About the Book
Most interesting and difficult problems in equilibrium statistical mechanics concern models which exhibit phase transitions. For graduate students and more experienced researchers this book provides an invaluable reference source of approximate and exact solutions for a comprehensive range of such models.
Part I contains background material on classical thermodynamics and statistical mechanics, together with a classification and survey of lattice models. The geometry of phase transitions is described and scaling theory is used to introduce critical exponents and scaling laws. An introduction is given to finite-size scaling, conformal invariance and Schramm--Loewner evolution.
Part II contains accounts of classical mean-field methods. The parallels between Landau expansions and catastrophe theory are discussed and Ginzburg--Landau theory is introduced. The extension of mean-field theory to higher-orders is explored using the Kikuchi--Hijmans--De Boer hierarchy of approximations.
In Part III the use of algebraic, transformation and decoration methods to obtain exact system information is considered. This is followed by an account of the use of transfer matrices for the location of incipient phase transitions in one-dimensionally infinite models and for exact solutions for two-dimensionally infinite systems. The latter is applied to a general analysis of eight-vertex models yielding as special cases the two-dimensional Ising model and the six-vertex model. The treatment of exact results ends with a discussion of dimer models.
In Part IV series methods and real-space renormalization group transformations are discussed. The use of the De Neef--Enting finite-lattice method is described in detail and applied to the derivation of series for a number of model systems, in particular for the Potts model. The use of Pad\'e, differential and algebraic approximants to locate and analyze second- and first-order transitions is described. The realization of the ideasof scaling theory by the renormalization group is presented together with treatments of various approximation schemes including phenomenological renormalization.
Part V of the book contains a collection of mathematical appendices intended to minimise the need to refer to other mathematical sources.
Table of Contents:
Part I Thermodynamics, Statistical Mechanical Models and Phase Transitions.- Introduction.- Thermodynamics.- Statistical Mechanics.- A Survey of Models.- Phase Transitions and Scaling Theory.- Part II Classical Approximation Methods.- Phenomenological Theory and Landau Expansions.- Classical Methods.- The Van der Waals Equation.- Landau Expansions with One Order Parameter.- Landau Expansions with Two Order Parameter.- Landau Theory for a Tricritical Point.- Landau_Ginzburg Theory.- Mean-Field Theory.- Cluster-Variation Methods.- Part III Exact Results.- Introduction.- Algebraic Methods.- Transformation Methods.- Edge-Decorated Ising Models.- 11 Transfer Matrices: Incipient Phase Transitions.- Transfer Matrices: Exactly Solved Models.- Dimer Models.- Part IV Series and Renormalization Group Methods.- Introduction.- Series Expansions.- Real-Space Renormalization Group Theory.- A Appendices.- References and Author Index.
Review :
Review MathSciNet MR1683784 (2001g:82001) Lavis, David A.(4-LNDKC); Bell, George M. Statistical mechanics of lattice systems. Vol. 1. (English summary) Closed-form and exact solutions. Second edition. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1999. xii+368 pp. ISBN: 3-540-64437-7 82-01 (82B05 82B20 82B23 82B26 82B27) (2001g:82001) Lavis, David A.(4-LNDKC); Bell, George M. Statistical mechanics of lattice systems. Vol. 1. (English summary) Closed-form and exact solutions. Second edition. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1999. xii+368 pp. ISBN: 3-540-64437-7 82-01 (82B05 82B20 82B23 82B26 82B27) PDF Doc Del Clipboard Journal Article Make Link This is the first of two volumes devoted to the pedagogical presentation of important methods for studying lattice systems in statistical mechanics. The volume focuses on mean-field and exact-solution methods for classical finite-spin systems. It starts with a useful summary of the motivating thermodynamic notions, and its material can be roughly classified in three categories: (1) one-dimensional models (Ising, lattice gas and relatives); (2) mean-field approximations and their many-site generalizations (cluster-variation methods); and (3) exact solutions for spin models in larger dimensions (Ising, models in decorated lattices, six-vertex models) based either on duality transformations or on transfer-matrix techniques. The book takes a physicist's viewpoint: experimental data, rather than mathematical control, is used to gauge the quality of approximation schemes. Finer mathematical points, for instance those associated with the infinite-volume limit, are only the subject of brief comments. The book manages to put together a wealth of information in a simple, accessible manner. While leaving aside some issues heavily studied in recent decades (e.g. correlation inequalities, systems with continuous spins, hierarchical models), the authors present a very rich variety of material, generous in applications (model for DNA denaturation, water-like models, mixture models, amphipathic monolayers, $\dots$), which allows them to introduce a broad taxonomy of critical and phase-transition phenomena (azeotropy, multicritical points, ferro-, para-, antiferro-, ferri-, meta-magnetism, etc.). The book is of immediate interest to practitioners and teachers in physical---as opposed to mathematical---statistical mechanics. It offers a self-contained survey both of the main notions of the field and of approximate methods that are usually the first (and often the only) resource to study complicated systems. It can also be a useful reference for more mathematically inclined people wishing to understand concepts and arguments of widespread use in physics and chemistry. ----------- Review from MathSciNet MR1683783 (2001g:82002) Lavis, David A.(4-LNDKC); Bell, George M. Statistical mechanics of lattice systems. Vol. 2. (English summary) Exact, series and renormalization group methods. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1999. xii+429 pp. ISBN: 3-540-64436-9 82-01 (82B05 82B20 82B23 82B26 82B27 82B28) (2001g:82002) Lavis, David A.(4-LNDKC); Bell, George M. Statistical mechanics of lattice systems. Vol. 2. (English summary) Exact, series and renormalization group methods. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1999. xii+429 pp. ISBN: 3-540-64436-9 82-01 (82B05 82B20 82B23 82B26 82B27 82B28) PDF Doc Del Clipboard Journal Article Make Link This is the second of two volumes devoted to the pedagogical survey of methods for studying lattice systems in statistical mechanics. As in the first volume [Statistical mechanics of lattice systems. Vol. 1, Second edition, Springer, Berlin, 1999; MR1683784 (2001g:82001); see the preceding review], the emphasis in most chapters is on the algebraic and algorithmic aspects of the methods, rather than on mathematical control (a conspicuous exception is Chapter 4, as I comment below). Existence and convergence issues are left aside or mentioned only briefly. All systems studied are classical and with finite spin, except for the application of cluster-expansion methods to the classical and quantum Heisenberg models. After a quick review of thermodynamical notions the book develops the following topics: (1) A rather complete presentation of scaling theory, at the heuristic level, including finite-size scaling and conformal invariance. (2) Landau and Landau-Ginzburg theories of critical behavior. (3) Rigorous methods, e.g., Peierls' contour argument, Lee-Yang theorems, transfer matrices and their relation with Perron-Frobenius theory, and a more recent method due to Wood and collaborators based also on the study of the loci of the complex zeroes of the partition function. In this chapter, unlike the rest of both volumes, the authors develop a rather rigorous mathematical theory, including theorems and proofs. (4) Exact solution of the 8-vertex model. (5) Real-space renormalization. This is an honest, informative and concise presentation of techniques based on the block-renormalization idea. In less than 50 pages, the general approach is clearly stated, most practical approximate implementations are reviewed and illustrated, and the limitations and potential problems of the usual approaches are spelled out (in particular Griffiths and Pearce's peculiarities [R. B. Griffiths and P. A. Pearce, J. Statist. Phys. 20 (1979), no. 5, 499--545; MR0533527 (81m:82005)], systematically studied later [A. C. D. van Enter, R. Fernandez and A. D. Sokal, J. Statist. Phys. 72 (1993), no. 5-6, 879--1167; MR1241537 (94m:82012)]). (6) Series expansions---low-temperature, high-temperature and more general cluster expansions---with applications to Ising, Potts and classical and quantum Heisenberg models. (7) Solutions for dimer systems, with applications to the exact solution of the two-dimensional Ising model (already studied in Volume 1) and to a model of amphipathic monolayers (different from the one presented in Volume 1). As the list shows, the authors offer in a single, largely self-contained volume a true catalogue of very important techniques. Together, both volumes cover almost all the main methods used in practice by theoretical physicists and chemists to study classical lattice models. They are useful on several grounds: for teaching purposes, as a learning resource for applied researchers, as a bibliographical source (especially for work published up to the mid-1980s), and as inspiration and reference for people studying mathematical and foundational issues.
