##
**Exactly solved models in statistical mechanics.**
*(English)*
Zbl 0538.60093

London - New York etc.: Academic Press. A Subsidiary of Harcourt Brace Jovanovich, Publishers. XII, 486 p. $ 81.00 (1982).

In this book the author gives a very interesting account of two- dimensional lattice models in statistical mechanics. In opposition to those physicists and chemists who reject lattice models as being unrealistic, the author, starting from the idea that any model, which permits concrete results, is useful and taking into account that, in particular, the three-dimensional Ising model is a realistic one, describes in his book in a very elegant manner some models of this type.

The author does not insist, in all chapters of the book, on the calculus details which, in many situations, are very long and technical and which may be found, by the interested reader, in the original papers cited in the references (at the end of the book). On the other hand, there are some situations, as, for example, the functions \(k(\cdot)\) and \(g(\cdot)\) (which appear in the section 8.13) for which the calculus is given in detail, due to the intrinsic importance of these objects as well as due to the fact that they give a methodology more general then those directly implied by the above mentioned section (they provide a clear example of how elliptic functions come into).

The author starts with a chapter in which he introduces the fundamental notions of statistical mechanics that are used in the remainder of the book. Further, the following six chapters describe the models and give methods of resolution for: the one-dimensional Ising-model, the mean field model, the spherical model, and the square-lattice Ising model. In the 8th chapter are delivered the Ice-type models and in the 9th chapter an alternative way of solving the Ice-type models is presented.

In chapters 10 and 11 are given two models of type eight-vertex and in the following three chapters Potts and Ashkin-Teller models, corner transfer matrices and the hard hexagon and other related models. In the last chapter are treated very concisely the fundamental properties of elliptic functions, which have been employed everywhere in the contents of the book.

The present book is a very good guide through the lattice-models in statistical mechanics. It is useful to students and researchers in physics, chemistry and mechanics who wish to see solutions of a large class of theoretical and practical problems which appear in statistical mechanics.

The author does not insist, in all chapters of the book, on the calculus details which, in many situations, are very long and technical and which may be found, by the interested reader, in the original papers cited in the references (at the end of the book). On the other hand, there are some situations, as, for example, the functions \(k(\cdot)\) and \(g(\cdot)\) (which appear in the section 8.13) for which the calculus is given in detail, due to the intrinsic importance of these objects as well as due to the fact that they give a methodology more general then those directly implied by the above mentioned section (they provide a clear example of how elliptic functions come into).

The author starts with a chapter in which he introduces the fundamental notions of statistical mechanics that are used in the remainder of the book. Further, the following six chapters describe the models and give methods of resolution for: the one-dimensional Ising-model, the mean field model, the spherical model, and the square-lattice Ising model. In the 8th chapter are delivered the Ice-type models and in the 9th chapter an alternative way of solving the Ice-type models is presented.

In chapters 10 and 11 are given two models of type eight-vertex and in the following three chapters Potts and Ashkin-Teller models, corner transfer matrices and the hard hexagon and other related models. In the last chapter are treated very concisely the fundamental properties of elliptic functions, which have been employed everywhere in the contents of the book.

The present book is a very good guide through the lattice-models in statistical mechanics. It is useful to students and researchers in physics, chemistry and mechanics who wish to see solutions of a large class of theoretical and practical problems which appear in statistical mechanics.

Reviewer: V.Tigoiu

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

82B05 | Classical equilibrium statistical mechanics (general) |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

### Keywords:

lattice models in statistical mechanics; three-dimensional Ising model; elliptic functions; Ice-type models; Potts and Ashkin-Teller models### Digital Library of Mathematical Functions:

§17.17 Physical Applications ‣ Applications ‣ Chapter 17 𝑞-Hypergeometric and Related Functions§20.12(ii) Uniformization and Embedding of Complex Tori ‣ §20.12 Mathematical Applications ‣ Applications ‣ Chapter 20 Theta Functions

§22.18(iii) Uniformization and Other Parametrizations ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions

2nd item ‣ §23.21(iv) Modular Functions ‣ §23.21 Physical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§26.20 Physical Applications ‣ Applications ‣ Chapter 26 Combinatorial Analysis