Introductory description

This will be an introduction to statistical mechanics, that deals with large systems of interacting particles. There will be a through review of the Ising model and its properties, including a mathematical description of its magnetic phase transition. The second part of the module will be spent on other models and other questions, such as systems with continuous symmetry; gaussian model; infinite-volumes Gibbs states; quantum spin systems.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Statistical mechanics describes physical systems with a huge number of particles. In physics, the goal is to describe macroscopic phenomena in terms of microscopic models and to give a meaning to notions such as temperature or entropy. Mathematically, it can be viewed as the study of random variables with spatial dependence. Models of statistical mechanics form the background for recent advances in probability theory and stochastic analysis, such as SLE and the theory of regularity structures. So, they form an important background for understanding these topics of modern mathematics.

The module will give a thorough mathematical introduction to the Ising model and to the gaussian free field on regular graphs, and to the theory of infinite volume Gibbs measures.

Learning outcomes

By the end of the module, students should be able to:

• Apply basic ideas of phase transitions and critical behaviour to lattice systems of statistical mechanics
• Understand the theory of infinite volume Gibbs measures
• Understand how large complex systems at equilibrium can be described from microscopic rules
• Have understood basic ideas of phase transitions and critical behaviour in the case of the Ising model and the gaussian free field; they will have mastered the theory of infinite volume Gibbs measures.

Indicative reading list

We will mainly follow Chapter 3 of the new introductory textbook:

Sacha Friedli and Yvan Velenik, Equilibrium Statistical Mechanics of Classical Lattice Systems: a Concrete Introduction. Available at: http://www.unige.ch/mth/folks/velenik/smbook/index.html

Interested students can also look in

David Ruelle, Statistical Mechanics: Rigorous Results, World Scientific, 1999.

James Sethna: Statistical Mechanics: Entopy, Order Parameters and Complexity Oxford Master Series in Physics, 2006.

Subject specific skills

We will review methods that allow to derive the macroscopic properties of large systems of interacting particles. We will do this in the specific case of models of statistical mechanics at equilibrium, but this is conceptually more general.

Transferable skills

Understand and deal with large systems of interacting agents.