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Entropy, Large Deviations, and Statistical Mechanics

Entropy, Large Deviations, and Statistical Mechanics
Catalogue Information
Field name Details
Dewey Class 621
Title Entropy, Large Deviations, and Statistical Mechanics ([EBook]) / by Richard S. Ellis.
Author Ellis, Richard S. , 1947-
Other name(s) SpringerLink (Online service)
Publication New York, NY : Springer , 1985.
Physical Details XIV, 365 pages : online resource.
Series Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 0072-7830 ; ; 271
ISBN 9781461385332
Summary Note This book has two main topics: large deviations and equilibrium statistical mechanics. I hope to convince the reader that these topics have many points of contact and that in being treated together, they enrich each other. Entropy, in its various guises, is their common core. The large deviation theory which is developed in this book focuses upon convergence properties of certain stochastic systems. An elementary example is the weak law of large numbers. For each positive e, P{ISn/nl 2: e} con­ verges to zero as n --+ 00, where Sn is the nth partial sum of indepen­ dent identically distributed random variables with zero mean. Large deviation theory shows that if the random variables are exponentially bounded, then the probabilities converge to zero exponentially fast as n --+ 00. The exponen­ tial decay allows one to prove the stronger property of almost sure conver­ gence (Sn/n --+ 0 a.s.). This example will be generalized extensively in the book. We will treat a large class of stochastic systems which involve both indepen­ dent and dependent random variables and which have the following features: probabilities converge to zero exponentially fast as the size of the system increases; the exponential decay leads to strong convergence properties of the system. The most fascinating aspect of the theory is that the exponential decay rates are computable in terms of entropy functions. This identification between entropy and decay rates of large deviation probabilities enhances the theory significantly.:
Contents note I: Large Deviations and Statistical Mechanics -- I. Introduction to Large Deviations -- II. Large Deviation Property and Asymptotics of Integrals -- III. Large Deviations and the Discrete Ideal Gas -- IV. Ferromagnetic Models on ? -- V. Magnetic Models on ?D and on the Circle -- II: Convexity and Proofs of Large Deviation Theorems -- VI. Convex Functions and the Legendre-Fenchel Transform -- VII. Large Deviations for Random Vectors -- VIII. Level-2 Large Deviations for I.I.D. Random Vectors -- IX. Level-3 Large Deviations for I.I.D. Random Vectors -- Appendices -- Appendix A: Probability -- A.1. Introduction -- A.2. Measurability -- A.3. Product Spaces -- A.4. Probability Measures and Expectation -- A.S. Convergence of Random Vectors -- A.6. Conditional Expectation, Conditional Probability, and Regular Conditional Distribution -- A.7. The Kolmogorov Existence Theorem -- A.8. Weak Convergence of Probability Measures on a Metric Space -- Appendix B: Proofs of Two Theorems in Section II.7 -- B.1. Proof of Theorem II.7.1 -- B.2. Proof of Theorem II.7.2 -- Appendix C: Equivalent Notions of Infinite-Volume Measures for Spin Systems -- C.1. Introduction -- C.2. Two-Body Interactions and Infinite-Volume Gibbs States -- C.3. Many-Body Interactions and Infinite-Volume Gibbs States -- C.4. DLR States -- C.5. The Gibbs Variational Formula and Principle -- C.6. Solution of the Gibbs Variational Formula for Finite-Range Interactions on ? -- Appendix D: Existence of the Specific Gibbs Free Energy -- D.1. Existence Along Hypercubes -- D.2. An Extension -- List of Frequently Used Symbols -- References -- Author Index.
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