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Statistical mechanics of lattice models. Volume 1: Closed form and exact theories of cooperative phenomena. (English) Zbl 0712.60105

Ellis Horwood Series in Mathematics and its Applications. Chichester: Ellis Horwood Limited; New York etc.: Halsted Press, a division of John Wiley & Sons. 368 p. £55.00 (1989).
Cooperative phenomena are due to the interactions of a very large number of particles. These interactions often lead to singularities in some thermodynamic variables. In other words, such systems often yield critical phenomena or phase transitions. The statistical mechanics of lattice models treated in this book is a rich field and has been studied in a great deal by physicists and mathematicians. This volume consists of ten chapters.
The book begins with some necessary thermodynamics theory and the required statistical mechanics formalism. Then the authors introduce the mean-field method, cluster variation methods and transformation methods successively. As an essential part of the book, the applications of these methods, such as phase diagram, critical point and multi-critical point, critical scaling and exponents, exponent renormalization, exact solutions and so on, are presented chapter by chapter. Of course, the central topic is the well-known Ising model. However, various modified versions of this model are studied in detail and a large number of interesting examples are discussed at the end of each chapter.
The book is valuable not only because of the importance of the material but also of its well-organization. Much work was previously available only in research journals.
Reviewer: Chen Mu-fa

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
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
82B26 Phase transitions (general) in equilibrium statistical mechanics
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