balance       ( May 23, 2018 )
•    5.D.3. Mean First Passage Time       ( May 24, 2018 )
•     5.E. Brownian Motion
•    5.E.1. Langevin Equation       ( Nov 23, 2017 )
•    5.E.2. The Spectral Density       ( May 24, 2018 )
•    Zip file for Chap.5       ( Aug 12, 2018 )

### Special Topics

•    S5.A. Time Periodic Markov Chain       ( Jun 04, 2018 )
•     S5.B. Master Equation for Birth-Death Processes
•    S5.B.1. The Master Equation       ( Jun 04, 2018 )
•    S5.B.2. Linear Birth-Death Processes       ( Jun 02, 2018 )
•    S5.B.3. Nonlinear Birth-Death Processes       ( Jun 03, 2018 )
•     S5.C. The Fokker-Planck Equation
•    S5.C.1. Probability Flow in Phase Space       ( Jun 04, 2018 )
•    S5.C.2. Probability Flow for Brownian Particle       ( Jun 04, 2018 )
•    S5.C.3. The Strong Friction Limit       ( Jun 03, 2018 )
•    S5.C.4. Solution of Fokker-Planck Equations with One Variable       ( Jun 03, 2018 )
•    S5.D. Approximations to the Master Equation       ( Jun 03, 2018 )
•    Zip file for Special Topics       ( Jun 04, 2018 )

### Chapter 6: The Foundations of Statistical Mechanics

•    6.B. The Classical Probability Density       ( Jun 05, 2018 )
•    6.C. Ergodic Theory and the Foundations of Statistical Mechanics       ( Dec 20, 2017 )
•    6.C.1. Examples       ( Jun 28, 2018 )
•    6.D. The Quantum Probability Density Operator       ( Dec 27, 2017 )
•    Zip file for Chap.6 (contains 'TwoBallsOnBoundedPlane.nb')       ( Jun 28, 2018 )

### Special Topics

•    S6.A. Reduced Probability Densities and the BBGKY Hierarchy       ( Dec 29, 2017 )
•    S6.B. Reduced Density Matrices and the Wigner Distribution       ( Jan 08, 2018 )
•    S6.C. Microscopic Balance Equations       ( Jan 26, 2018 )
•    S6.D. Mixing Flow       ( Jan 29, 2018 )
•    S6.E. Anharmonic Oscillator Systems       ( Feb 27, 2018 )
•    Zip file for Special Topics (contains 'WalkerFord.nb')       ( Feb 27, 2018 )

## Part III.   Equilibrium Statistical Mechanics

### Chapter 7: Equilibrium Statistical Mechanics

•    7.B. The Microcanonical Ensemble       ( Aug 22, 2019 )
•    7.C. Einstein Fluctuation Theory       ( Aug 24, 2019 )
•     7.D. The Canonical Ensemble
•    7.D.1. Probability Density Operator       ( Aug 25, 2019 )
•    7.D.2. Systems of Indistinguishable Particles       ( Aug 28, 2019 )
•    7.D.3. Systems of Distinguishable Particles       ( Aug 28, 2019 )
•    7.E. Heat Capacity of a Debye Solid       ( Aug 30, 2019 )
•    7.F. Order-Disorder Transitions       ( Sep 02, 2019 )
•    7.G. The Grand Canonical Ensemble       ( Sep 04, 2019 )
•     7.H. Ideal Quantum Gases
•    7.H.0. Introduction       ( Sep 09, 2019 )
•    7.H.1. Bose-Einstein Gases       ( Sep 12, 2019 )
•    7.H.2. Fermi-Dirac Ideal Gases       ( Sep 12, 2019 )
•    Zip file for Chap.7       ( Sep 12, 2019 )

### Special Topics

•    S7.A. Heat Capacity of Lattice Vibrations on a 1-D Lattice       ( Sep 16, 2019 )
•    S7.B. Momentum Condensation in an Interacting Fermi Fluid       ( Sep 25, 2019 )
•    S7.C. The Yang-Lee Theory of Phase Transition       ( Sep 28, 2019 )
•    Zip file for Special Topics       ( Sep 28, 2019 )

### Chapter 8: Order-Disorder Transitions and Renormalization Group

•    8.B. Static Correlation Functions & Response Functions       ( Mar 01, 2018 )
•    8.C. Scaling       ( Mar 09, 2018 )
•    8.D. Microscopic Calculation of Critical Exponents       ( Apr 10, 2018 )
•    Zip file for Chap.8 (contains 'block.nb')       ( Apr 10, 2018 )

### Special Topics

•    S8.A. Critical Exponents for the S4 Model       ( Apr 30, 2018 )
•    S8.A.1. Contributions from the Second Cumulant       ( Apr 30, 2018 )
•    Zip file for Special Topics (contains 'graphs.nb')       ( Apr 30, 2018 )

### Chapter 9: Interacting Fluids

•    9.B. Thermodynamics and the Radial Distribution Function       ( Apr 12, 2018 )
•     9.C. Virial Expansion of the Equation of State
•    9.C.1. Virial Expansion and Cluster Functions       ( May 01, 2018 )
•    9.C.2. The Second Virial Coefficient       ( May 02, 2018 )
•    9.C.3. Higher Order Virial Coefficients       ( May 03, 2018 )
•    Zip file for Chap.9 (contains '9.C.*_Code.nb')       ( May 03, 2018 )

### Special Topics

•    S9.A. The Pressure and Compressibility Equations       ( May 04, 2018 )
•    S9.B. Ornstein-Zernicke Equation       ( May 12, 2018 )
•    S9.C. Third Virial Coefficient       ( May 13, 2018 )
•    S9.D. Virial Coefficients for Quantum Gases       ( May 18, 2018 )
•    Zip file for Special Topics       ( May 18, 2018 )

## Part IV.   Non-Equilibrium Statistical Mechanics

### Chapter 10: Hydrodynamic Processes Near Equilibrium

10.B. Navier-Stokes Hydrodynamic Equations
•    10.B.1. Balance Equations       ( Oct 18, 2018 )
•    Appendix A. Balance Equations       ( May 27, 2018 )
•    10.B.2. Entropy Source & Entropy Current       ( Jul 08, 2018 )
•    10.B.3. Transport Coefficients       ( Jan 24, 2019 )
•     10.C. Linearized Hydrodynamic Equations
•    10.C.1. Linearization of the Hydrodynamic Equations       ( Jul 15, 2018 )
•    10.C.2. Transverse Hydrodynamic Modes       ( Jul 15, 2018 )
•    10.C.3. Longitudinal Hydrodynamic Modes       ( Jul 16, 2018 )
•     10.D. Dynamic Equilibrium Fluctuations & Transport Processes
•    Time Reversal Symmetry       ( Aug 06, 2018 )
•    10.D.1. Onsager's Relations       ( Sep 23, 2018 )
•    10.D.2. Wiener-Khintchine Theorem       ( Aug 16, 2018 )
•     10.E. Linear Response Theory & the Fluctuation-Dissipation Theory
•    10.E.1. The Response Matrix       ( Aug 25, 2018 )
•    10.E.2. Causality       ( Aug 25, 2018 )
•    10.E.3. The Fluctuation-Dissipation Theorem       ( Oct 29, 2018 )
•    10.E.4. Power Absorption       ( Aug 27, 2018 )
•     10.F. Transport Properties of Mixtures
•    10.F.1. Entropy Production in Multi-Component Systems       ( Sep 09, 2018 )
•    10.F.2. Fick's Law for Diffusion       ( Sep 09, 2018 )
•    10.F.3. Thermal Diffusion       ( Sep 09, 2018 )
•    10.F.4. Electrical Conductivity & Diffusion in Fluids       ( Sep 08, 2018 )
•    Zip file for Chap.10       ( Jan 24, 2019 )

### Special Topics

•    S10.A. Onsager's Relations When a Magnetic Field is Present       ( Sep 13, 2018 )
•    S10.B. Microscopic Linear Response Theory       ( Sep 16, 2018 )
•    S10.C. Light Scattering       ( Sep 20, 2018 )
•    S10.D. Thermoelectricity       ( Sep 25, 2018 )
•    S10.E. Entropy Production in Discontinuous Systems       ( Sep 29, 2018 )
•    S10.F. Stochastic Hydrodynamics       ( Oct 06, 2018 )
•     S10.G. Long-Time Tails
•    S10.G.1. Fluid Flow Around the Brownian Particle       ( Oct 10, 2018 )
•    S10.G.2. Drag Force on the Brownian Particle       ( Oct 12, 2018 )
•    S10.G.3. Velocity Autocorrelation Function       ( Oct 15, 2018 )
•     S10.H. Superfluid Hydrodynamics
•    S10.H.1. Superfluid Hydrodynamic Equations       ( Oct 22, 2018 )
•    S10.H.2. Sound Modes       ( Oct 23, 2018 )
•    S10.I. General Definition of Hydrodynamic Modes       ( Oct 29, 2018 )
•    Zip file for Special Topics (contains 'graph.nb')       ( Oct 29, 2018 )

### Chapter 11: Transport Theory

11.B. Elementary Transport Theory
•    11.B.1. The Maxwell-Boltzmann Distribution       ( Oct 30, 2018 )
•    11.B.2. The Mean Free Path       ( Oct 30, 2018 )
•    11.B.3. The Collision Frequency       ( Nov 01, 2018 )
•    11.B.4. Self-Diffusion       ( Nov 03, 2018 )
•    11.B.5. The Coefficients of Viscosity and Thermal Conductivity       ( Nov 03, 2018 )
•    11.B.6. The Rate of Reaction       ( Nov 04, 2018 )
•    11.C. The Boltzmann Equation       ( Nov 05, 2018 )
•           11.C.1. Two-Body Scattering
•         11.C.1.1. Dynamics of the Scattering Process       ( Dec 23, 2018 )
•         11.C.1.2. The Scattering Cross Section       ( Nov 18, 2018 )
•       11.C.2. Derivation of the Boltzmann Equation       ( Dec 02, 2018 )
•       11.C.3. Boltzmann's H Theorem       ( Dec 02, 2018 )
•      11.D. Linearized Boltzmann & Lorentz-Boltzmann Equations
•       11.D.1. Kinetic Equations for a Two-Component Gas       ( Dec 06, 2018 )
•       11.D.2. Collision Operators       ( Dec 18, 2018 )
•      11.E. Coefficiant of Self-Diffusion
•       11.E.1. Derivation of the Diffusion Equation       ( Dec 07, 2018 )
•       11.E.2. Eigen-frequencies of the Lorentz-Boltzmann Equation       ( Dec 16, 2018 )
•      11.F. Coefficiants of Viscosity & Thermal Conductivity
•       11.F.1. Derivation of the Hydrodynamic Equations       ( Dec 13, 2018 )
•       11.F.2. Eigen-frequencies of the Boltzmann Equation       ( Dec 17, 2018 )
•       11.F.3. Shear Viscosity and Thermal Conductivity       ( Dec 17, 2018 )
•    11.G._ComputationOfTransportCoefficients       ( Dec 17, 2018 )
•       11.G.1. Sonine Polynomials       ( Dec 17, 2018 )
•       11.G.2._DiffusionCoefficient       ( Dec 24, 2018 )
•       11.G.3. Thermal Conductivity       ( Dec 25, 2018 )
•       11.G.4. Shear Viscosity       ( Dec 25, 2018 )
•    Zip file with Mathematica code for Chap.11 (contains 'graphics.nb' & 'scattering.nb')       ( Nov 28, 2018 )

Zip file for Chap.11       ( Dec 25, 2018 )

### Special Topics

•    S11.A. Beyond the Boltzmann Equation       ( Jan 02, 2019 )

• ### Chapter 12: Nonequilibrium Phase Transitions

•    12.B. Nonquilibrium Stability Criteria       ( Jan 03, 2019 )
•       12.B.1. Stability Conditions Near Equilibrium       ( Jan 12, 2019 )
•       Appendix C. Stability of Solutions to Nonlinear Equations       ( Jan 10, 2019 )
•      12.B.2. Stability Conditions Far from Equilibrium
•         12.B.2.1. General Arguments       ( Jan 12, 2019 )
•         12.B.2.2. Stability of Nonlinear Chemical Reactions       ( Jan 14, 2019 )
•    12.C. The Schlogl Model       ( Jan 17, 2019 )
•      12.D. The Brusselator
•       12.D.1. TheBrusselator: A Nonlinear Chemical Model       ( Jan 18, 2019 )
•       12.D.2. Boundary Conditions       ( Jan 24, 2019 )
•       12.D.3. Linear Stability Analysis       ( Jan 23, 2019 )
•      12.E. The Rayleigh-Benard Instability
•       12.E.1. Hydrodynamic Equations and Boundary Conditions       ( Jan 27, 2019 )
•       12.E.2. Linear Stability Analysis       ( Jan 28, 2019 )
•    Zip file for Chap.12       ( Jan 28, 2019 )

### Special Topics

•    S12.A.1. Fluctuations in the Rayleigh-Benard System       ( Mar 18, 2019 )
•    S12.A.2. Fluctuations in the Brusselator       ( Mar 18, 2019 )
•    Zip file for Special Topics       ( Oct 17, 2019 )

### Appendices

•    Appendix A. BalanceEquations       ( May 28, 2018 )
•      Appendix B. Systems of Identical Particles
•    B.1. Position and Momentum Eigenstates       ( Sep 29, 2019 )
•    B.2. Symmetrized N-Particle Position and Momentum Eigenstates       ( Oct 03, 2019 )
•          B.3. The Number Representation
•    B.3.0. Introduction       ( Oct 16, 2019 )
•    B.3.1. The Number Representation for Bosons       ( Oct 16, 2019 )
•    B.3.2. The Number Representation for Fermions       ( Oct 15, 2019 )
•    B.3.3. Field Operators       ( Oct 16, 2019 )
•    Appendix C. Stability of Solutions to Nonlinear Equations       ( Jan 10, 2019 )
•    Zip file for Appendices       ( Oct 16, 2019 )

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