**Prerequisites**

No prior knowledge in statistical physics is required. The student should have a strong background in probability theory, stochastic processes and measure theory (a measure theory course in the Bachelor is suffiecient).

**Aims of the course**

Statistical mechanics (SM) is a branch of physics aiming at understanding the laws of the macroscopic behaviour of systems composed by many microscopic components. Critical phenomena, such as phase transitions, involve a drastic change in the macroscopic state by tuning model parameters.

Critical phenomena are extremely universal far beyond physics, for instance in chemistry, biology or complex systems. In this course we aim at giving a mathematical foundation for the study of many component systems on the lattice and in the continuum space. Moreover, we would like to motivate the theory of Gibbs measures starting from basic principles in classical mechanics.

The knowledge of SM is fundamental for understanding the thermodynamic properties (e.g., phase transitions) of systems consisting with a large number of degrees of freedom (e.g., particles) starting from their microscopic description. Therefore, SM is crucial for modeling critical phenomena of physical systems. This course will provide the students with the rigorous mathematical foundation of modern SM, developed at the crossroad between mathematical physics and probability theory.

We will introduce the notion of Gibbs measures and phase transitions. At the end of the course, the student will be able to understand and characterise critical phenomena in terms of extremal probability measures and their variational representations, including standard techniques for proving them.

In particular, we will treat:

- Gibbs ensembles and thermodynamic limits.
- Infinite volume Gibbs measures and DLR formalism in the lattice and in the continuum. Variational characterisation.
- Peierls contour method.
- Cluster expansion and polymer models.
- Piragov Sinai theory and its application to the Blume-Capel model.
- Particle Systems in the Continuum:
- Ideal Gas; Poisson Point Processes.
- Gibbs measures and DLR equations for stable and superstable interactions.
- Phase transition for Widow-Rowlinson model

**Lecturers:**

Wioletta Ruszel and Cristian Spitoni (Utrecht University)

- Docent: Jochem Hoogendijk
- Docent: Wioletta Ruszel
- Docent: Cristian Spitoni