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401-8822-23L 6 Credits BSC , MSC D-PHYS , D-MATH

Introduction to the Statistical Mechanics of Lattice Systems (University of Zurich)

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Last Updated: 2026-02-05 16:22:22

Abstract

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…

Objective

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

Content

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.

Resources

Literature

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.

General Information

Language
English
Levels
BSC , MSC

Examination

Type
graded semester performance
Registration modalities, date and venue of this performance assessment are specified solely by the UZH.Because this course unit has some overlaps with previous course units 401-3822-17L (taught in the Autumn Semesters 2017, 2018 and 2021) and maybe future such course units on the Ising model, only one of those course units is eligible for credits.

Course Components

Type Title Time & Place Hours
lecture Introduction to the Statistical Mechanics of Lattice Systems (University of Zurich) No time listed 2 h weekly
exercise Introduction to the Statistical Mechanics of Lattice Systems (University of Zurich) No time listed 2 h weekly

Offered In