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Introduction to Mathematical Physics, Spring 2012

Introduction to Statistical Mechanics


Antti Kupiainen


10 cu.


Advanced studies


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 

Boltzmann equation

Hydrodynamic limit


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.

Lecture notes

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.


Lecture notes 


Did you forget to register? What to do.

Exercise groups

Note! First exercise session is on Thursday February 2

Note! The time and place for the exercises changes starting from Friday March 16.










Christian Webb

Exercise 1       Solutions 1

Exercise 2       Solutions 2

Exercise 3       Solutions 3

Exercise 4       Solutions 4

Exercise 5       Solutions 5    

Exercise 6       Solutions 6

Exercise 7       Solutions 7

Exercise 8       Solutions 8

Exercise 9       Solutions 9

Exercise 10     Solutions 10

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