INTRODUCTION TO MATHEMATICAL PHYSICS, MATHEMATICS 541, FALL 2004
I. General principles of statistical physics.
Ensembles, thermodynamic potentials, high and low temperature behavior,
phase transitions, critical phenomena. The purpose of this part of the
course is to make students familiar with the ideas from physics underlying
the mathematical models, questions and methods discussed in the course.
2. Basic tools from probability theory.
Probability spaces, random variables, expected values, characteristic
functions, laws of large numbers, central limit theorems, ergodicity.
In this part we will review the basic concepts and results of probability
theory used in the course. More special tools will be introduced later
as needed.
3. Equlibirium statstical mechanics of classical lattice systems.
General lattice models, existence of the thermodynamic limit, Gibbs states,
equivalence of ensembles. In this part we will introduce the mathematical
description of equlibrium states of a general class of infinite systems
with an emphasis on precise mathematical description of phase transitions.
4. Ising and Potts models.
This is the most important class of lattice models and we will discuss
their specific properties using a variety of tools, including correlation
inequalities, contour representations, complex analytic methods and cluster
expansions.
5. Systems with continuous symmetry
We will study the classical Heisenberg and plane rotor models and discuss
presence of continuous symmetry breaking depending on the dimension of the
system.
6. Disordered systems
This part will review the most important models of random media: the random
field models and spin glasses.
7. Percolation theory
We will prove basic results about percolation models in an arbitrary
number of dimensions and then study in more detail two-dimensional
models, including recent Smirnov's proof of Cardy's formula.