Introduction to Statistical Mechanics

Time and place: Tuesdays 17-19 in B3.02, Fridays 11-12 in MS.05.

Support classes: Thursdays 12-13 in B1.01 by Costanza Benassi.

Assessment: Three-hour examination (100%).

Description: 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 a few other topics.

Material: The main reference is the excellent new book Statistical Mechanics of Lattice Systems by Sacha Friedli and Yvan Velenik (Cambridge University Press, 2017). We will cover Chapter 3 about the Ising model in the first 5-6 weeks. Then the class will decide whether to learn about infinite-volume Gibbs measures (Chapter 6), the Gaussian free field (Chapter 8), models with continuous symmetry (Chapters 9/10), or quantum spin systems (in which case notes will be provided).

Recommended exercises: In the book of Friedli & Velenik, 3.1; 3.2; 3.5; 3.7; 3.8; 3.9; 3.10; 3.13; 3.15; 3.16; 3.17; 3.18; 3.21; 3.22; 3.24; 3.25; 3.26; and all exercises in Chapter 8.

Further reading:
David Ruelle, Statistical Mechanics: Rigorous Results, World Scientific, 1999.
James Sethna: Statistical Mechanics: Entropy, Order Parameters and Complexity, Oxford Master Series in Physics, 2006.