Mathematical statistical mechanics (MAT685)


Important notice: Due to the Covid-19 pandemic, the lecture will be given in a podcast format with video lectures and a two hour discussion group via Zoom. A link to the video of the weekly lecture will be sent to all participants on Thursdays at the time of the lecture. In the first week at the time of the lecture there will also be a meeting of all participants via Zoom. During this meeting we will agree upon a time for the discussion group. If you are interested in participating you can either formally register for the course if you need the credits, or you just write me an email if this is not the case and I will make all necessary information available for you.


Time: Thursday from 3:00 - 4:45 pm (from 25.02.2021 to 03.06.2021)


General description: The aim of statistical mechanics is to understand the thermodynamic properties (e.g. phase transitions) of systems consisting of a huge number of degrees of freedom (e.g. particles) starting from their microscopic description (e.g. classical/quantum mechanics or toy models). To define phase transitions in a mathematically precise sense, one needs to take a thermodynamic limit, that is, an infinite volume limit at constant density. In the lecture we will introduce mathematical tools in order to study this infinite volume limit and to prove the (non-)existence of phase transitions. Topics to be covered are: Existence of the thermodynamic limit for classical and quantum lattice systems, equivalence of ensembles, discussion of thermodynamic stability for classical continuum systems, Lee-Yang theorem and consequences (existence and non-existence of phase transitions for certain classical lattice systems), Mermin-Wagner theorem (Absence of phase transition in the 2d quantum Heisenberg model). Depending on the remaining time and on the background of the audience, we will also construct and investigate infinite volume Gibbs measures for the 2d Ising model or discuss the proof of the Kosterlitz-Thouless phase transition in the classical xy model.


Prior Knowledge: Apart from analysis and linear algebra, the knowledge of one course in either probability (from a mathematics point of view) or statistical mechanics (from a physics point of view). The lecture will be adapted to the background of the audience.


Course Material: Handwritten lecture notes, videos in podcast format.


Literature:

a) D. Ruelle, Statistical Mechanics, Rigorous Results, Imperial College Press and World Scientific Publishing (1999). (A selection of the material in Chapters 1, 2 and 4 will be relevant for the course.)
b) S. Friedli, Y, Velenik, Statistical Mechanics of Lattice Systems, A Concrete Mathematical Introduction, Cambridge University Press (2017). (Chapter 3 and Chaper 6 are relevant for our study of the Ising model (depending on the audience).)
c) V. Kharash, R. Peled, The Fröhlich-Spencer Proof of the Berezinskii-Kosterlitz-Thouless Transition, arXiv:1711.04720 [math-ph]. (Relevant for our study of the BKT phase transition depending on the audience and if time permits.)
d) B. Simon, The Statistical Mechanics of Lattice Gases, Part I, Princeton University Press (1993). (Additional reading with a slightly different point of view. Apart from content interesting for our course, the book has chapters on convexity and on exactly solvable models, e.g. the 2d Ising model, that can be used by motivated students for additional reading.)
e) L. D. Landau, E. M. Lifschitz, Statistical Physics, Part I, Elsevier (2011). (Statistical mechanics from a physics point of view. Interesting for mathematician that would like to understand the different ensembles of statistical mechanics.)
f) H. O. Georgii, Gibbs Measures and Phase Transitions, De Gruyter (2011). (Additional reading in statistical mechanics appropriate for students with a background in probability or with in interest in this direction.)
g) R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Dover Publications (2008). (Exactly solved models play in important role in statistical mechanics. Additional reading suitable for all students.)
h) E. H. Lieb, R. Seiringer, The stability of matter in quantum mechanics, Cambridge University press. (Complementary reading on the stability of matter for realistic quantum mechanical systems with Coulomb interactions.)


Lecture notes (Disclaimer: The notes are not self-contained and certainly full of minor mistakes.):

  1. Introduction (01.04.21)
  2. Thermodynamic limit for lattice systems (01.04.21)
  3. Thermodynamic limit for continuum systems (29.04.21)