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Statistical Mechanics 3 d rB rB 3 d rA rA Statistical Mechanics Daniel F. Styer December 2007 Statistical Mechanics Dan Styer Department of Physics and Astronomy Oberlin College Oberlin, Ohio 44074-1088 [email protected] http://www.oberlin.edu/physics/dstyer December 2007 Although, as a matter of history, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light. | J. Willard Gibbs Elementary Principles in Statistical Mechanics Contents 0 Preface 1 1 The Properties of Matter in Bulk 4 1.1 What is Statistical Mechanics About? . 4 1.2 Outline of Book . 4 1.3 Fluid Statics . 5 1.4 Phase Diagrams . 7 1.5 Additional Problems . 7 2 Principles of Statistical Mechanics 10 2.1 Microscopic Description of a Classical System . 10 2.2 Macroscopic Description of a Large Equilibrium System . 14 2.3 Fundamental Assumption . 15 2.4 Statistical Definition of Entropy . 17 2.5 Entropy of a Monatomic Ideal Gas . 19 2.6 Qualitative Features of Entropy . 25 2.7 Using Entropy to Find (Define) Temperature and Pressure . 34 2.8 Additional Problems . 44 3 Thermodynamics 46 3.1 Heat and Work . 46 3.2 Heat Engines . 50 i ii CONTENTS 3.3 Thermodynamic Quantities . 52 3.4 Multivariate Calculus . 55 3.5 The Thermodynamic Dance . 60 3.6 Non-fluid Systems . 67 3.7 Thermodynamics Applied to Fluids . 68 3.8 Thermodynamics Applied to Phase Transitions . 76 3.9 Thermodynamics Applied to Chemical Reactions . 76 3.10 Thermodynamics Applied to Light . 81 3.11 Additional Problems . 85 4 Ensembles 89 4.1 The Canonical Ensemble . 89 4.2 Meaning of the Term \Ensemble" . 90 4.3 Classical Monatomic Ideal Gas . 91 4.4 Energy Dispersion in the Canonical Ensemble . 94 4.5 Temperature as a Control Variable for Energy (Canonical Ensemble) . 97 4.6 The Equivalence of Canonical and Microcanonical Ensembles . 98 4.7 The Grand Canonical Ensemble . 98 4.8 The Grand Canonical Ensemble in the Thermodynamic Limit . 99 4.9 Summary of Major Ensembles . 101 4.10 Quantal Statistical Mechanics . 102 4.11 Ensemble Problems I . 104 4.12 Ensemble Problems II . 111 5 Classical Ideal Gases 113 5.1 Classical Monatomic Ideal Gases . 113 5.2 Classical Diatomic Ideal Gases . 116 5.3 Heat Capacity of an Ideal Gas . 116 5.4 Specific Heat of a Hetero-nuclear Diatomic Ideal Gas . 121 5.5 Chemical Reactions Between Gases . 124 5.6 Problems . 124 CONTENTS iii 6 Quantal Ideal Gases 127 6.1 Introduction . 127 6.2 The Interchange Rule . 127 6.3 Quantum Mechanics of Independent Identical Particles . 128 6.4 Statistical Mechanics of Independent Identical Particles . 135 6.5 Quantum Mechanics of Free Particles . 140 6.6 Fermi-Dirac Statistics . 141 6.7 Bose-Einstein Statistics . 143 6.8 Specific Heat of the Ideal Fermion Gas . 148 6.9 Additional Problems . 155 7 Harmonic Lattice Vibrations 157 7.1 The Problem . 157 7.2 Statistical Mechanics of the Problem . 157 7.3 Normal Modes for a One-dimensional Chain . 158 7.4 Normal Modes in Three Dimensions . 158 7.5 Low-temperature Heat Capacity . 158 7.6 More Realistic Models . 160 7.7 What is a Phonon? . 160 7.8 Additional Problems . 160 8 Interacting Classical Fluids 162 8.1 Introduction . 162 8.2 Perturbation Theory . 164 8.3 Variational Methods . 168 8.4 Distribution Functions . 169 8.5 Correlations and Scattering . 172 8.6 The Hard Sphere Fluid . 172 iv CONTENTS 9 Strongly Interacting Systems and Phase Transitions 175 9.1 Introduction to Magnetic Systems and Models . 175 9.2 Free Energy of the One-Dimensional Ising Model . 175 9.3 The Mean-Field Approximation . 178 9.4 Correlation Functions in the Ising Model . 178 9.5 Computer Simulation . 184 9.6 Additional Problems . 193 A Series and Integrals 199 B Evaluating the Gaussian Integral 200 C Clinic on the Gamma Function 201 D Volume of a Sphere in d Dimensions 203 E Stirling's Approximation 206 F The Euler-MacLaurin Formula and Asymptotic Series 208 G Ramblings on the Riemann Zeta Function 209 H Tutorial on Matrix Diagonalization 214 H.1 What's in a name? . 214 H.2 Vectors in two dimensions . 215 H.3 Tensors in two dimensions . 217 H.4 Tensors in three dimensions . 219 H.5 Tensors in d dimensions . 220 H.6 Linear transformations in two dimensions . 221 H.7 What does \eigen" mean? . 223 H.8 How to diagonalize a symmetric matrix . 223 H.9 A glance at computer algorithms . 229 H.10 A glance at non-symmetric matrices and the Jordan form . 230 CONTENTS v I Catalog of Misconceptions 234 J Thermodynamic Master Equations 236 K Useful Formulas 237 Index 238 Chapter 0 Preface This is a book about statistical mechanics at the advanced undergraduate level. It assumes a background in classical mechanics through the concept of phase space, in quantum mechanics through the Pauli exclusion principle, and in mathematics through multivariate calculus. (Section 9.2 also assumes that you can can diagonalize a 2 × 2 matrix.) The book is.
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