Introduction to Mathematical Physics, Spring 2012
Introduction to Statistical Mechanics
Statistical mechanics was developed in the 19th century to study mechanical systems
such as gases with a very large number of degrees of freedom. A detailed microscopic
study of such systems is out of question and one has to resort to a probabilistic description.
In the 20th century powerful methods were developed for the study of phenomena such
as phase transitions and non-equilibrium states. The framework of statistical mechanics
is quite universal and can be applied to a variety of problems ranging from quantum
field theory to biological and social systems as well as questions in pure mathematics.
The common feature of all these is the presence of a large number of elementary units
interacting with each other and giving rise to interesting collective behaviour.
The course provides an introduction to some basic topics and techniques of statistical
mechanics from the mathematical point of view. The first half deals with equilibrium
statistical mechanics of "spin systems" where in the simplest case the elementary units
are binary variables. We study low and high temperature behavior as well as critical
behaviour with elements of the renormalization group. The second half adresses
nonequlibrium behavior and introduces Boltzmann equation and hydrodynamic limit.
The course is meant for mathematics students interested in probability who want
to learn the basics of statistical mechanics without physics background and physics
students who want to have a more conceptual and mathematical treatment than
in physics courses.
Independent random variables (infinite temperature): ensembles
Ising model: low temperature and high temperature expansion
1st order phase transition
General spin systems
Application to chaotic dynamical systems
2nd order phase transition, critical point
Introduction to renormalization group and scaling limit
Disordered systems: Ising spin glass and random field Ising model
Introduction to non-equlibrium
For the main bulk of the course the mathematics is developed on the spot and
no physics background is assumed. A few lectures deal with more advanced issues
and are not required for the exam.
Weeks 3-9 and 11-18, Tuesday 14-16, Thursday 14-16 in room C123.
Easter holiday 5.-11.4.
Extra material on dynamical systems
NOTE! - The lecture notes have been updated!
NEW! There will be no lecture on Tuesday April 24!
If you are interesting in taking the exam for the course, let the assistant know what would be a suitable date for you.
Did you forget to register? What to do.
Note! First exercise session is on Thursday February 2
Note! The time and place for the exercises changes starting from Friday March 16.