Introduction to the statistical mechanics of lattice systems (MAT778, 6 ECTS)

Lecturer: Andreas Deuchert
TA exercise group: Marco Caporaletti

Time lecture: Wednesday from 3:00 - 4:45 pm
Time exercise group: Tuesday from 3:00 - 4:45 pm

General description: Statistical mechanics was originally introduced to provide a microscopic justification of equilibrium thermodynamics, the physical theory of heat. In the last 70 years, it also developed into a well-established branch of mathematics and its ideas and methods have had an important impact on several other fields of mathematics, such as probability, analysis, geometry…
The goal of the course is to give an introduction to statistical mechanics from a mathematical point of view. Topics to be covered by the course are:
-) Ising model. The Ising model is one of the most important models in statistical mechanics. Introduced to describe the ferromagnetic phase transition, it is an ideal testing ground for new mathematical techniques because of its simplicity. We will use it to discuss the concepts of thermodynamic functions, thermodynamic limit (infinite volume limit), infinite volume states, and phase transition.
-) Cluster expansions. Cluster expansions are a powerful tool in the study of statistical mechanics that allow for the rigorous implementation of perturbative arguments. We will introduce a general framework for cluster expansions and afterward provide applications to the Ising model.
-) Depending on the background of the audience, the third part of the lecture will either be focusing on the construction of infinite volume Gibbs measures (approach by Dobrushin, Lanford, Ruelle (DLR)) or on Pirogov-Sinai theory. The former aims at constructing a probability measure (with the example of the Ising model in mind) that yields a more detailed description of states in the thermodynamic limit, and therefore of infinite systems. The latter is a general framework to establish the possible macroscopic behaviors of a class of statistical mechanics models that share some key features with the Ising model.

Target group: Bachelor and Master students in mathematics and physics at UZH and ETH Zurich.

Prior Knowledge: Analysis, linear algebra. An introduction to probability theory is helpful but not required.

Learning Outcome: Knowledge of mathematical techniques suitable for the study of classical lattice models describing phase transitions.

Assessment: Exam.

Notes: Due to an overlap of topics, it is not possible to validate this course if the course ``Ising Model’’ taught by Prof. Vincent Tassion in Autumn 2021 (see course webpage) has already been validated.

Course Material: The course follows Chapters 3 (Ising model), 5 (Cluster expansion), 6 (Infinite volume Gibbs measures), and 7 (Pirogov-Sinai theory) in the book ``Statistical mechanics of lattice systems’’ by Sascha Friedli and Yvan Velenik, Cambridge University Press, Cambridge, 2018, that is available online (link will be provided). Handwritten lecture notes will also be available.

Lecture notes:

  1. Chapter 1: Ising Model
  2. Chapter 2: Mean-Field Theory