Spin Systems and Phase Transitions

Lecture (2SWS) and Seminar (2SWS)

Summer Semester 2020 at TU Berlin

 

Due to the pandemics, we provide our lectures via videos presented on this website. The videos are uploaded on the dates of the regular meetings. For discussions, several options are thinkable: chats, conference calls, emails, etc. .

Please contact us via email regarding any questions: Lorenzo Taggi, Benedikt Jahnel

Updates:

Lecturer:

Dr. Lorenzo Taggi and Dr. Benedikt Jahnel

E-mail:

Lorenzo.Taggi@wias-berlin.de and Benedikt.Jahnel@wias-berlin.de

Address:

Room 413, Mohrenstra├če 39, 10117 Berlin


Lecture

In the lecture, in the first part of the semester, we present the general probabilistic framework of statistical mechanics in infinite volume. That is, we consider systems of infinite configurations of components in space that carry some random local information, their spins. The probability to see a configuration then depends on the mutual energy of the system, introducing dependencies. Apart from its original intention to describe phenomena in physics, like spontaneous magnetization and other macroscopic phase-transition phenomena, the approach of statistical mechanics has found numerous applications in various parts of applied mathematics, for example probabilistic biology, stochastic geometry or even data science. We will focus on the essential parts of the general theory, e.g., existence and uniqueness results, which we illustrate via paradigmatic models such as the famous Ising model and more general spin systems with continuous symmetries. As a prerequisite for the course we need a solid foundation in measure-theoretic probability. The lecture will be given as a 4SWS lecture in the first half of the summer semester 2020. In the second half of the semester we offer a seminar on the same topic, for which the lecture provides the necessary background.


Seminar

In the seminar, in the second part of the semester, we present and discuss topics of statistical mechanics in infinite volume, more precisely, spin systems and associated macroscopic phase transitions. Statistical mechanics, apart from its original intention to describe phenomena in physics, like spontaneous magnetization and other phase-transition phenomena, has found numerous applications in various parts of applied mathematics, for example probabilistic biology, stochastic geometry or even data science. This seminar builds on the 2SWS lecture with the same title that will be given by the instructors in the first half of the summer semester 2020. The individual topics will be distributed in early May 2020 and presentations should be delivered in the second half of the semester.


Mode of Operation and Dates

The original plan was to have a classical lecture and seminar on the following dates:
  • Tuesdays 10:00 - 12:00 Uhr MA 748
  • Fridays 10:00 - 12:00 Uhr MA 645

  • The first video will be made available on Tuesday 21 April 2020.
    The lecture phase is from 21 April 2020 until 05 June 2020.
    The seminar phase is from 09 June 2020 until approximately 21 July 2020.


    Videos

      24 April 2020:
    • 5. Ising model - pressure
    • 6. Ising model - magnetization
    • 7. Ising model - one dimension
    • Slides from the lecture
    • Comments and corrections:
      • Video 5: The parallelpipeds Lambda and Delta have to be disjoint; misspelling of edge set at 18:36 - B^b must be E^b
      • Video 6: Misspelling in the corollary about properties of the magentization item 1): It holds for all h not in B_beta
      • Video 7: When I write "Laplace principle", I here mean the elementary property that can be found for example in Lemma 1.2.15 (Large Deviations Techniques and Applications - Dembo / Zeitouni Springer '98)


    References

    The lecture will be based on the following book:
  • Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction (Sacha Friedli and Yvan Velenik) (book can be downloaded here for free)

  • We also recommend these books on the topic, which are also available as ebooks:
  • Gibbs Measures and Phase Transitions (Hans-Otto Georgii) (book can be downloaded here from within the TU network)
  • A Course on Large Deviations With an Introduction to Gibbs Measures (Firas Rassoul-Agha and Timo Seppalainen) (book can be downloaded here from within the TU network)

  • Those three books are also blocked in the mathematical library of the TU Berlin for our course, see primo search portal


    Seminar topics

    Please send us a first draft of a three-page PDF summary (using LaTex) one week before the date. The presentation should not longer than 1 hour. Then we have up to 30 minutes for discussion. Finally, we expect the final version of the three-page PDF summary approximately two weeks after the presentation.
  • June 30, 10:00-11:00am (Luisa): Cluster expansion for polymer models (5.1 - 5.3 in Friedli and Velenik)
  • June 30, 11:30-12:30am (Anastasija): Convergence in the cluster expansion (5.4 in Friedli and Velenik)
  • July 03, 10:00-11:00am (Yankaren): Cluster expansion for the Ising model in strong magnetic field (5.7.1 in Friedli and Velenik)
  • July 03, 11:30-12:30am (Nils): Cluster expansion for the symmetric Ising model as low temperature (5.7.4 in Friedli and Velenik)
  • July 07, 10:00-11:00am (Dorothea): The random cluster representation of the Ising model and consequences (3.10.6 in Friedli and Velenik)
  • ...
  •