250123 VO Dynamische Systeme (Thermodynamischer Formalismus

Lecturer: Prof. Henk Bruin

Please contact H. Bruin for further information about this course.

Announcements

Class Details

Description of topic

Thermodynamic formalism is a mathematical approach to the theory, in statistical physics, on the way many-particle systems strive to configurations (called equilibrium states) of maximal entropy and minimal potential energy. It is used to explain phase transitions in materials, such as liquid vs. solid and magnitized vs. non-magnitized. Whereas in physics, these ideas go back to Boltzmann in the late 19th century, it was only in the 1970s that they were given a firm mathematical framework by Sinai, Ruelle and Bowen. Topics of this course may include:
- Symbolic dynamics and subshifts of finite type.
- Entropy and the Variational Principle.
- The pressure function and phase transitions.
- Measures of maximal entropy.
- The Ising model and related models.
- Transfer operators and their spectral properties.
- Thermodynamics and fractal dimensions.

References

• Peter Walters, An Introduction to Ergodic Theory, Springer-Verlag 1975.
• Gerhard Keller, Equilibrium States in Ergodic Theory, Cambridge Univ. Press. 1998
• Rufus Bowen, Equilibrium states and ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Lect. Notes in Mathematics, 470, 1975.
• Omri Sarig, Lecture Notes on Thermodynamic Formalism for Topological Markov Shifts. Penn State, 2009.

• David Ruelle, Thermodynamic Formalism, Cambridge University Press, Cambridge, 2004.
• Vivianne Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific Press 2000.

Assessment

Assesment is based on an (oral) exam.

Exercise sheets

These will appear here in due time.

Course material (hand-outs/assignments)

• Class notes for the course pdf. This file may be corrected and extended over time.
• Lecture notes by Omri Sarig as PDF file
• Ising's 1925 paper "Beitrag zur Theorie des Ferromagnetismus" as PDF
• Notes by Hofbauer and Schmidt on Ergodic Theory (in German) PDF