Entropy, Large Deviations, and Statistical Mechanics

  • Richard S. Ellis

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 271)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Large Deviations and Statistical Mechanics

    1. Front Matter
      Pages 1-1
    2. Richard S. Ellis
      Pages 3-29
    3. Richard S. Ellis
      Pages 59-87
    4. Richard S. Ellis
      Pages 88-137
    5. Richard S. Ellis
      Pages 138-207
  3. Convexity and Proofs of Large Deviation Theorems

    1. Front Matter
      Pages 209-209
    2. Richard S. Ellis
      Pages 229-249
  4. Back Matter
    Pages 293-365

About this book

Introduction

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.

Keywords

Large Maxwell-Boltzmann distribution Mechanics entropy statistical mechanics system thermodynamics

Authors and affiliations

  • Richard S. Ellis
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4613-8533-2
  • Copyright Information Springer-Verlag New York 1985
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-8535-6
  • Online ISBN 978-1-4613-8533-2
  • Series Print ISSN 0072-7830
  • About this book
Industry Sectors
Oil, Gas & Geosciences