Quantitative Theory of Critical Phenomena

Quantitative Theory of Critical Phenomena

1st Edition - July 28, 1990

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  • Author: George A. Jr. Baker
  • eBook ISBN: 9780323153157

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Quantitative Theory of Critical Phenomena details in a self-contained manner the most popular and extensively practiced methods for the quantitative study of critical phenomena. The text is divided into three parts. Part I deals with the general theory of critical phenomena — its thermodynamic aspects, statistical mechanical framework, classical model, and inequalities. Part II tackles the combinatorial theory of series generation. Part III covers the quantitative analysis of series expansions, which includes topics such as the complex variable theory, the algebraic aspects and numerical evaluation of Padé approximants, and special continuation methods. The book is recommended for mathematicians and physicists who would like to know more about critical phenomena, its theories, and the methods for its quantitative study.

Table of Contents

  • Illustrations


    Part I. General Theory of Critical Phenomena

    Chapter 1. Thermodynamic Aspects

    a. Phase Diagrams

    b. Thermodynamic Potentials

    c. Behavior at Critical Points

    d. Stability and Convexity

    e. Thermodynamic Inequalities

    Chapter 2. Statistical Mechanical Framework

    a. Regular Assemblies

    b. One Dimensional Assemblies

    d. Two-Point Correlation Functions

    Chapter 3. Classical Models

    a. Mean Field Theory

    b. Ornstein-Zernike Theory of the Two Point Correlation Function

    c. The Gaussian Model

    d. The Spherical Model

    e. Landau-Ginsburg Theory

    Chapter 4. Inequalities

    a. Correlation Function Inequalities

    b. The Yang-Lee Theorem

    c. Application of the Inequalities to the Critical Region

    Chapter 5. Two dimensional Ising Model

    Chapter 6. General Approaches

    a. Series Expansions

    b. Homogeneity Theory

    c. Scaling Theory

    d. Renormalization Group Theory

    e. The E-Expansion

    f. Hierarchical Model

    g. Implementation of Renormalization Group Ideas by Means of Field Theory Methods

    h. Universality

    i. Conformal Invariance

    Part II. Combinatorial Theory of Series Generation

    Chapter 7. Elementary combinatorics

    a. Preliminaries

    b. Basic Cluster Theorem

    Chapter 8. Finite Cluster Method

    Chapter 9. Star Graph Expansion

    Chapter 10. Linked Cluster Expansion

    Chapter 11. Expansion about the Gaussian (or Spherical) Model

    a. Coupling Constant Expansion

    b. 1/n -expansion

    Chapter 12. Numerical Computation of Combinatorics

    Part III. Quantitative Analysis of Series Expansions

    Chapter 13. Complex Variable Theory

    a. Preliminaries

    b. Exact Analytic Continuation

    c. Riemann's Monodromy Theorem

    d. Carleman's Criterion

    Chapter 14. Padé Approximants, Algebraic Aspects

    a. Approximate Analytic Continuation

    b. Padé Table

    c. Invariance Theorems

    d. Bigradients

    e. Continued Fractions

    f. Some Exactly Known Padé Approximants

    Chapter 15. Numerical Evaluation of Padé Approximants

    a. Requirements

    b. Gaussian Elimination

    c. Viskovatov-Butheel Algorithm

    d. Recursion Methods

    Chapter 16. Padé Approximants to Series of Stieltjes

    a. Yang-Lee Theorem and Series of Stieltjes

    b. Characterization of Series of Stieltjes

    c. Location of Poles of the Padé Approximant

    d. Padé Bounds on the Function

    e. Convergence Theory

    Chapter 17. Padé Approximants, General Convergence Theory

    a. Counter-Examples

    b. Faithfulness Theorem

    c. Error Formula

    d. Point Wise Convergence

    e. Convergence Almost Everywhere

    Chapter 18. Interpretation of Padé Approximants

    a. Representation of Singularity Structures

    b. Apparent Error Estimation

    Chapter 19. Integral Approximant Theory

    a. Definition of Integral Approximants

    b. Invariance Properties

    c. Separation Property

    d. Convergence Properties

    e. Numerical Examples

    Chapter 20. Special Continuation Methods

    a. Ratio Methods

    b. Gammel's Emphasis Method

    c. Baker-Gammel Approximants

    d. Darboux's Theorems Methods

    e. Uniformization Transformation Methods

    Chapter 21. Series Analysis for Critical Properties

    a. Monodromic Dimensions 1, High- and Low-Temperature Series Estimates for Critical Points, Exponents, etc.

    b. Series Estimates for Higher Monodromic Dimensions

    c. Renormalization Group Series Estimates for Critical Indices



Product details

  • No. of pages: 382
  • Language: English
  • Copyright: © Academic Press 1990
  • Published: July 28, 1990
  • Imprint: Academic Press
  • eBook ISBN: 9780323153157

About the Author

George A. Jr. Baker

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