Interacting Particle Systems in the Lattice and Continuum - M2 - 8EC

1) Aims of the course
This course will provide a solid knowledge about the mathematical foundation of statistical mechanics, which is the basis of many problems related to critical phenomena of complex systems. In particular 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 (Peierls contour, cluster expansion).

Content

Statistical mechanics 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.
In particular we will treat:

  1. Gibbs ensembles and thermodynamic limits.
  2. Infinite volume Gibbs measures and DLR formalism in the lattice and in the continuum.
  3. Variational characterisation of Gibbs measures.
  4. Peierls contour method for proving phase transitions.
  5. Cluster expansion and polymer models.
  6. Piragov Sinai theory and its application to the Blume-Capel model.
  7. Particle Systems in the continuum: Gibbsian formalism for superstable interactions.
  8. Systems in the  continuum as point processes.
  9. Example: Widom-Rowlinson model.

2) Prerequisites
No prior knowledge in statistical physics is required. The student has to have a strong background in probability theory, stochastic processes and measure theory.

Learning goals
After completion of the course, the student is able to:

  1.  Convert material from part of a graduate-level textbook or a scientific paper into a coherent and comprehensible presentation for fellow students and mathematicians in general.
  2. Choose appropriate means of communicating theoretical mathematics to fellow students and mathematicians, in written and oral form.
  3. Explain specific topics from the content list of the seminar to fellow students, and put them in perspective as far as their relevance to wider mathematics is concerned,

3) Rules about Homework/Exam
The contribution of each student is twofold:

80% - individual project consisting of 40% written thesis and 40% oral presentation
20% - Homework assignments
Final Project guidelines:

length: 15-25 pages; 
clear relation with the course;
personal editorial line, add computations, do not copy and paste the paper;
indicate clearly your own contribution;
discuss with your tutor what elements should be there.

Presentations:

  • 30 minutes presentation, 
  • 15 minutes about the project,
  • 15 minutes answering questions about the project and theory related to the project.

4) Lecture Notes/Literature

Textbooks used:

  1. S. Friedli, Y. Velenik, Statistical Mechanics of Lattice Systems: A concrete mathematical introduction, https://www.unige.ch/math/folks/velenik/smbook/
  2. Lecture notes of Prof. Sabine Jansen, Gibbsian Point Processes, http://www.mathematik.uni-muenchen.de/~jansen/gibbspp.pdf