Statistical Mechanics Made Simple

by Mattis, Daniel C.; Swendsen, Robert H.

• ISBN13: 9789812779090
• ISBN10: 9812779094
• Edition: 2nd
• Format: Paperback
• Copyright: 2008-03-30
• Publisher: World Scientific Pub Co Inc

Summary

This second edition extends and improves on the first, already an acclaimed and original treatment of statistical concepts insofar as they impact theoretical physics and form the basis of modern thermodynamics. This book illustrates through myriad examples the principles and logic used in extending the simple laws of idealized Newtonian physics and quantum physics into the real world of noise and thermal fluctuations.

Table of Contents

Preface to Second Edition	p. xi
Preface to First Edition	p. xiii
Introduction: Theories of Thermodynamics, Kinetic Theory and Statistical Mechanics	p. xvii
Elementary Concepts in Statistics and Probability	p. 1
Random Variables and Their Distributions	p. 1
The Binomial Distribution	p. 2
Length of a Winning Streak	p. 4
Brownian Motion and the Random Walk	p. 5
Poisson versus Normal (Gaussian) Distributions	p. 6
Central Limit Theorem (CLT)	p. 10
Multinomial Distributions, Statistical Thermodynamics	p. 13
The Barometer Equation	p. 15
Other Distributions	p. 16
The Ising Model and the Lattice Gas	p. 19
Physical Applications of the Binary Model	p. 19
Some Background and Motivation	p. 20
First-Principles Statistical Theory of Paramagnetism	p. 21
More on Entropy and Energy	p. 24
Some Other Relevant Thermodynamic Functions	p. 25
Mean-Field Theory, Stable and Metastable Solutions	p. 26
The Lattice Gas	p. 30
The Nearest-Neighbor Chain: Thermodynamics in 1D	p. 31
The Disordered Ising Chain	p. 32
Other Magnetic Systems in One Dimension	p. 32
Elements of Thermodynamics	p. 36
The Scope of Thermodynamics	p. 36
Equations of State and Some Definitions	p. 37
Maxwell Relations	p. 40
Three Important Laws of Thermodynamics	p. 40
The Second Derivatives of the Free Energy	p. 43
Phase Diagrams for the van der Waals Gas	p. 44
Clausius-Clapeyron Equation	p. 48
Phase Transitions	p. 50
The Carnot Cycle	p. 54
Superconductivity	p. 57
Statistical Mechanics	p. 60
An Axiomatic Approach and the Ergodic Hypothesis	p. 60
The Formalism - and a False Start	p. 62
Gibbs' Paradox and Its Remedy	p. 65
The Gibbs Factor	p. 68
The Grand Ensemble	p. 69
Non-Ideal Gas and the 2-Body Correlation Function	p. 71
The Virial Equation of State	p. 73
Weakly Non-Ideal Gas	p. 74
Two-body Correlations	p. 77
Configurational Partition Function in 1D	p. 82
One Dimension versus Two	p. 84
Two Dimensions versus Three: The Debye-Waller Factors	p. 86
Specific Heat of Quasi-Ideal Dilute Atomic and Diatomic Gases	p. 90
Nanophysics and Inhomogeneity	p. 93
The World of Bosons	p. 94
Quantum "Statistics"	p. 94
Two Types of Bosons and Their Operators	p. 94
Number Representation and the Many-Body Problem	p. 97
The Adiabatic Process and Conservation of Entropy	p. 99
Many-Body Perturbations	p. 100
Photons	p. 101
Phonons	p. 105
Ferromagnons	p. 107
Conserved Bosons and the Ideal Bose Gas	p. 110
Nature of "Ideal" Bose-Einstein Condensation	p. 113
Ideal Bose-Einstein Condensation in Low Dimensions	p. 115
Consequences of a Hard Core Repulsion in 1D	p. 118
Bosons in 3D Subject to Weak Two-Body Forces	p. 121
Superfluid Helium (He II)	p. 126
All About Fermions: Theories of Metals, Superconductors, Semiconductors	p. 131
Fermi-Dirac Particles	p. 131
Slater Determinant: The Ground State	p. 132
Ideal Spinless Fermi-Dirac Gas	p. 133
Ideal Fermi-Dirac Gas with Spin	p. 135
Fermi Integrals	p. 136
Thermodynamic Functions of an Ideal Metal	p. 137
Quasiparticles and Elementary Excitations	p. 140
Semiconductor Physics: Electrons and Holes	p. 142
n-Type Semiconductor Physics: The Statistics	p. 143
Correlations and the Coulomb Repulsion	p. 144
Miscellaneous Properties of Semiconductors	p. 147
Aspects of Superconductivity: Cooper Pairs	p. 149
Aspects of BCS Theory	p. 152
Contemporary Developments in Superconductivity	p. 158
Kinetic Theory	p. 160
Scope of This Chapter	p. 160
Quasi-Equilibrium Flows and the Second Law	p. 161
The Collision Integral	p. 162
Approach to Equilibrium of a "Classical" Non-Ideal Gas	p. 165
A New Look at "Quantum Statistics"	p. 168
Master Equation: Application to Radioactive Decay	p. 169
Boltzmann Equation	p. 171
Electrical Currents in a Low-Density Electron Gas	p. 174
Diffusion and the Einstein Relation	p. 176
Electrical Conductivity of Metals	p. 177
Exactly Solved "Backscattering" Model	p. 178
Electron-Phonon Scattering	p. 180
Approximating the Boltzmann Equation	p. 181
Crossed Electric and Magnetic Fields	p. 183
Propagation of Sound Waves in Fluids	p. 184
The Calculations and Their Result	p. 189
The Transfer Matrix	p. 195
The Transfer Matrix and the Thermal Zipper	p. 195
Opening and Closing a "Zipper Ladder" or Polymer	p. 198
The Full Zipper (N > 2)	p. 203
The Transfer Matrix and Gaussian Potentials	p. 204
Transfer Matrix in the Ising Model	p. 205
The Ising Ladder or Polymer	p. 207
Ising Model on the Isotropic Square Lattice (2D)	p. 209
The Phase Transition	p. 214
A Question of Long-Range Order	p. 216
Ising Model in 2D and 3D	p. 218
Antiferromagnetism and Frustration	p. 219
Maximal Frustration	p. 222
Separable Model Spin-Glass without Frustration	p. 224
Critical Phenomena and Critical Exponents	p. 225
Potts Models	p. 229
Monte Carlo and Other Computer Simulation Methods	p. 232
Numerical Methods in Statistical Mechanics	p. 232
Molecular Dynamics	p. 232
Stochastic Processes	p. 234
Transition Rates and the Master Equation	p. 235
Monte Carlo Moves	p. 237
Applications of ARM and DOMC	p. 241
Monte Carlo Estimates of Thermal Averages	p. 241
Equilibration Times	p. 243
Discrete Variables	p. 245
Continuous Thermodynamic Functions and Histogram Methods	p. 246
Phase Transitions	p. 249
MC Calculation of Critical Exponents: Finite-Size Scaling	p. 249
Critical Phenomena and the Renormalization Group	p. 254
The Problem with Mean-Field Theories	p. 254
Block-Spin Transformations for the Ising Model	p. 256
Effect of Renormalization on the Spin Configurations	p. 258
Renormalization of the Correlation Length	p. 260
Mathematical Form of the RG Transformation	p. 260
Effective or "Renormalized" Interactions	p. 261
A Simple Example: The One-Dimensional Ising Model	p. 263
Renormalization Trajectories	p. 265
RG Trajectories and Critical Properties	p. 267
Mathematical Analysis of RG Trajectories Near a Fixed Point	p. 268
Calculating the Other Critical Exponents from the Eigenvalue Exponents	p. 270
Exponent Identities	p. 271
Calculating Values for the Eigenvalue Exponents Using MC Simulations	p. 272
First-Order Transitions and Discontinuity Fixed Points	p. 276
Calculation of the Renormalized Couplings	p. 277
Some Uses of Quantum Field Theory in Statistical Physics	p. 279
Outline of the Chapter	p. 279
Diffusion on a Lattice: Standard Formulation	p. 280
Diffusion as Expressed in QFT	p. 282
Diffusion plus One-Body Recombination Processes	p. 285
Diffusion and Two-Body Recombination Processes	p. 286
Questions Concerning Long-Range Order	p. 288
Mermin-Wagner Theorem	p. 290
Proof of Bogolubov Inequality	p. 293
Correlation Functions and the Free Energy	p. 294
Introduction to Thermodynamic Green's Functions	p. 297
Solutions to Selected Problems	p. 305
Bibliography	p. 327
Index	p. 331
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