Statistical Mechanics of Lattice Systems: Volume 1: Closed-Form and Exact Solutions (Theoretical and Mathematical Physics)

Most of the interesting and difficult problems in statistical mechanics arise when the constituent particles of the system interact with each other with pair or multi particle energies. The types of behaviour which occur in systems because of these interactions are referred to as cooperative phenomena giving rise in many cases to phase transitions. This book and its companion volume (Lavis and Bell 1999, referred to in the text simply as Volume 2) are princi­ pally concerned with phase transitions in lattice systems. Due mainly to the insights gained from scaling theory and renormalization group methods, this 1 subject has developed very rapidly over the last thirty years. In our choice of topics we have tried to present a good range of fundamental theory and of applications, some of which reflect our own interests. A broad division of material can be made between exact results and ap­ proximation methods. We have found it appropriate to include some of our discussion of exact results in this volume and some in Volume 2. The other main area of discussion in this volume is mean-field theory leading to closed­ form approximations. Although this is known not to give reliable results close to a critical region, it often provides a good qualitative picture for phase dia­ grams as a whole. For complicated systems some kind of mean-field method is often the only tractable method available.

This two-volume work provides a comprehensive study of the statistical mechanics of lattice models. It introduces the reader to the main areas in statistical mechanics and the theory of phase transitions. The development is built on a firm mathematical and physical basis. Volume 1 contains an account of mean-field and cluster variation methods successfully used in many applications in solid-state physics and theoretical chemistry as well as an account of exact results for the Ising and six-vertex models and those derivable by transformation methods. Volume 2 includes extensive treatments of scaling theory, algebraic and real-space renormalization methods and the eight-vertex model. It also includes an account of series methods and a treatment of dimer assemblies.