Lecture on Statistical mechanics of interacting particle systems

Wintersemester 2026/27, TU Berlin

Contact information:

Dr. Elena Magnanini, magnanini (at) wias-berlin.de

Content:

Statistical mechanics provides a mathematical framework for understanding how macroscopic behaviour emerges from the interaction of a large number of microscopic components. Even when the microscopic rules are simple, the collective behaviour of the system can be highly non-trivial, giving rise to interesting phase transitions.
In this course, we will develop a rigorous probabilistic approach to equilibrium statistical mechanics, with particular emphasis on interacting spin systems. We will begin by introducing the basic framework of equilibrium statistical mechanics: Gibbs distributions, entropy, partition functions, and limiting free energy. We will then use probabilistic tools, in particular large-deviation methods, to study the macroscopic behaviour of the system, with the empirical magnetization as the main observable. The main classical models under consideration will be the Curie-Weiss and Ising models, which will serve as the main examples for the study of phase transitions. In the final part of the course, we will introduce the Sherrington-Kirkpatrick model as a paradigmatic model of spin glasses.

Topics covered:

• Gibbs measures and thermodynamic quantities: microstates, Hamiltonians, entropy, partition functions, free energy.
• Probabilistic tools and large deviations: (recap of) law of large numbers, central limit theorem, Cramér's theorem, large deviations, and Laplace principle with applications to statistical mechanics.
• Ferromagnetic spin systems: magnetization, susceptibility, correlations, critical temperature, phase transitions, and critical behaviour.
• Curie-Weiss model: mean-field interaction, free energy in the thermodynamic limit, spontaneous magnetization, and fluctuations away from and at criticality.
• Finite-dimensional Ising models: the one-dimensional model and the transfer-matrix method; phase transitions in dimension d ≥ 2.
• Further topics: partition-function zeros and the Lee-Yang theorem; introduction to spin glasses and the Sherrington-Kirkpatrick model.

A background in probability may be helpful.

Lecture information:

The lecture takes place once a week throughout the semester.

• Times:
  • Lecture periods: 15.10-17.12 (2026) and 07.01-11.02 (2027).
  • Thursdays, 10:15-11:45.
• Assessment: Oral exams. The appointments are made during the lecture or via email. Once we have agreed on the appointment, please fill out this form and send it to me via e-mail.
• Language: English
• Thesis in this field: Thesis (Bachelor and Master) in one of the topics concerning statistical mechanics and interacting particle systems are very welcome, under the supervision of myself and Prof. Dr. Wolfgang König.
• ISIS course: can be found here.
• Credit points: 5 CP
• Lecture notes: Lecture notes will be provided during the course.

References

• F. den Hollander, Large Deviations, Fields Institute Monographs, American Mathematical Society, 2000.
• R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Springer-Verlag, Berlin, 2006 (Reprint of the 1985 Edition).
• S. Friedli and Y. Velenik, Statistical mechanics of lattice systems, Cambridge Univ. Press, Cambridge, 2018. The book is accesible online on the author's personal website.