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Equilibrium Statistical Mechanics of Lattice Models

  • David A. Lavis

Part of the Theoretical and Mathematical Physics book series (TMP)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Thermodynamics, Statistical Mechanical Models and Phase Transitions

    1. Front Matter
      Pages 1-3
    2. David A. Lavis
      Pages 5-11
    3. David A. Lavis
      Pages 13-27
    4. David A. Lavis
      Pages 29-87
    5. David A. Lavis
      Pages 89-166
  3. Classical Approximation Methods

    1. Front Matter
      Pages 167-168
    2. David A. Lavis
      Pages 169-204
    3. David A. Lavis
      Pages 205-228
    4. David A. Lavis
      Pages 229-252
  4. Exact Results

    1. Front Matter
      Pages 253-257
    2. David A. Lavis
      Pages 259-282
    3. David A. Lavis
      Pages 283-310
    4. David A. Lavis
      Pages 311-343
    5. David A. Lavis
      Pages 345-380
    6. David A. Lavis
      Pages 381-493
    7. David A. Lavis
      Pages 495-516
  5. Series and Renormalization Group Methods

    1. Front Matter
      Pages 517-520
    2. David A. Lavis
      Pages 521-566
    3. David A. Lavis
      Pages 567-616
  6. Mathematical Appendices

    1. Front Matter
      Pages 617-617
    2. David A. Lavis
      Pages 619-657
    3. David A. Lavis
      Pages 659-702
    4. David A. Lavis
      Pages 703-756
  7. Back Matter
    Pages 757-793

About this book

Introduction

Most interesting and difficult problems in equilibrium statistical mechanics concern models which exhibit phase transitions. For graduate students and more experienced researchers this book provides an invaluable reference source of approximate and exact solutions for a comprehensive range of such models. Part I contains background material on classical thermodynamics and statistical mechanics, together with a classification and survey of lattice models. The geometry of phase transitions is described and scaling theory is used to introduce critical exponents and scaling laws. An introduction is given to finite-size scaling, conformal invariance and Schramm—Loewner evolution. Part II contains accounts of classical mean-field methods. The parallels between Landau expansions and catastrophe theory are discussed and Ginzburg—Landau theory is introduced. The extension of mean-field theory to higher-orders is explored using the Kikuchi—Hijmans—De Boer hierarchy of approximations. In Part III the use of algebraic, transformation and decoration methods to obtain exact system information is considered. This is followed by an account of the use of transfer matrices for the location of incipient phase transitions in one-dimensionally infinite models and for exact solutions for two-dimensionally infinite systems. The latter is applied to a general analysis of eight-vertex models yielding as special cases the two-dimensional Ising model and the six-vertex model. The treatment of exact results ends with a discussion of dimer models. In Part IV series methods and real-space renormalization group transformations are discussed. The use of the De Neef—Enting finite-lattice method is described in detail and applied to the derivation of series for a number of model systems, in particular for the Potts model. The use of Padé, differential and algebraic approximants to locate and analyze second- and first-order transitions is described. The realization of the ideas of scaling theory by the renormalization group is presented together with treatments of various approximation schemes including phenomenological renormalization. Part V of the book contains a collection of mathematical appendices intended to minimise the need to refer to other mathematical sources.

Keywords

Catastrophe theory Classical approximation method Classical approximation methods Dimer Dimer model Dimer models Equilibrium Equilibrium statistical mechanics Equilibrium statistical mechanics of lattice sytems Finite lattice method Finite lattice methods High temperature series Landau expansion and catastrophe theory Lattice model Lattice models Lattice system Lattice systems Low temperature series Mean field approximations Mean-field approximation Neef-Enting method Phenomenological scaling theory Potts model Renormalization group transformation Renormalization group transformations Scaling theory Series method Series methods Statistical mechanics Statistical mechanics of lattice models Theory of phase transition Theory of phase transitions

Authors and affiliations

  • David A. Lavis
    • 1
  1. 1.King's College London, Department of MathematicsLondonUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-9430-5
  • Copyright Information Springer Science+Business Media Dordrecht 2015
  • Publisher Name Springer, Dordrecht
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-94-017-9429-9
  • Online ISBN 978-94-017-9430-5
  • Series Print ISSN 1864-5879
  • Series Online ISSN 1864-5887
  • Buy this book on publisher's site