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Equilibrium Statistical Mechanics of Lattice Models (Paperback, Softcover reprint of the original 1st ed. 2015)
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Equilibrium Statistical Mechanics of Lattice Models (Paperback, Softcover reprint of the original 1st ed. 2015)
Series: Theoretical and Mathematical Physics
Expected to ship within 10 - 15 working days
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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 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.
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