Mathematical aspects of phase transitions WS 2014/2015  

V5B5-Advanced Topics in Analysis and Calculus of Variations

Time and Room	Monday and Wednesday, 12-14, Math/SemR 1.007
Start	6. October 2014

Content

This course will give an introduction to a number of techniques to
study integrals over many variables, in the limit  as the number of variables
diverges. Such integrals arise in statistical mechanics and theoretical physics
and their properties  give information on the underlying physical system.
In this context, multiscale analysis and renormalization is a set of tools that allow to
partially factorize the integrals over a product of ``small'' integrals (over a finite
set of variables), a bit like  the linearization procedure in a partial differential equation. We will see some classical techniques used to control  rigorously this factorization procedure as well as some applications. According to time we will also consider some examples where these classical techniques do not apply.

 

Program and lecture notes:

• Chapter 1:  introduction (definition of phase transition, mean field Ising)                                                                                                                
• Chapter 2: high temperature region                                                                                                                                                                                                               
• Chapter 3: low temperature region (preliminary notes)

 

Some bibliography:

 

D. Brydges "A short course on cluster expansions" Les Houches 1984

 

A. Abdesselam, V. Rivasseau  "Trees, forests and jungles: A botanical garden for cluster expansions" Constructive physics (Palaiseau, 1994), 7--36, Lecture Notes in Phys., 446, Springer, Berlin, 1995

 

S. Friedli, Y. Velenik ”Equilibrium Statistical mechanics of classical lattice
systems: a concrete introduction.”