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Physics - Complexity | Equilibrium Statistical Mechanics of Lattice Models

Equilibrium Statistical Mechanics of Lattice Models

Lavis, David

2015, XVII, 793 p. 101 illus.

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  • 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

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 Padé, differential and algebraic approximants to locate and analyze second- and first-order transitions is described. The realization of the ideas of 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.

Content Level » Research

Keywords » Catastrophe theory - Classical approximation method - Classical approximation methods - Dimer - Dimer model - Dimer models - Equilibrium - Equilibrium statistical mechanics - Equilibrium statistical mechanics of lattice sytems - Finite lattice method - Finite lattice methods - High temperature series - Landau expansion and catastrophe theory - Lattice model - Lattice models - Lattice system - Lattice systems - Low temperature series - Mean field approximations - Mean-field approximation - Neef-Enting method - Phenomenological scaling theory - Potts model - Renormalization group transformation - Renormalization group transformations - Scaling theory - Series method - Series methods - Statistical mechanics - Statistical mechanics of lattice models - Theory of phase transition - Theory of phase transitions

Related subjects » Complexity - Materials - Probability Theory and Stochastic Processes - Theoretical, Mathematical & Computational Physics

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.

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