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| 艾里斯，Richard S.Ellis，received his B.A. degree in mathematics and German literature from Harvard University in 1969 and his Ph.D. degree in mathematics from New York University in 1972. After spending three years at Northwestern University, he moved to the University of Massachusetts, Amherst, where he is a Professor in the Department of Mathematics and Statistics and Adjunct Professor in the Depart-ment of Judaic and Near Eastern Studies. His research interests in mathematics focus on the theory of large deviations and on applica-tions to statistical mechanics and other areas. |
| Preface Comments on the Use of This Book PART 1: LARGE DEVIATIONS AND STATISTICAL MECHANICS Chapter 1. Introduction to Large Deviations Overview Large Deviations for 1.I.D. Random Variables with a Finite State Space Levels-1 and 2 for Coin Tossing Levels-1 and 2 for I.I.D. Random Variables with a Finite State Space Level-3: Empirical Pair Measure Level-3: Empirical Process Notes Problems Chapter 2. Large Deviation Property and Asymptotics of Integrals Introduction Levels-l, 2, and 3 Large Deviations for I.I.D. Random Vectors The Definition of Large Deviation Property Statement of Large Deviation Properties for Levels-l, 2, and 3 Contraction Principles Large Deviation Property for Random Vectors and Exponential Convergence Varadhan's Theorem on the Asymptotics of Integrals Notes Problems Chapter 3. Large Deviations and the Discrete Ideal Gas Introduction Physics Prelude: Thermodynamics The Discrete Ideal Gas and the Microcanonical Ensemble Thermodynamic Limit, Exponential Convergence, and Equilibrium Values The Maxwell-Boltzmann Distribution and Temperature The Canonical Ensemble and Its Equivalence with the Microcanonical Ensemble A Derivation of a Thermodynamic Equation The Gibbs Variational Formula and Principle Notes Problems Chapter 4. Ferromagnetic Models on Z Introduction An Overview of Ferromagnetic Models Finite-Volume Gibbs States on Z Spontaneous Magnetization for the Curie-Weiss Model Spontaneous Magnetization for General Ferromagnets on Z Infinite-Volume Gibbs States and Phase Transitions The Gibbs Variational Formula and Principle Notes Problems Chapter 5. Magnetic Models on Zn and on the Circle Introduction Finite-Volume Gibbs States on ZD, D> 1 Moment Inequalities Properties of the Magnetization and the Gibbs Free Energy Spontaneous Magnetization on ZD, D>2, Via the Peierls Argument Infinite-Volume Gibbs States and Phase Transitions Infinite-Volume Gibbs States and the Central Limit Theorem Critical Phenomena and the Breakdown of the Central Limit Theorem Three Faces of the Curie-Weiss Model The Circle Model and Random Waves A Postscript on Magnetic Models Notes Problems PART 2: CONVEXITY AND PROOFS OF LARGE DEVIATION THEOREMS Chapter 6. Convex Functions and the Legendre-Fenchel Transform Introduction Basic Definitions Properties of Convex Functions …… APPENDICES |
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