Equilibrium and Nonequilibrium Statistical Mechanics

Description

PREFACE CONTENTS PART 1. GENERAL CONCEPTS OF STATISTICAL MECHANICS Chapter 1. Review of Hamiltonian Dynamics 1.1. Individual and Global Behavior 1.2. Hamiltonian Description of Classical Mechanics 1.3. Hamiltonian Description of Quantum Mechanics 1.4. Pure States in Quantum Mechanics: Bosons and Fermions 1.5. The Second Quantization Formalism Chapter 2. Statistical Ensembles 2.1. Macroscopic Physics and Microscopic Physics 2.2. Classical Ensembles: The Phase-Space Distribution Function 2.3. Quantum Ensembles: The von Neumann Density Operator 2.4. Hamiltonian and Liouvillian of a System of Interacting Particles Chapter 3. Reduced Distribution Functions 3.1. The Classical Distribution Vector 3.2. The Principle of Macroscopic Equivalence 3.3. The Thermodynamic Limit 3.4. Evolution in Time of the Classical Distribution Vector 3.5. Homogeneous and Inhomogeneous Systems: Correlations in Classical Systems 3.6. The Quantum Distribution Vector: The Wigner Functions 3.7. Evolution in time of the Quantum Distribution Vector 3.8. Special Properties of Wigner Functions: Quantum Correlations PART 2. EQUILIBRIUM STATISTICAL MECHANICS Chapter 4. Equilibrium Ensembles and Thermodynamics 4.1. Equilibrium Solutions of the Liouville Equation 4.2. The Microcanonical Ensemble 4.3. The Canonical Ensemble 4.4. Connection Between Statistical Mechanics and Thermodynamics 4.5. The Grand Canonical Ensemble 4.6. Equivalence of the Equilibrium Ensembles: Fluctuations *4.7. Dynamics and Thermodynamics: Existence of the Thermodynamic Limit Appendix: The Method of the Most Probable Distribution Chapter 5. Equilibrium Properties of Ideal Systems 5.1. Definition of Ideal Systems 5.2. Ideal Systems in the High-Temperature Limit: The Boltzmann Gas 5.3. Molecular Structure and Thermodynamics 5.4. Ideal Systems of Bosons or Fermions 5.5. The Bose-Einstein and the Fermi-Dirac Distributions 5.6. Highly Degenerate Ideal Fermi Gases 5.7. Highly Degenerate Ideal Bose Gases Chapter 6. Slightly Nonideal Systems in Equilibrium 6.1. Perturbation Expansions in Statistical Mechanics 6.2. The λ Expansion of the Configuration Integral 6.3. The λ Expansion of the Free Energy 6.4. The Density Expansion of the Free Energy: The Virial Coefficients 6.5. Classical Plasmas in Equilibrium Chapter 7. Reduced Distribution Functions in Equilibrium 7.1. Basic Definitions 7.2. Thermodynamic Quantities Expressed in Terms of Reduced Distribution Functions 7.3. Reduced Distribution Functions of Ideal Systems in Equilibrium 7.4. Equilibrium Hierarchy for the Reduced Distribution Functions 7.5. Connection Between the Partition Function and the Reduced Distribution Functions Chapter 8. Dense Fluids in Equilibrium 8.1. Pair Correlation Function and Scattering Phenomena 8.2. Density Expansion of the Pair Distribution Function 8.3. The Percus-Yevick and the Hypernetted-Chains Equation *8.4. Solution of the Percus-Yevick Equation for Hard Spheres 8.5. The Monte Carlo and the Molecular-Dynamics Method 8.6. Present State of the Theory of Dense Fluids Chapter 9. Phase Transitions 9.1. Qualitative Description of Phase Transitions 9.2. Weiss' Mean Field Theory of Ferromagnetism 9.3. The van der Waals Theory of Condensation *9.4. Weak Infinite-Range Interactions and the van der Waals-Maxwell Equation 9.5. Macroscopic Properties Near the Critical Point 9.6. Correlations Near the Critical Point Chapter 10. Modern Theories of Critical Phenomena 10.1. Methods of Investigation of Critical Phenomena 10.2. Model Systems 10.3. Exponent Inequalities 10.4. The Scaling-Law Hypothesis 10.5. Kadanoff's Theory of the Scaling Laws 10.6. Wilson's Formulation of the Kadanoff Theory 10.7. The Renormalization-Group Equations and the Partition Function 10.8. Dimensionality as a Continuous Parameter PART 3. NONEQUILIBRIUM STATISTICAL MECHANICS Chapter 11. Intuitive Treatment of Nonequilibrium Phenomena 11.1. Introduction 11.2. Theory of the Brownian Motion: The Langevin Equation 11.3. Random Processes: The Fokker-Planck Equation 11.4. Dilute Gases: The Boltzmann Equation 11.5. Assumptions Made in the Derivation of the Boltzmann Equation 11.6. Homogeneous Weakly Coupled Gases: The Landau Equation 11.7. Inhomogeneous Plasmas: The Vlassov Equation 11.8. Connection Between the Landau and the Fokker-Planck Equations Chapter 12. Kinetic Equations and Hydrodynamics 12.1. Moment Equations of a Kinetic Equation 12.2. Boltzmann's H Theorem and the Entropy Balance: Irreversibility and Dissipativity 12.3. The Collisional Invariants 12.4. The Hydrodynamical Balance Equations 12.5. The Phenomenological Treatment of the Hydrodynamical Equations 12.6. The Hydrodynamical Normal Modes 12.7. The Plasmadynamical Normal Modes Appendix: Uniqueness Theorem for the Collisional Invariants Chapter 13. Eigenvalues of the Kinetic Equations and the Theory of Transport Coefficients 13.1. Eigenvalues of the Linearized Kinetic Equation for a Homogeneous Gas 13.2. The "Normal" State of an Inhomogeneous Gas 13.3. Eigenvalues of the Linearized Kinetic Equation for an Inhomogeneous Gas 13.4. The Microscopic Expressions of the Transport Coefficients 13.5. Explicit Calculation of a Transport Coefficient 13.6. Eigenvalues of the Linearized Kinetic Equation for an Inhomogeneous Plasma Chapter 14. Dynamics of Correlations 14.1. Time Evolution of the Classical Correlation Patterns 14.2. Classical Dynamics as a Dynamics of Correlations 14.3. Quantum Dynamics of Correlations Chapter 15 Dynamics and Subdynamics of Ideal Systems 15.1. Solution of the Unperturbed Liouville Equation 15.2. The Resolvent Operator 15.3. Vacuum and Correlations: The Concept of Subdynamics Chapter 16 Dynamics and Subdynamics of Interacting Systems 16.1. Formal Solution of the Liouville Equation 16.2. Program for a Kinetic Theory of Irreversibility 16.3. Further Transformation of the Resolvent Operator: The Irreducible Evolution Operator ε(z)—The Master Equation 16.4. Construction of the Operator Π Appendix 1: Proof of the Relation Σ(t)= Π U(t) Appendix 2: Proof of the Relation Π^2 = Π Chapter 17 Subdynamics and Kinetic Theory 17.1. The Kinetic Component of the Correlations 17.2. The General Kinetic Equation 17.3. Time-Independent Functional Relations 17.4. Evolution of the Nonkinetic Component of the Distribution Vector 17.5. The Initial-Value Problem in Subdynamics 17.6. Spatially Homogeneous States 17.7. Equilibrium States 17.8. Nonequilibrium Stationary States Chapter 18 The Evolution in Time of Weakly Coupled Gases 18.1. The Classical Kinetic Equation 18.2. The Classical Kinetic Correlations 18.3. The Spatially Homogeneous Classical Weakly Coupled Gas 18.4. The Inhomogeneous Classical Weakly Coupled Gas 18.5. The Nonkinetic Component of the Distribution Vector 18.6. The Quantum Kinetic Equation and Kinetic Correlations 18.7. The Spatially Homogeneous Quantum Weakly Coupled Gas 18.8. Properties of the Quantum Kinetic Equation Chapter 19 Translation of the General Formalism into Explicit Equations 19.1. Reduction of the Kinetic Equation 19.2. The Diagram Representation of the Irreducible Evolution Operator 19.3. The Diagram Representation of the Kinetic Equation and of the Correlation Functions Chapter 20 Special Kinetic Equations 20.1. Rearrangements of the Perturbation Expansion: Divergences 20.2. Dilute Gas of Strongly Interacting Particles 20.3. Two-Body Scattering Theory and the Boltzmann Equation 20.4. Kinetic Equations for Moderately Dense Gases 20.5. Classical Plasmas out of Equilibrium 20.6. Kinetic Equation for Classical Plasmas 20.7. Other Soluble Problems Chapter 21 Dynamics of Fluctuations and of Correlations 21.1. Definition and Classification of Correlation Functions 21.2. The Green-Kubo Formulas for the Transport Coefficients 21.3. The Fluctuation-Dissipation Theorem 21.4. The Thermal Transport Coefficients 21.5. "Long Tails" in the Correlation Functions 21.6. Many-Times-Reduced Distribution Functions 21.7. Dynamics of Two-Time Distribution Functions Chapter 22 Conclusion Appendix: The Ergodic Problem A.l. Trajectories and Integrals of the Motion A.2. Two Harmonic Oscillators A.3. The Kolmogorov-Arnold-Moser Theorem A.4. Numerical Experiments on Simple Dynamical Systems A.5. Measure in Phase Space A.6. Ergodicity and Mixing: Sinai's Theorem A.7. Ergodic Theory and Statistical Mechanics Author Index Subject Index