On this page, you can find the latest preprint version of our introductory book on statistical mechanics, published by Cambridge University Press. As there are already many such books, both for physicists and mathematicians, we believe it is important to describe what we aimed to achieve when we started with this project. Our goal was to write a book that is

• of limited scope: although we cover many important topics, we decided to limit the scope of the book. Among the most important topics that are not discussed (except, occasionally, in short notes), let us list: quantum models, non-lattice models, disordered systems, critical phenomena, stochastic dynamics.
• student-friendly: we tried to make the book as readable as possible, accessible to advanced undergraduates (say, what corresponds to master students in Europe), both in mathematics and physics. To achieve this end, we have refrained from stating and proving the most general claims possible, but discussed instead in detail particularly important (classes of) examples (among which the Ising model, the Curie-Weiss model, the Blume-Capel model, the Gaussian Free Field, the XY model, the Heisenberg model, etc). We also wanted the book to be more or less self-contained, so that we have included a (rather long) appendix introducing the required mathematical tools, and provided solutions to most of the exercises.
• based on the probabilistic approach to (rigorous) equilibrium statistical mechanics, even though we have not refrained from using other types of tools when needed. In particular, there is no discussion of explicit solutions for integrable models (apart from brief comments).

The book starts with a brief introduction to thermodynamics and statistical physics and the relations between these two theories, as well as an informal discussion of phase transitions, the role of the thermodynamic limit, etc. In chapter 2, the focus then moves to the Curie-Weiss model, as one example of application of the mean-field approach; this also allows a first rigorous discussion of an order-disorder phase transition, while only relying on elementary tools. Chapter 3, which forms a core part of the book, discusses the Ising model. The detailed analysis of its phase diagram serves as a motivation for introducing many fundamental concepts of statistical physics (pressure, magnetization, Gibbs states, etc), as well as several important techniques (consequences of the convexity of the thermodynamic potentials, correlation inequalities, low- and high-temperature expansions, etc) and classical results (Peierls argument, Lee-Yang theorem, etc). Chapter 4 is then devoted to the lattice gas, as a mean to a rigorous description of the liquid-vapor equilibrium. Cluster expansion, which is one of the main technical tools used in this area, is then introduced in the Chapter 5. It allows a detailed analysis in perturbative regimes, as we illustrate with several applications to the Ising model and to the lattice gas. This is then followed by a long chapter introducing a general approach to the description of infinite lattice spin systems: the Dobrushin-Lanford-Ruelle theory. This chapter includes important topics such as Dobrushin's uniqueness theorem, extremal decomposition, properties of extremal Gibbs measures and the variational principle. Chapter 7 presents an introduction to one of the general approaches to proving existence of first-order phase transitions in systems without symmetries: the Pirogov-Sinai theory. The latter is illustrated on the case of the Blume-Capel model. The remaining chapters cover systems of continuous spins. In Chapter 8, the Gaussian Free Field on $\mathbb{Z}^d$ is introduced, as well as the associated random walk representation. In Chapter 9, the consequences of the presence of a continuous symmetry on the behavior of low-dimensional systems are explored; among others, a proof of a strong form of the Mermin-Wagner theorem is provided. In Chapter 10, another far-reaching tool is presented: Reflection Positivity. After a discussion of its two main consequences (the chessboard estimate and the infrared bound), several applications are given (in particular, the existence of a phase transition in the low-temperature anisotropic XY model in dimension $2$ and higher, and the existence of a phase transition for low-temperature $O(N)$ models in dimensions $3$ and higher. The book concludes with several appendices, and in particular a long appendix introducing a variety of mathematical tools required in the main chapters.

We plan to keep on this page an up-to-date list of corrections to the book. So, please, do not hesitate to communicate to us any typo or error you spot. All these should be sent to us at introsmbook@gmail.com.

## News

Note [May 11, 2015]: Claudio Landim (IMPA) has recently given a course based on Chapter 3 (The Ising model). Videos of the lectures can be found here (see lectures 1 to 9); the lectures are in Portuguese.

Note [January 19, 2016]: One of us (Y.V.) has recently given a one-semester course, entitled Introduction to Statistical Mechanics, based on Chapters 3, 7, 8 and 10. Recordings of the lectures (as well as other lectures in the same program) have been uploaded here.

Note [March 22, 2016]: As you may have noted, the title of the book has been shortened, after discussion with the book's publisher. This is the final title.

Note [February 20, 2017]: As you may have noted, the numbering of the theorems, exercises, etc, has been changed, so that each category has its own separate numbering. This was done at the request of the book's publisher. We apologize for any inconvenience that this change might cause.

Note [April 25, 2017]: As you may have noted, the numbering of the theorems, exercises, etc, has been changed once more, so that all categories, except the exercises, share a common numbering. This was done at the request of the book's publisher. We apologize for any inconvenience that this change might cause. THIS WILL NOT CHANGE ANYMORE.

Note [May 12, 2017]: The book is now officially announced by Cambridge University Press.

Note [July 22, 2017]: While reviewing the galley proofs, we have found quite a few (minor) errors or imprecisions that we have now corrected. The revised (final) version of the book can be downloaded below.

Note [November 3, 2017]: The book will be published on November 23! More information can be found on the official book page at Cambridge University Press.

## Files

You can find here the final draft of the book. Its content should be nearly identical to the version published by Cambridge University Press, although the latter may have fewer grammatical and stylistic errors, thanks to their editorial work. The numbering of the chapters, sections, equations, theorems, etc, is identical, but the page numbering is not.

In this chapter, we briefly (and somewhat heuristically) introduce equilibrium thermodynamics and equilibrium statistical mechanics, explaining the links between these two theories. We also provide an informal discussion of the van der Waals–Maxwell theory of condensation and of the paramagnet/ferromagnet phase transition. Finally, we make some general remarks on the role of the thermodynamic limit and the use of simple models. The chapter concludes with a roadmap to the book and an overview of the literature.

Mean-field models play a useful role, both from the physical and mathematical point of view, as first approximation to more realistic ones. This chapter provides a detailed account of the Curie–Weiss model, which can be seen as the mean-field version of the Ising model. The advantage is that this model exhibits a phase transition between paramagnetic and ferromagnetic behaviors that can be described with elementary tools. The approach is mostly combinatorial, but alternatives are described in complements.

The Ising model is possibly the simplest "realistic" model which exhibits a non-trivial collective behavior. As such, it has played, and continues to play, a central role in statistical mechanics. This chapter uses it to introduce several central notions (e.g., the thermodynamic limit and infinite-volume Gibbs states) and precise definitions (e.g., the analytic and the probabilistic definitions of first-order phase transitions and their equivalence). The complete phase diagram of the model is then constructed, in all dimensions. The analysis makes use of convexity properties of the pressure, correlation inequalities, low- and high-temperature expansions, Peierls' argument and the Lee–Yang theorem, all of which are discussed in detail. The chapter concludes with an extensive list of complements covering a wide range of topics.

Historically, the liquid-vapor equilibrium played a central role in the first theoretical studies of phase transitions. In this chapter, the mathematical description of the lattice gas is exposed in detail, as well as its mean-field and nearest-neighbor (Ising) versions. The mean-field (Kać) limit is also studied in a simple case, providing a rigorous justification of the van der Waals–Maxwell theory of condensation. The chapter concludes with several complements discussing more advanced questions.

The cluster expansion remains the most important perturbative technique in mathematical statistical mechanics. In this chapter, it is presented in a simple fashion and several applications to the Ising model and the lattice gas are presented. This tool is also used several times later in the book and plays, in particular, a central role in the implementation of the Pirogov–Sinai theory of Chapter 7.

In this chapter, we present a probabilistic description of infinite systems of particles at equilibrium, which is known nowadays as the theory of Gibbs measures or as the DLR (Dobrushin–Lanford–Ruelle) formalism. The theory is developed from scratch, restricting attention to models with finite single-spin-space to keep the exposition as simple as possible. The Ising model serves as a guiding example throughout the chapter. Several important aspects, such as Dobrushin's Uniqueness Theorem, spontaneous symmetry breaking, the properties of extremal measures and the extremal decomposition, are exposed in detail. At the end of the chapter, the variational principle is also introduced; the latter is closely linked with the basic concepts of equilibrium thermodynamics.

The Pirogov–Sinai theory is one of the very few general approaches to the rigorous study of first-order phase transitions. It yields, under rather weak assumptions, a sharp description of such phase transitions in perturbative regimes. In this chapter, this theory is first introduced in a rather general setting and then implemented in detail on one specific three-phase model: the Blume–Capel model.

In this chapter, the lattice version of the Gaussian Free Field is analyzed. Several features related to the non-compactness of its single-spin-space are discussed, exploiting the Gaussian nature of the model. We also derive the random walk representation that characterizes its mean and covariance structure. The recurrence properties of this random walk turn out to be crucial to analyze the thermodynamic limit.

An important class of models with a continuous symmetry, including the XY and Heisenberg models, is studied in this chapter. The emphasis is on the implications of the presence of the continuous symmetry on long-range order in these models in low dimensions. In particular, a strong form of the celebrated Mermin–Wagner theorem is proved in an simple way. We then complement that with a discussion of correlations decay.

Reflection positivity is another tool that plays a central role in the rigorous study of phase transitions. We first expose it in detail, proving its two central estimates: the infrared bound and the chessboard estimate. We then apply the latter to obtain several results of importance. In particular, we prove the existence of a phase transition in the anisotropic XY model in dimensions $d\geq 2$, as well as in the (isotropic) $O(N)$ model in dimensions $d\geq 3$. Combined with the results of Chapter 9, this provides a detailed description of this type of systems in the thermodynamic limit.

This appendix regroups short notes that are sometimes referred to in the text.

In this appendix, we introduce various mathematical topics used throughout the book, which might not be part of all undergraduate curricula. For example: elementary properties of convex functions, some aspects of complex analysis, measure theory, conditional expectation, random walks, etc., are briefly introduced, not always in a self-contained manner, often without proofs, but with references to the literature.

Hints or detailed solutions for most of the exercises of the book can be found in this appendix.