Statistical Mechanics: Part A: Equilibrium Techniques

ISBN-13: 9781468425550 / Angielski / Miękka / 2012 / 242 str.

Bruce Berne

Opis

The last decade has been marked by a rapid growth in statistical mechanics, especially in connection with the physics and chemistry of the fluid state. Our understanding in these areas has been considerably advanced and enriched by the discovery of new techniques and the sharpening of old techniques, ranging all the way from computer simulation to mode-mode coupling theories. Statistical mechanics brings together under one roof a broad spectrum of mathematical techniques. The aim of these volumes is to provide a didactic treatment of those techniques that are most useful for the study of problems of current interest to theoretical chemists. The emphasis throughout is on the techniques themselves and not on reviewing the enormous literature in statistical mechanics. Each author was charged with the following task. Given N pages, (a) pose the problem, (b) present those aspects of the particular technique that clearly illustrate its internal workings, (c) apply the technique to the solution of several illustrative examples, and (d) write the chapter so that it will enable the reader to approach key citations to the literature intelligently. These volumes are designed for graduate students and research workers in statistical mechanics. Nevertheless, because of the range of techniques and their general utility, they should be useful in other areas as well.

Spis treści

of Volume 5.- 1. Cluster Methods in Equilibrium Statistical Mechanics of Fluids.- 1. Introduction.- 2. Graph-Theoretic Definitions.- 2.1. Definition of a Graph and the Ideas of Topological Equivalence and Connectivity.- 2.2. Point Functions, Bond Functions, and the Definition of the Value of a Graph.- 3. Partition Function, Pair Correlation Function, and Their Graphical Representation.- 3.1. Statistical-Mechanical Definitions.- 3.2. Cluster Expansion Expressions for A and g.- 4. Topological Reduction.- 5. General Philosophy of the Renormalization Method.- 5.1. Motives for the Use of Renormalization.- 5.2. Strategies for Dealing with the Cluster Series for a Particular Fluid.- 6. Applications of Cluster Theory.- 6.1. Low-Density Virial Series for Nonpolar Fluids.- 6.2. Mayer Theory of Ionic Solutions.- 6.3. Blip Function Theory.- 6.4. Perturbation Theory of Fluids.- 6.5. ?-Ordering and F-Ordering.- 6.6. Optimized Cluster Theory.- 6.7. Hydrogen-Bonded Fluids.- 7. Concluding Remarks.- References.- 2. Fluids with Long-Range Forces: Toward a Simple Analytic Theory.- 1. Introduction.- 2. Some General Features of the Fluids under Consideration.- 3. Large ?-Small ? Interpolation; Padé Approximants.- 4. Large r-Small r Interpolation; The Mean Spherical Approximation.- 5. Beyond the MSA.- 5.1. ? Ordering, Nodal Contraction, Nodal Ordering, Nodal Approximation.- 5.2. Mixed Perturbation Theory.- 6. Summary of Results.- References.- 3. Electrolyte Solutions at Equilibrium.- 1. Introduction.- 2. Models for Ionic Solutions.- 2.1. Hamiltonian Models.- 2.2. Choice of Level.- 2.3. Principal Features of Interionic Forces.- 2.4. Additional Contributions to Interionic Forces.- 2.5. Current Research on Interionic Forces in Solution.- 3. Measurable Properties of Solutions at Equilibrium.- 3.1. Solvation Coefficients.- 3.2. Thermodynamic Excess Functions. Solutions of a Single Electrolyte.- 3.3. Thermodynamic Excess Functions. Mixed Electrolytes.- 3.4. Thermodynamic Excess Functions. Mixtures of Electrolytes with Nonionic Solutes.- 3.5. Partial Structure Factors.- 4. McMillan-Mayer Theory.- 4.1. Basic Grand Ensemble Equations.- 4.2. McMillan-Mayer Theory.- 4.3. More General Results.- 4.4. General Significance of the MM Theory.- 5. Thermodynamic Aspects of the McMillan-Mayer Theory.- 5.1. Solvation Thermodynamics.- 5.2. Thermodynamic Excess Functions. LR to MM Conversions.- 5.3. Thermodynamic Excess Functions in the MM System.- 6. Cluster Expansions.- 6.1. Graph Theory.- 6.2. Graphical Representation of the Grand Partition Function.- 6.3. Ursell Functions of the WN.- 6.4. Cluster Functions.- 6.5. Rooted Ursell Functions.- 6.6. Cluster Expansion of Aex.- 6.7. Discussion of the Cluster Expansion of Aex.- 6.8. Mayer Resummation.- 7. Integral Equations.- 7.1. Ornstein-Zernike Equation.- 7.2. Integral Equations for Nonionic and Ionic Systems.- 7.3. Mean Spherical Approximation.- 7.4. Other Approximation Methods.- 7.5. Quality Tests.- References.- 4. A Guide to Monte Carlo for Statistical Mechanics: 1. Highways.- 1. Introduction.- 2. The Monte Carlo Method.- 2.1. The Need for Refined Monte Carlo Sampling.- 2.2. Importance Sampling.- 2.3. The Metropolis Sampling Scheme.- 2.4. Choice of the Transition Matrix.- 3. Some Practical Details.- 3.1. Typical Procedures.- 3.2. Other Ensembles.- 3.3. Nonfluid Problems.- 4. Boundary Conditions.- 4.1. Avoiding Surfaces: Periodic Boundary Conditions.- 4.2. Size and Shape of the Sample.- 4.3. Configurational Energy Estimation.- 5. Conclusion.- Appendix A: Random Number Generators.- Appendix B: Ewald Potential Technique.- References.- 5. A Guide to Monte Carlo for Statistical Mechanics: 2. Byways.- 1. Introduction.- 2. Estimations of Free Energy and Entropy.- 2.1. Salsburg and Others.- 2.2. Coldwell.- 2.3. Thermodynamic Integration.- 2.4. Communal Free-Energy Estimation.- 2.5. Widom’s Particle Insertion Method.- 2.6. Grand Canonical Approach.- 2.7. McDonald and Singer: Energy Density Functions.- 2.8. Multistage Sampling.- 2.9. Umbrella Sampling Methods.- 3. Quantum Mechanical Calculations.- 3.1. Variational Calculations.- 3.2. Numerical Solution of Schrödinger’s Equation.- 4. Microscopic Studies.- 4.1. Gas-Liquid Interface.- 4.2. Interionic Mean Forces.- 5. Conclusion.- References.- 6. Nucleation Theory.- 1. Introduction.- 2. Mathematical Formalism.- 2.1. The Nucleation Problem.- 2.2. McDonald’s Trick.- 2.3. An Alternative Trick.- 2.4. Some Remarks.- 3. Homogeneous Gas Phase Nucleation.- 3.1. Capture Rates.- 3.2. Equilibrium Cluster Concentrations—Statistical-Mechanical Considerations.- 3.3. The Drop Model.- 3.4. Classical Nucleation Theory.- 3.5. Criticisms of the Drop Model and Classical Nucleation Theory.- 3.6. Statistical-Mechanical Emendations.- 3.7. Modification of the Surface Free Energy.- 3.8. A Microscopic Approach to the Cluster Free Energy.- 3.9. A Proposed New Approach.- 4. Condensation of Water on Ions.- 5. Void Nucleation in Nuclear Reactor Materials.- References.- Author Index.