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Physics - Classical Continuum Physics | Elementary Lectures in Statistical Mechanics

Elementary Lectures in Statistical Mechanics

Phillies, George D.J.

2000, XVI, 431 p.

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This volume is based on courses on Statistical Mechanics which I have taught for many years at the Worcester Polytechnic Institute. My objective is to treat classical statistical mechanics and its modem applications, especially interacting particles, correlation functions, and time-dependent phenomena. My development is based primarily on Gibbs's ensemble formulation. Elementary Lectures in Statistical Mechanics is meant as a (relatively sophis­ ticated) undergraduate or (relatively straightforward) graduate text for physics students. It should also be suitable as a graduate text for physical chemistry stu­ dents. Physicists may find my treatment of algebraic manipulation to be more explicit than some other volumes. In my experience some of our colleagues are perhaps a bit over-enthusiastic about the ability or tendency of our students to complete gaps in the derivations. I emphasize a cyclic development of major themes. I could have begun with a fully detailed formal treatment of ensemble mechanics, as found in Gibbs's volume, and then given material realizations. I instead interleave formal discussions with simple concrete models. The models illustrate the formal definitions. The approach here gives students a chance to identify fundamental principles and methods before getting buried in ancillary details.

Content Level » Graduate

Keywords » classical systems - correlation - equilibrium properties - mechanics - quantum systems - statistical mechanics

Related subjects » Classical Continuum Physics - Physical & Information Science

Table of contents 

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 Theory.- 27.1 Introduction.- 27.2 Linear Response Theory.- 27.3 Electrical Conductivity.- 27.4 Discussion.- Problems.- References.- V A Research Problem.- Aside G. Scattering of Light, Neutrons, X-Rays, and Other Radiation.- G.1 Introduction.- G.2 Scattering Apparatus; Properties of Light.- G.3 Time Correlation Functions.- Problems.- References.- Lecture 28. Diffusion of Interacting Particles.- 28.1 Why Should We Care About this Research Problem—.- 28.2 What Shall We Calculate—.- 28.3 Model for Particle Dynamics.- 28.4 First Cumulant for g(1)(k, t).- 28.5 Summary.- Problems.- References.- Lecture 29. Interacting Particle Effects.- 29.1 Reduction to Radial Distribution Functions.- 29.2 Numerical Values for K1 and K1s.- 29.3 Discussion.- Problems.- References.- Lecture 30. Hidden Correlations.- 30.1 Model-Independent Results.- 30.2 Evaluation of the Derivatives.- 30.3 Resolution of the Anomaly.- 30.4 Discussion.- Problems.- References.

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