Statistical Mechanics Made Simple (2nd Edition)

by Daniel C Mattis, Robert H Swendsen

Overview

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.In response to the many helpful comments by users of the first edition, important features have been added in this second, new and revised edition. These additions allow a more coherent picture of thermal physics to emerge. Benefiting from the expertise of the new co-author, the present edition includes a detailed exposition — occupying two separate chapters — of the renormalization group and Monte-Carlo numerical techniques, and of their applications to the study of phase transitions. Additional figures have been included throughout, as have new problems. A new Appendix presents fully worked-out solutions to representative problems; these illustrate various methodologies that are peculiar to physics at finite temperatures, that is, to statistical physics.This new edition incorporates important aspects of many-body theory and of phase transitions. It should better serve the contemporary student, while offering to the instructor a wider selection of topics from which to craft lectures on topics ranging from thermodynamics and random matrices to thermodynamic Green functions and critical exponents, from the propagation of sound in solids and fluids to the nature of quasiparticles in quantum liquids and in transfer matrices.

Product Details

ISBN-13:	9789812779083
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/05/2008
Edition description:	2nd ed.
Pages:	360
Product dimensions:	6.70(w) x 9.80(h) x 1.00(d)

Table of Contents

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