Gibbs measures and phase transitions
type: Item
instance of: scholarly article
title: Gibbs measures and phase transitions
zbMATH Open document ID: 0657.60122
DOI: 10.1515/9783110850147
author: Hans-Otto Georgii
published in: De Gruyter Studies in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: This book is much more than an introduction to the subject of its title. It covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics and as an up to date reference in its chosen topics it is a work of outstanding scholarship. It is in fact one of the author's stated aims that this comprehensive monograph should serve both as an introductory text and as a reference for the expert. In its latter function it informs the reader about the state of the art in several directions. It is introductory in the sense that it does not assume any prior knowledge of statistical mechanics and is accessible to a general readership of mathematicians with a basic knowledge of measure theory and probability. As such it should contribute considerably to the further growth of the already lively interest in statistical mechanics on the part of probabilists and other mathematicians. Before reviewing the contents it may be worth explaining what the book is and what it does not purport to be or do. It is a mathematician's rather than a physicist's book. It treats the rigorous mathematical theory of lattice models of classical equilibrium statistical mechanics, i.e. lattice-indexed systems of interacting random spins in statistical equilibrium. It does not deal with non-equilibrium systems or systems evolving in time, particle systems in a continuum, quantum systems or systems with random interactions. Ferromagnetic correlation inequalities feature essentially only in the bibliographical notes, as does the Pirogov-Sinai theory of phase diagrams. However, the topics covered are dealt with systematically, with the development progressing from basics to quite advanced and fascinating material in several subareas. The book is divided into four parts. Part I (Chapters 1 to 9) deals with the basic theory. The first chapter presents the fundamental concept, going back to Dobrushin, Lanford and Ruelle, of a specification, i.e. of a consistent syst…
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 82-02
zbMATH DE Number: 193244
zbMATH Keywords: mathematical theory of phase transition
zbMATH Keywords: statistical mechanics
zbMATH Keywords: lattice models of classical equilibrium statistical mechanics
zbMATH Keywords: Gibbs measure
zbMATH Keywords: symmetry breaking
zbMATH Keywords: extreme point decompositions
zbMATH Keywords: Gaussian fields as Gibbs measures
zbMATH Keywords: Entropy
zbMATH Keywords: breaking of rotational symmetries
zbMATH Keywords: pair potential
zbMATH Keywords: quadratic decay
reviewed by: F. Papangelou
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110850147
OpenAlex ID: W4205240937
description: scientific article; zbMATH DE number 193244