Elementary Lectures in Statistical Mechanics

This textbook for graduates and advanced undergraduates in physics and physical chemistry covers the major areas of statistical mechanics and concludes with the level of current research. It begins with the fundamental ideas of averages and ensembles, focusing on classical systems described by continuous variables such as position and momentum, and using the ideal gas as an example. It then turns to quantum systems, beginning with diatomic molecules and working up through blackbody radiation and chemical equilibria. The discussion of equilibrium properties of systems of interacting particles includes such techniques as cluster expansions and distribution functions and uses non-ideal gases, liquids, and solutions. Dynamic behavior -- treated here more extensively than in other texts -- is discussed from the point of view of correlation functions. The text concludes with the problem of diffusion in a suspension of interacting hard spheres and what can be learned about such a system from scattered light. Intended for a one-semester course, the text includes several "asides" on topics usually omitted from introductory courses, as well as numerous exercises.

Contenu

I Fundamentals: Separable Classical Systems.- Lecture 1. Introduction.- 1.1 Historical Perspective.- 1.2 Basic Principles.- 1.3 Author's Self-Defense.- 1.4 Other Readings.- References.- Lecture 2. Averaging and Statistics.- 2.1 Examples of Averages.- 2.2 Formal Averages.- 2.3 Probability and Statistical Weights.- 2.4 Meaning and Characterization of Statistical Weights.- 2.5 Ideal Time and Ensemble Averages.- 2.6 Summary.- Problems.- References.- Lecture 3. Ensembles: Fundamental Principles of Statistical Mechanics.- 3.1 Ensembles.- 3.2 The Canonical Ensemble.- 3.3 Other Ensembles.- 3.4 Notation and Terminology: Phase Space.- 3.5 Summary.- Problems.- References.- Lecture 4. The One-Atom Ideal Gas.- 4.1 The Classical One-Atom Ensemble.- 4.2 The Average Energy.- 4.3 Mean-Square Energy.- 4.4 The Maxwell-Boltzmann Distribution.- 4.5 Reduced Distribution Functions.- 4.6 Density of States.- 4.7 Canonical and Representative Ensembles.- 4.8 Summary.- Problems.- References.- Aside A. The Two-Atom Ideal Gas.- A.1 Setting Up the Problem.- A.2 Average Energy.- A.3 Summary.- Problems.- Lecture 5. N-Atom Ideal Gas.- 5.1 Ensemble Average for N-Atom Systems.- 5.2 Ensemble Averages of E and E2.- 5.3 Fluctuations and Measurements in Large Systems.- 5.4 Potential Energy Fluctuations.- 5.5 Counting States.- 5.6 Summary.- Problems.- References.- Lecture 6. Pressure of an Ideal Gas.- 6.1 P from a Canonical Ensemble Average.- 6.2 P from the Partition Function.- 6.3 P from the Kinetic Theory of Gases.- 6.4 Remarks.- Problems.- References.- Aside B. How Do Thermometers Work-The Polythermal Ensemble.- B.1 Introduction.- B.2 The Polythermal Ensemble.- B.3 Discussion.- Problems.- References.- Lecture 7. Formal Manipulations of the Partition Function.- 7.1 The Equipartition Theorem.- 7.2 First Generalized Equipartition Theorem.- 7.3 Second Generalized Equipartition Theorem.- 7.4 Additional Tests; Clarification of the Equipartition Theorems.- 7.5 Parametric Derivatives of the Ensemble Average.- 7.6 Summary.- Problems.- References.- Aside C. Gibbs's Derivation of.- References.- Lecture 8. Entropy.- 8.1 The Gibbs Form for the Entropy.- 8.2 Special Cases.- 8.3 Discussion.- Problems.- References.- Lecture 9. Open Systems; Grand Canonical Ensemble.- 9.1 The Grand Canonical Ensemble.- 9.2 Fluctuations in the Grand Canonical Ensemble.- 9.3 Discussion.- Problems.- References.- II Separable Quantum Systems.- Lecture 10. The Diatomic Gas and Other Separable Quantum Systems.- 10.1 Partition Functions for Separable Systems.- 10.2 Classical Diatomic Molecules.- 10.3 Quantization of Rotational and Vibrational Modes.- 10.4 Spin Systems.- 10.5 Summary.- Problems.- References.- Lecture 11. Crystalline Solids.- 11.1 Classical Model of a Solid.- 11.2 Einstein Model.- 11.3 Debye Model.- 11.4 Summary.- Problems.- References.- Aside D. Quantum Mechanics.- D.1 Basic Principles of Quantum Mechanics.- D.2 Summary.- Problems.- References.- Lecture 12. Formal Quantum Statistical Mechanics.- 12.1 Choice of Basis Vectors.- 12.2 Replacement of Sums over All States with Sums over Eigenstates.- 12.3 Quantum Effects on Classical Integrals.- 12.4 Summary.- Problems.- References.- Lecture 13. Quantum Statistics.- 13.1 Introduction.- 13.2 Particles Whose Number Is Conserved.- 13.3 Noninteracting Fermi-Dirac Particles.- 13.4 Photons.- 13.5 Historical Aside: What Did Planck Do-.- 13.6 Low-Density Limit.- Problems.- References.- Aside E. Kirkwood-Wigner Theorem.- E.1 Momentum Eigenstate Expansion.- E.2 Discussion.- Problems.- References.- Lecture 14. Chemical Equilibria.- 14.1 Conditions for Chemical Equilibrium.- 14.2 Equilibrium Constants of Dilute Species from Partition Functions.- 14.3 Discussion.- Problems.- References.- III Interacting Particles and Cluster Expansions.- Lecture 15. Interacting Particles.- 15.1 Potential Energies; Simple Fluids.- 15.2 Simple Reductions; Convergence.- 15.3 Discussion.- Problems.- References.- Lecture 16. Cluster Expansions.- 16.1 Search for an Approach.- 16.2 An Approximant.- 16.3 Flaws of the Approximant.- 16.4 Approximant as a Motivator of Better Approaches.- Problems.- References.- Lecture 17. ? via the Grand Canonical Ensemble.- 17.1 ? and the Density.- 17.2 Expansion for P in Powers of z or ?.- 17.3 Graphical Notation.- 17.4 The Pressure.- 17.5 Summary.- Problems.- References.- Lecture 18. Evaluating Cluster Integrals.- 18.1 B2; Special Cases.- 18.2 More General Techniques.- 18.3 g-Bonds.- 18.4 The Law of Corresponding States.- 18.5 Summary.- Problems.- References.- Lecture 19. Distribution Functions.- 19.1 Motivation for Distribution Functions.- 19.2 Definition of the Distribution Function.- 19.3 Applications of Distribution Functions.- 19.4 Remarks.- 19.5 Summary.- Problems.- Lecture 20. More Distribution Functions.- 20.1 Introduction.- 20.2 Chemical Potential.- 20.3 Charging Processes.- 20.4 Summary.- Problems.- References.- Lecture 21. Electrolyte Solutions, Plasmas, and Screening.- 21.1 Introduction.- 21.2 The Debye-Huckel Model.- 21.3 Discussion.- Problems.- References.- IV Correlation Functions and Dynamics.- Lecture 22. Correlation Functions.- 22.1 Introduction; Correlation Functions.- 22.2 The Density Operator: Examples of Static Correlation Functions.- 22.3 Evaluation of Correlation Functions via Symmetry: Translational Invariance.- 22.4 Correlation Functions of Vectors and Pseudovectors; Other Symmetries.- 22.5 Discussion and Summary.- Problems.- References.- Lecture 23. Stability of the Canonical Ensemble.- 23.1 Introduction.- 23.2 Time Evolution: Temporal Stability of the Canonical Ensemble.- 23.3 Application of the Canonical Ensemble Stability Theorem.- 23.4 Time Correlation Functions.- 23.5 Discussion.- Problems.- References.- Aside F. The Central Limit Theorem.- F.1 Derivation of the Central Limit Theorem.- F.2 Implications of the Central Limit Theorem.- F.3 Summary.- Problems.- References.- Lecture 24. The Langevin Equation.- 24.1 The Langevin Model for Brownian Motion.- 24.2 A Fluctuation-Dissipation Theorem on the Langevin Equation.- 24.3 Mean-Square Displacement of a Brownian Particle.- 24.4 Cross Correlation of Successive Langevin Steps.- 24.5 Application of the Central Limit Theorem to the Langevin Model.- 24.6 Summary.- Problems.- References.- Lecture 25. The Langevin Model and Diffusion.- 25.1 Necessity of the Assumptions Resulting in the Langevin Model.- 25.2 The Einstein Diffusion Equation: A Macroscopic Result.- 25.3 Diffusion in Concentrated Solutions.- 25.4 Summary.- Problems.- References.- Lecture 26. Projection Operators and the Mori-Zwanzig Formalism.- 26.1 Time Evolution of Phase Points via the Liouville Operator.- 26.2 Projection Operators.- 26.3 The Mori-Zwanzig Formalism.- 26.4 Asides on the Mori-Zwanzig Formalism.- Problems.- References.- Lecture 27. Linear Response T…