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 SchrammLoewner 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 NeefEnting 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.

From one author of the successful two-volume work on statistical mechanics by Lavis and Bell (Springer, 1999) Includes accounts of mean-field, exact, renormalization group and series methods for critical phenomena An accurate and scholarly book useful for consultation Includes supplementary material: sn.pub/extras

Autorentext

David A. Lavis david.lavis@kcl.ac.uk Address: Department of Mathematics King's College London The Strand London WC2R 2LS, UK David Lavis is an Emeritus Senior Lecturer in the Department of Mathematics at King's College London (KCL) and a Research Associate in the Centre for Philosophy of Natural and Social Science at the London School of Economics. During his career, which has been mainly based at KCL, he has spent periods of leave as a lecturer at Makerere University, Uganda and as a researcher at the Institut Laue-Langevin, Genoble, France and at the Universities of Waterloo and Manitoba, Canada. His main research work and his extensive publications have been devoted to the investigation of phase transitions in complex lattice systems using mean-field, renormalization group and transfer matrix methods. The well-known Bell-Lavis model originated in a paper he wrote with G. M. Bell in 1970, with whom the first two of his three books on the statistical mechanics of lattice systems were co-authored. More recently he has researched and published papers on the conceptual foundations of thermodynamics and statistical mechanics with a particular concentration on the relationship between them. Roman Frigg R.P.Frigg@lse.ac.uk Work Address: Department of Philosophy London School of Economics Houghton Street London WC2A 2AE UK England Roman Frigg is Professor of Philosophy in the Department of Philosophy, Logic and Scientific Method and a permanent visiting professor in the Munich Centre for Mathematical Philosophy of the Ludwig-Maximilians-University Munich. He is the winner of the Friedrich Wilhelm Bessel Research Award of the Alexander von Humboldt Foundation. His research interests lie in general philosophy of science and philosophy of physics. He has published papers on scientific representation, modelling, statistical mechanics, randomness, chaos, climate change, quantum mechanics, complexity, probability, scientific realism, computer simulations, reductionism, confirmation, and the relation between art and science. His research in the philosophy of physics focusses on statistical mechanics and thermodynamics. He is the author the Cambridge Element "Foundations of Statistical Mechanics" and he has published a large number of research papers on various aspects of statistical mechanics.

Inhalt

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.