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Statistical Mechanics of Lattice Systems

Sacha Friedli and Yvan Velenik
Publisher: 
Cambridge University Press
Publication Date: 
2018
Number of Pages: 
622
Format: 
Hardcover
Price: 
69.99
ISBN: 
9781107184824
Category: 
Textbook
[Reviewed by
William J. Satzer
, on
07/31/2018
]

Statistical mechanics is a branch of physics that applies probabilistic methods to describe the macroscopic behavior of systems with large numbers of particles using microscopic information about the particles. Equilibrium statistical mechanics focuses on macroscopic systems whose dynamics has reached a steady state.

Equilibrium statistical mechanics as a subject is more than a century old that arguably began with the work of Ludwig Boltzmann. The closely related study of equilibrium thermodynamics also developed during the nineteenth century. While equilibrium thermodynamics is primarily a phenomenological theory, equilibrium statistical mechanics has a distinctly mathematical favor. The subject began to get considerable attention in the 1970s from mathematicians like Ruelle and Sinai in the context of dynamical systems and ergodic theory.

The authors’ goal with this book is to develop the subject in a mathematically rigorous way and to focus on discrete models where the particle positions are restricted to a cubic lattice. This is most definitely a work of mathematical physics, and that means — according to the authors — analysis of problems that arise in physics treated with the rigor of mathematics. (Their arguments that rigor is not just nitpicking — probably directed mostly to physicists — are compelling.)

They begin by discussing basic ideas of statistical mechanics and thermodynamics. A critical element of this is the Gibbs distribution that describes the probability of observing a particular microscopic state of a system when that system is in equilibrium at a fixed temperature. Over the course of the book the authors demonstrate the importance of this distribution by studying its appearance in several important models with diverse behaviors.

The first five chapters discuss models whose state variables are discrete and take values in a finite set. The authors begin with two magnetic system models consisting of magnetic spins on a lattice (where each lattice point has a “spin up” or “spin down” orientation.) The aim is to understand the phase transitions between paramagnetic (randomly aligned) and ferromagnetic (strongly aligned) behavior. The authors first describe the Curie-Weiss model that uses a mean field approximation and demonstrates that a phase transition actually occurs using simple mathematical tools. The Curie-Weiss model is an approximation of the Ising model, treated next, that is the simplest almost realistic model showing nontrivial collective behavior arising from locally interacting units.

The last three chapters consider models with continuous variables. These include the lattice version of the Gaussian Free Field model and classes of models with continuous symmetry.

The authors have created an impressive book that shows its strengths in several ways: thoughtful organization and a well-designed presentation, real attention to the needs of the reader, and a very nice guide to the existing literature. It could be a model of how mathematical physics should be presented.

Exercises are interspersed throughout the text and closely integrated with it. The authors provide solutions for many of them in an appendix; some solutions have full details and others offer sketches of an approach or simple suggestions.

This book is probably best suited to graduate students with a decent background in analysis and probability. Knowledge of basic undergraduate physics is also desirable. Well-prepared undergraduates would find it appealing but rather challenging. The authors provide about fifty pages of mathematical appendices; these include short pieces on measure theory, complex analysis, convex functions, and probability.


Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Preface
Convention
1. Introduction
2. The Curie–Weiss model
3. The Ising model
4. Liquid-vapor equilibrium
5. Cluster expansion
6. Infinite-volume Gibbs measures
7. Pirogov–Sinai theory
8. The Gaussian free field on Zd
9. Models with continuous symmetry
10. Reflection positivity
A. Notes
B. Mathematical appendices
C. Solutions to exercises
Bibliography
Index.