ISBN: 3-540-64883-6
TITLE: Statistical Mechanics
AUTHOR: Gallavotti, Giovanni
TOC:

1. Classical Statistical Mechanics 1 
1.1 Introduction 1 
1.2 Microscopic Dynamics 3 
1.3 Time Averages and the Ergodic Hypothesis 10 
1.4 Recurrence Times and Macroscopic Observables 14 
1.5 Statistical Ensembles or "Monodes" and Models 
of Thermodynamics. Thermodynamics Without Dynamics 17 
1.6 Models of Thermodynamics. Microcanonical and Canonical 
Ensembles and the Ergodic Hypothesis 20 
1.7 Critique of the Ergodic Hypothesis 24 
1.8 Approach to Equilibrium and Boltzmann's Equation. 
Ergodicity and Irreversibility 27 
1.9 A Historical Note. The Etymology of the Word "Ergodic" 
and the Heat Theorems 36 
Appendix 1.A1 Monocyclic Systems, Keplerian Motions 
and Ergodic Hypothesis 44 
Appendix 1.A2 GradBoltzmann Limit and Lorentz's Gas 48 
2. Statistical Ensembles 57 
2.1 Statistical Ensembles as Models of Thermodynamics 57 
2.2 Canonical and Microcanonical Ensembles: Orthodicity 60 
2.3 Equivalence Between Canonical 
and Microcanonical Ensembles 68 
2.4 Non-equivalence of the Canonical and Microcanonical 
Ensembles. Phase Transitions. Boltzmann's Constant 73 
2.5 The Grand Canonical Ensemble 
and Other Orthodic Ensembles 76 
2.6 Some Technical Aspects 84 
3. Equipartition and Critique 89 
3.1 Equipartition and Other Paradoxes and Applications 
of Statistical Mechanics 89 
3.2 Classical Statistical Mechanics 
When Cell Sizes Are Not Negligible 95 
3.3 Introduction to Quantum Statistical Mechanics 103 
3.4 Philosophical Outlook on the Foundations 
of Statistical Mechanics 107 
4. Thermodynamic Limit and Stability 111 
4.1 The Meaning of the Stability Conditions 111 
4.2 Stability Criteria 114 
4.3 Thermodynamic Limit 117 
5. Phase Transitions 129 
5.1 Virial Theorem, Virial Series and van der Waals Equation 129 
5.2 The Modern Interpretation 
of van der Waals' Approximation 136 
5.3 Why a Thermodynamic Formalism? 142 
5.4 Phase Space in Infinite Volume 
and Probability Distributions on It. Gibbs Distributions 144 
5.5 Variational Characterization 
of Translation Invariant Gibbs Distributions 147 
5.6 Other Characterizations of Gibbs Distributions. 
The DLR Equations 153 
5.7 Gibbs Distributions and Stochastic Processes 155 
5.8 Absence of Phase Transitions: d = 1. Symmetries: d = 2 157 
5.9 Absence of Phase Transitions: High Temperature 
and the KS Equations 161 
5.10 Phase Transitions and Models 167 
Appendix 5.A1 Absence of Phase Transition 
in Non-Nearest-Neighbor One-Dimensional Systems 172 
6. Coexistence of Phases 175 
6.1 The Ising Model. Inequivalence of Canonical 
and Grand Canonical Ensembles 175 
6.2 The Model. Grand Canonical and Canonical Ensembles. 
Their Inequivalence 176 
6.3 Boundary Conditions. Equilibrium States 178 
6.4 The Ising Model in One and Two Dimensions 
and Zero Field 180 
6.5 Phase Transitions. Definitions 182 
6.6 Geometric Description of the Spin Configurations 184 
6.7 Phase Transitions. Existence 188 
6.8 Microscopic Description of the Pure Phases 189 
6.9 Results on Phase Transitions in a Wider Range 
of Temperature 192 
6.10 Separation and Coexistence of Pure Phases. 
Phenomenological Considerations 196 
6.11 Separation and Coexistence of Phases. Results 198 
6.12 Surface Tension in Two Dimensions. 
Alternative Description of the Separation Phenomena 199 
6.13 The Structure of the Line of Separation. 
What a Straight Line Really Is 200 
6.14 Phase Separation Phenomena and Boundary Conditions. 
Further Results 202 
6.15 Further Results, Some Comments 
and Some Open Problems 205 
7. Exactly Soluble Models 209 
7.1 Transfer Matrix in the Ising Model: Results in d = 1,2 209 
7.2 Meaning of Exact Solubility 
and the Two-Dimensional Ising Model 211 
7.3 Vertex Models 214 
7.4 A Nontrivial Example of Exact Solution: 
The Two-Dimensional Ising Model 221 
7.5 The Six-Vertex Model and Bethe's Ansatz 227 
8. Brownian Motion 233 
8.1 Brownian Motion and Einstein's Theory 233 
8.2 Smoluchowski's Theory 239 
8.3 The UhlenbeckOrnstein Theory 242 
8.4 Wiener's Theory 246 
9. Coarse Graining and Nonequilibrium 253 
9.1 Ergodic Hypothesis Revisited 253 
9.2 Timed Observations and Discrete Time 257 
9.3 Chaotic Hypothesis. Anosov Systems 260 
9.4 Kinematics of Chaotic Motions. Anosov Systems 265 
9.5 Symbolic Dynamics and Chaos 270 
9.6 Statistics of Chaotic Attractors. SRB Distributions 278 
9.7 Entropy Generation. Time Reversibility 
and Fluctuation Theorem. Experimental Tests 
of the Chaotic Hypothesis 281 
9.8 Fluctuation Patterns 287 
9.9 "Conditional Reversibility" and "Fluctuation Theorems" 288 
9.10 Onsager Reciprocity and GreenKubo's Formula 292 
9.11 Reversible Versus Irreversible Dissipation. 
Nonequilibrium Ensembles? 294 
Appendix 9.A1 Mcanique statistique hors quilibre: 
l'hritage de Boltzmann 298 
Appendix 9.A2 Heuristic Derivation of the SRB Distribution 308 
Appendix 9.A3 Aperiodic Motions Can Be Regarded 
as Periodic with Infinite Period! 310 
Appendix 9.A4 Gauss' Least Constraint Principle 311 
Bibliography 313 
Name Index 331 
Subject Index 333 
END
