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Statistical Mechanics Second edition
(John Wiley, New York, 1987) 493 pages
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FROM THE PREFACE
In this book the starting point of statistical mechanics is taken to be certain phenomenological postulates, whose relation to quantum mechanics I try to state as clearly as I can, and whose physical consequences I try to derive as simply and directly as I can. Before the subject of statistical mechanics proper is presented, a brief but self-contained discussion of thermodynamics and the classical kinetic theory of gases is given. The order of this evelopment is imperative, from a pedagogical point of view, for two reasons. First, thermodynamics has successfully described a large part of macroscopic xperience, which is the concern of statistical mechanics.It has done so not on the basis of molecular dynamics but on the basis of a few simple and intuitive postulates stated in everyday terms. If we first familiarize ourselves with thermodynamics, the task of statistical mechanics reduces to the explanation of thermodynamics. Second, the classical kinetic theory of gases is the only known special case in which thermodynamics can be derived nearly from first principles, i.e., molecular dynamics. A study of this special case will help us understand why statistical mechanics works. A large part of this book is devoted to selected applications of statistical mechanics. The selection is guided by the interest of the topic to physicists, its value as an illustration of calculating techniques, and my personal taste. To read the first half of the book the reader needs a good knowledge of classical mechanics and some intuitive feeling for thermodynamics and kinetic theory. To read the second half of the book he needs to have a working knowledge of quantum mechanics. The mathematical knowledge required of the reader does not exceed what he should have acquired in his study of classical mechanics and quantum mechanics.
1 THE LAWS OF THERMODYNAMICS Preliminaries The First Law of Thermodynamics The Second Law of Thermodynamics Entropy Some Immediate Consequences of the Second Law Thermodynamic Potentials The Third Law of Thermodynamics 2 SOME APPLICATIONS OF THERMODYNAMICS Thermodynamic Description of Phase Transitions Surface Effects in Condensation Van der Waals Equation of State Osmotic Pressure The Limit of Thermodynamics 3 THE PROBLEM OF KINETIC THEORY Formulation of the Problem Binary Collisions The Boltzmann Transport Equation The Gibbsian Ensemble The BBGKY Hierarchy 4 THE EOUILIBRIUM STATE OF A DILUTE GAS Boltzmann's H Theorem The Maxwell-Boltzmann Distribution The Method of the Most Probable Distribution Analysis of the H Theorem The Poincare Cycle 5 TRANSPORT PHENOMENA The Mean Free Path Effusion The Conservation Laws The Zero-Order Approximation The First-Order Approximation Viscosity Viscous Hydrodynamics The Navier-Stokes Equation Examples in Hydrodynamics 6 CLASSICAL STATISTICAL MECHANICS The Postulate of Classical Statistical Mechanics Microcanonical Ensemble Derivation of Thermodynamics Equipartition Theorem Classical Ideal Gas 8 Gibbs Paradox 7 CANONICAL ENSEMBLE AND GRAND CANONICAL ENSEMBLE Canonical Ensemble Energy Fluctuations in the Canonical Ensemble Grand Canonical Ensemble Density Fluctuations in the Grand Canonical Ensemble The Chemical Potential Equivalence of the Canonical Ensemble and the Grand Canonical Ensemble Behavior of W(N) The Meaning of the Maxwell Construction 8 QUANTUM STATISTICAL MECHANICS The Postulates of Quantum Statistical Mechanics Density Matrix Ensembles in Quantum Statistical, Mechanics The Third Law of Thermodynamics The Ideal Gases: Microcanonical Ensemble The Ideal Gases: Grand Canonical Ensemble Foundations of Statistical Mechanics 9 GENERAL PROPERTIES OF THE PARTITION FUNCTION The Darwin-Fowler Method Classical Limit of the Partition Function Singularities and Phase Transitions The Lee-Yang Circle Theorem 10 APPROXIMATE METHODS Classical Cluster Expansion Quantum Cluster Expansion The Second Virial Coefficient Variational Principles Imperfect Gases at Low Temperatures 11 FERMI SYSTEMS The Equation of State of an Ideal Fermi Gas The Theory of White Dwarf Stars Landau Diamagnetism The De Haas-Van Alphen Effect The Quantized Hall Eflect Pauli Paramagnetism Magnetic Properties of an Imperfect Gas 12 BOSE SYSTEMS Photons Phonons in Solids Bose-Einstein Condensation An Imperfect Bose Gas The Superfluid Order Parameter 13 SUPERFLUIDS Liquid Helium Tisza's Two-Fluid Model The Bose-Einstein Condensate Landau's Theory Superfluid Velocity Superfluid Flow The Phonon Wave Function Dilute Bose Gas 14 THE SING MODEL Definition of the Ising Model Equivalence of the Ising Model to Other Models Spontaneous Magnetization The Bragg-Williams Approximation The Bethe-Peierls Approximation The One-Dimensional Ising Model 15 THE ONSAGER SOLUTION Formulation of the Two-Dimensional Ising Model Mathematical Digression4 The Solution 16 CRITICAL PHENOMENA The Order Parameter The Correlation Function and the Fluctuation~Dissipation Theorem Critical Exponents The Scaling Hypothesis Scale Invariance Goldstone Excitations The Importance of Dimensionality 17 THE LANDAU APPROACH The Landau Free Energy Mathematical Digression Derivation in Simple Models Mean-Field Theory The Van der Waals Equation of State The Tricritical Point The Gaussian Model The Ginzburg Criterion Anomalous Dimensions 18 RENORMALIZATION GROUP Block Spins The One-Dimensional Ising Model Renormalization-Group Transformation Fixed Points and Scaling Fields Momentum-Space Formulation The Gaussian Model The Landau-Wilson Model APPENDIX N-BODY SYSTEM OF IDENTICAL PARTICLES The Two Kinds of Statistics N-Body Wave Functions Method of Quantized Fields Longitudinal Sum Rules