Summer term 2024

mathematics Seminar

First meeting: April 18

When: Thursdays, 14:00—16:00

Where: Room MAR 4.064, TU Berlin

TU Links: Moses; ISIS.

Gibbs measures in statistical mechanics

In this seminar, we will explore several topics related to Gibbs measures in statistical mechanics, from the classical results of D. Ruelle and R.L. Dobrushin in the ‘60s to the state of the art in Gibbs point process theory.

After the seminar, interested students will be able to ask Professor W. König and me for a Bachelor’s/Master’s thesis.

  • The possible topics include:

    1. The Ising model: finite-volume Gibbs measures and thermodynamic limit

    2. First-order phase transition in the Ising model

    3. Infinite-volume Gibbs measures on the lattice: DLR equations

    4. Infinite-volume Gibbs measures on the lattice: the variational principle

    5. Continuous-spin models: absence of symmetry breaking

    6. The Poisson point process: geometrical properties

    7. The Poisson point process: Palm measure and Mecke equation

    8. Gibbs measures in the continuum: existence for finite-range interactions

    9. The Gibbs variational principle for finite-range interactions

    10. Phase transitions: the Widom—Rowlinson model

    Additional topics may be selected at the choice of the students.

  • The topics are mostly taken from the following references.

    Books and lecture notes:

    [1] Statistical mechanics of lattice systems (2017) S. Friedli and I. Velenik

    [2] Gibbs measures and phase transitions (2011) H.O. Georgii

    [3] Introduction to the theory of Gibbs point processes (2019) D. Dereudre

    [4] Lectures on the Poisson Process (2017) G. Last, M. Penrose

    Papers:

    [5] Variational principle for Gibbs point processes with finite range interaction (2016) D. Dereudre

    [6] The analysis of the Widom—Rowlinson model by stochastic geometric methods (1995) J.T. Chayes, L. Chayes, R. Kotecký

Contact me for questions and further details/topics.

    • 2020 (Potsdam University) Exercise classes for the Master’s course of Advanced Probability

    • 2018 (Potsdam University) Lecture and exercise class on the Poisson point process, Ringvorlesung of the Data Science Master’s

    • 2018 (Potsdam University) Exercise classes for the Master’s course of Stochastic Analysis

    • 2020, Master’s thesis of P. Tschisgale, Introduction to the Glauber Dynamics for the Curie-Weiss Potts Model

    • 2018, Bachelor’s thesis of P. Tschisgale, An introduction to the Potts model, winner of the 2019 Berliner Mathematische Gesellschaft (BMG) best Bachelor Thesis award

    • 2018, Bachelor’s thesis of C. Hansch, Some findings on the Ising model

    • 2018, High school pupil projects at the Berlin Französische Gymnasium as part of the popularisation project Math en Jeans