Introduction to Statistical Mechanics

by Walecka, John Dirk

• ISBN13: 9789814366212
• ISBN10: 9814366218
• Format: Paperback
• Copyright: 2011-09-18
• Publisher: World Scientific Pub Co Inc

Summary

The science of statistical mechanics is concerned with defining the thermodynamic properties of a macroscopic sample in terms of the properties of the microscopic systems of which it is composed. The aim of this book is to provide a clear, logical, and self-contained treatment of equilibrium statistical mechanics starting from Boltzmann's two statistical assumptions, and to present a wide variety of applications to diverse physical assemblies. The coverage is enhanced and extended through an extensive set of accessible problems. An appendix provides an introduction to non-equilibrium statistical mechanics through the Boltzmann equation and its extensions. The book assumes introductory courses in classical and quantum mechanics, as well as familiarity with multi-variable calculus and the essentials of complex analysis. Some knowledge of thermodynamics is assumed, although the book starts with an appropriate review of that topic. The targeted audience is first-year graduate students, and advanced undergraduates, in physics, chemistry, and the related physical sciences. The goal of this text is to help the reader obtain a clear working knowledge of the very useful and powerful methods of equilibrium statistical mechanics and to enhance the understanding and appreciation of the more advanced texts.

Table of Contents

Preface	p. vii
Introduction	p. 1
Review of Thermodynamics	p. 2
First Law	p. 2
Second Law	p. 3
Free Energies	p. 7
Equilibrium	p. 9
Third Law	p. 12
Basic Statistical Hypotheses	p. 12
Some Definitions	p. 12
Statistical Assumptions	p. 12
The Microcanonical Ensemble	p. 17
Independent Localized Systems	p. 17
The Boltzmann Distribution	p. 21
The Partition Function	p. 25
Einstein's Theory of the Specific Heat	p. 27
Method of Steepest Descent	p. 29
Independent Non-Localized Systems	p. 39
Perfect Gas of Structureless Particles	p. 41
Validity	p. 48
Transition to Classical Dynamics	p. 50
Classical Mechanics	p. 50
Quantum Mechanics	p. 51
Compute ¿(E, V, N)	p. 55
Applications of the Microcanonical Ensemble	p. 63
Internal Partition Function	p. 63
Molecular Spectroscopy	p. 64
Diatomic Molecules	p. 64
Born-Oppenheimer Approximation	p. 66
Partition Function	p. 70
Heat Capacity	p. 72
Symmetry of the Wave Function	p. 73
Ortho- and Para-Hydrogen H2	p. 79
Typical Spectrum	p. 80
Selection Rules	p. 81
Polyatomic Molecules	p. 85
Symmetric Top	p. 85
Partition Function	p. 87
Hindered Rotation	p. 90
Comparison of Spectroscopic and Calorimetric Entropies	p. 92
Sources of kB ln ¿0	p. 93
Paramagnetic and Dielectric Assemblies	p. 96
Classical Gas of Permanent Dipoles	p. 97
Magnetic Moments in Quantum Mechanics	p. 101
Polarization in a Dielectric Medium	p. 104
Paramagnetic Susceptibility	p. 109
Thermodynamics	p. 111
Chemical Equilibria	p. 112
Some Preliminaries	p. 112
Chemical Reactions and the Law of Mass Action	p. 114
Chemical Potentials	p. 118
Solid in Equilibrium with Its Vapor	p. 120
Surface Adsorption	p. 123
The Canonical Ensemble	p. 127
Constant-Temperature Partition Function	p. 127
Independent Localized Systems	p. 130
Independent Non-Localized Systems	p. 132
Classical Limit	p. 132
Energy Distribution	p. 135
Summary of Results So Far	p. 137
Microcanonical Ensemble	p. 138
Canonical Ensemble	p. 138
Applications of the Canonical Ensemble	p. 141
Solids	p. 141
Einstein Model	p. 141
Normal Modes	p. 142
Debye Model	p. 145
Normal-Model Spectrum	p. 145
Thermodynamics	p. 148
Discussion	p. 149
Improved Normal-Mode Spectrum	p. 151
Longitudinal Waves in a Rod	p. 151
Lattice Model	p. 154
Imperfect Gases	p. 158
Configuration Integral	p. 159
Second Virial Coefficient	p. 160
General Analysis of Configuration Integral	p. 165
Linked-Cluster Expansion	p. 165
Summation of Series	p. 169
Interpretation	p. 171
Virial Expansion	p. 171
Law of Corresponding States	p. 173
Derivation	p. 176
The Grand Canonical Ensemble	p. 181
Grand Partition Function	p. 182
Relation to Previous Results	p. 186
Independent Non-Localized Systems	p. 186
Independent Localized Systems	p. 187
Imperfect Gases	p. 188
Fluctuations	p. 189
Distribution of Energies in the Canonical Ensemble	p. 189
Distribution of Particle Numbers in the Grand Canonical Ensemble	p. 190
Applications of the Grand Canonical Ensemble	p. 195
Boltzmann Statistics	p. 195
Quantum Statistics	p. 196
Grand Partition Function	p. 197
Bose Statistics	p. 197
Fermi Statistics	p. 198
Distribution Numbers	p. 198
Energy	p. 199
Bosons	p. 199
Electromagnetic Radiation	p. 200
Normal Modes	p. 200
Chemical Potential	p. 201
Spectral Weight	p. 202
Equation of State	p. 203
Discussion	p. 205
Bose Condensation	p. 207
Non-Relativistic Equation of State	p. 207
Transition Temperature	p. 209
Discontinuity in Slope of CV	p. 214
Liquid 4He	p. 217
Fermions	p. 218
General Considerations	p. 219
Non-Relativistic Equation of State	p. 219
Distribution Numbers	p. 221
Zero Temperature	p. 222
Low-Temperature CV	p. 224
Pauli Spin Paramagnetism	p. 228
Grand Partition Function	p. 229
Magnetization	p. 230
Landau Diamagnetism	p. 233
Charged Particle in a Magnetic Field	p. 234
Counting of States	p. 238
Grand Partition Function and Magnetization	p. 240
High-Temperature Limit	p. 240
Low-Temperature Limit	p. 242
Special Topics	p. 249
Solutions	p. 249
Perfect Solutions	p. 249
Canonical Partition Function	p. 249
Helmholtz Free Energy	p. 251
Regular Solutions	p. 254
Improved Model of Localized Systems	p. 255
Configuration Partition Function	p. 257
Bragg-Williams Approximation	p. 258
Quasi-Chemical Approximation	p. 260
Order-Disorder Transitions in Crystals	p. 261
¿-Point Transitions	p. 261
Configuration Partition Function	p. 263
Bragg-Williams Approximation	p. 264
Ising Solution for Z = 2	p. 268
The Ising Model	p. 270
Heisenberg Hamiltonian	p. 270
One-Dimensional Ising Model	p. 271
Canonical Partition Function	p. 272
Matrix Solution	p. 272
Two-Dimensional Ising Model (Z = 4)	p. 277
Mean Field Theory	p. 278
Numerical Methods	p. 282
Lattice Gauge Theory	p. 283
The Standard Model	p. 283
Quantum Electrodynamics (QED)	p. 284
Partition Function in Field Theory	p. 284
U(1) Lattice Gauge Theory	p. 285
Mean Field Theory (MFT)	p. 289
Numerical Monte Carlo	p. 291
Strong-Coupling Limit	p. 291
Improved Analytic Approximations	p. 292
Non-Abelian Theory SU(n)	p. 295
Problems	p. 297
Non-Equilibrium Statistical Mechanics	p. 335
Boltzmann Equation	p. 335
One-Body Dynamics	p. 335
Boltzmann Collision Term	p. 337
Vlasov and Boltzmann Equations	p. 340
Equilibrium	p. 340
Molecular Dynamics	p. 342
Nordheim-Uehling-Uhlenbeck Equation	p. 342
Example-Heavy-Ion Reactions	p. 343
Bibliography	p. 345
Index	p. 349
Table of Contents provided by Ingram. All Rights Reserved.