Theory of Phase Transitions: Rigorous Results is inspired by lectures on mathematical problems of statistical physics presented in the Mathematical Institute of the Hungarian Academy of Sciences, Budapest. The aim of the book is to expound a series of rigorous results about the theory of phase transitions. The book consists of four chapters, wherein the first chapter discusses the Hamiltonian, its symmetry group, and the limit Gibbs distributions corresponding to a given Hamiltonian. The second chapter studies the phase diagrams of lattice models that are considered at low temperatures. The notions of a ground state of a Hamiltonian and the stability of the set of the ground states of a Hamiltonian are also introduced. Chapter 3 presents the basic theorems about lattice models with continuous symmetry, and Chapter 4 focuses on the second-order phase transitions and on the theory of scaling probability distributions, connected to these phase transitions. Specialists in statistical physics and other related fields will greatly benefit from this publication.
Inhalt
Preface
Chapter I. Limit Gibbs Distributions
1. Hamiltonians
2. Examples of Hamiltonians
3. The Definition of Limit Gibbs Distributions
4. Examples
5. Existence of Limit Gibbs Distributions
6. Limit Gibbs Distributions for Continuous Fields and for Point Fields
Historical Notes and References to Chapter I
Chapter II. Phase Diagrams for Classical Lattice Systems. Peierls's Method of Contours
1. Introduction
2. Ground States
3. Ground States of the Perturbed Hamiltonian
4. Phase Transitions in the Two-Dimensional Ising Ferromagnet
5. The Main Theorem and its Consequences
6. Contours
7. Contour Models
8. Correlation Functions of Infinite Contour Models
9. Surface Tension in Contour Models
10. Proof of the Main Theorem
11. Some Further Remarks
Historical Notes and References to Chapter II
Chapter III. Lattice Systems with Continuous Symmetry
1. Introduction
2. Absence of Breakdown of Continuous Symmetry in Two-Dimensional Models
3. The Fröhlich-Simon-Spencer Theorem on the Existence of Spontaneous Magnetization in the d-Dimensional Classical Heisenberg Model, d=3
Historical Notes and References to Chapter III
Chapter IV. Phase Transitions of the Second Kind and the Renormalization Group Method
1. Introduction
2. Dyson's Hierarchical Models
3. Gaussian Solutions
4. The Domain c
5. Scaling Probability Distributions
6. Gaussian Scaling Distributions
7. The Space of Hamiltonians and the Definition of the Linearized Renormalization Group
8. The Linearized Renormalization Group and its Spectrum in the Case of Gaussian Scaling Distributions
9. Bifurcation Points, Non-Gaussian Scaling Distributions, e-Expansions
Historical Notes and References to Chapter IV
Epilogue
References
Index
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