# zbMATH — the first resource for mathematics

The statistical mechanics of lattice gases. Vol. I. (English) Zbl 0804.60093
Princeton Series in Physics. Princeton, NJ: Princeton University Press. xi, 522 p. (1993).
This long-awaited book is, in effect, a fundamental introduction into the subject known in mathematical physics as lattice models (the next, second volume, is reserved for applications of what is accumulated in the Volume 1 as basic notions). This bifurcation of original project into two-volume plan has defined character of the Volume 1 as an “encyclopedy of basic notions and concepts”.
Chapter 1 introduces the space of the “Models to be discussed”, as well as the list of “Models not to be discussed”, plus fundamental tools of the mathematical formalism unavoidable for the further exposition: Convexity inequalities, Linear functional on infinite-dimensional spaces, Legendre transforms, and States on $$C^*$$-algebras. Chapter 2 begins with the introduction of the pressure formalism, including the pressure for Coulomb interactions. Then the transfer matrices method is explained and illustrated by the famous solution of the two-dimensional Ising model. Core of this chapter is the family of limiting theorems including the quasiclassical Lieb method, 1/D expansion, mean-field and Potts limits.
Next two chapters (Chapters 3 and 4) are central for explanation of the thermodynamics formalism in the classical and the quantum case. Chapter 3 introduces and studies the notion of states: The classical case. Well- known basis for that is the Dobrushin-Lanford-Ruelle equation for limiting Gibbs measures. Then the author passes to analysis of translation invariant equilibrium states, entropy and the Gibbs variational principle. Of course, it is overture to fundamental question of uniqueness/nonuniqueness of the equilibrium state, i.e., to the problem of phase transitions. The rest of this section, starting on the subsection “Pure states and pure phases”, is dedicated to this problem. In Chapter 4 basic notions of quantum equilibrium states are introduced. The author examines two fundamental types of them: The equilibrium states defined by the Kubo-Martin-Schwinger conditions and the equilibrium states defined by the Gibbs conditions. The rest of this chapter is concentrated on Bogolyubov-type inequalities and on the absence of continuous symmetry breaking in two dimensions.
The final Chapter 5 is devoted to the study of high-temperature and low density domain where different types of uniqueness theorems give exhaustive information about the behaviour of the Gibbs random field.
As it is mentioned in the Introduction to Volume 1 “many of the ‘sexiest’ topics – infrared bonds, the Fröhlich-Spencer theory, the Lee-Yang theory, and correlation inequalities – have been postponed until Volume 2”. But vis à vis de tout, Volume 1 is a comprehensive guide on the cross-road between statistical physics, probability theory and functional analysis.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics 60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory 82B03 Foundations of equilibrium statistical mechanics