Entropy, Large Deviations, and Statistical Mechanics

1. Front Matter

   Pages i-xiv

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2. Large Deviations and Statistical Mechanics

   1. Front Matter

      Pages 1-1

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   2. Introduction to Large Deviations

      • Richard S. Ellis

      Pages 3-29

   3. Large Deviation Property and Asymptotics of Integrals

      • Richard S. Ellis

      Pages 30-58

   4. Large Deviations and the Discrete Ideal Gas

      • Richard S. Ellis

      Pages 59-87

   5. Ferromagnetic Models on ℤ

      • Richard S. Ellis

      Pages 88-137

   6. Magnetic Models on ℤ^D and on the Circle

      • Richard S. Ellis

      Pages 138-207

3. Convexity and Proofs of Large Deviation Theorems

   1. Front Matter

      Pages 209-209

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   2. Convex Functions and the Legendre-Fenchel Transform

      • Richard S. Ellis

      Pages 211-228

   3. Large Deviations for Random Vectors

      • Richard S. Ellis

      Pages 229-249

   4. Level-2 Large Deviations for I.I.D. Random Vectors

      • Richard S. Ellis

      Pages 250-268

   5. Level-3 Large Deviations for I.I.D. Random Vectors

      • Richard S. Ellis

      Pages 269-291

4. Back Matter

   Pages 293-365

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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.

• Large
• Maxwell-Boltzmann distribution
• Mechanics
• entropy
• statistical mechanics
• system
• thermodynamics

• Department of Mathematics and Statistics, University of Massachusetts, Amherst, USA

  Richard S. Ellis

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